David Masclet
email: david.masclet@univ-rennes1.fr

Marie-Claire Villeval
email: villeval@gate.cnrs.fr

Punishment, Inequality and Emotions

Keywords: A13, C92, D63 Inequality Aversion, Negative Emotions, Free-Riding, Cooperation, Experiment

. However, the rationale for costly punishment remains unclear. This paper reports the results of an experiment investigating inequality aversion and negative emotions as possible determinants of punishment. We compare two treatments of a public good game, one in which costly punishment reduces the immediate payoff inequality between the punisher and the target, and one in which it does not affect inequality. We show that while inequality-aversion prevents some subjects from punishing in the equal cost treatment, negative emotions are the primary motive for punishment. Results also indicate that the intensity of punishment increases with the level of inequality, and reduces earnings inequality over time.

I. INTRODUCTION

Individuals sometimes cooperate with the authorities in reporting criminal activity, and witnesses accept to testify in favor of or against an unknown person even though they cannot expect any future benefits from doing so. The availability of costly sanctions has been shown to help enforce a social norm of cooperation among unrelated individuals (Fehr andGächter, 2000 and2002). 1 2 However, the forces driving subjects to make sacrifices in order to punish others, especially when they do not expect to become involved in long-run interactions, remain unclear.

This paper contributes to the literature on the behavioral determinants of altruistic punishment by examining the relative importance of inequality-aversion as an explanation of punishment in the framework of a public good experiment. The desire to induce higher contributions to the public good, and thus ensure a higher individual return in future periods, constitutes a strategic motive for punishment. However, since subjects also punish when they are certain that they will not interact again with the same person in the future, non-strategic motives, such as emotions and distributional concerns, have been suggested as alternative explanations.

In a cooperation game, negative emotions such as anger or disapproval are one non-strategic motive for individuals to sacrifice in order to punish others, even when there are no strategic reputation gains from doing so. Cooperators punish those who violate the pro-social norm of cooperation, or defectors punish those who try to establish such a norm (Fehr and Gächter 2000;[START_REF] Gintis | Strong Reciprocity and Human Sociality[END_REF][START_REF] Masclet | Monetary and Non-Monetary Punishment in the Voluntary Contributions Mechanism[END_REF]Carpenter and Matthews 2004;[START_REF] Carpenter | Why Punish? Social Reciprocity and the Enforcement of Pro-Social Norms[END_REF][START_REF] Gächter | Norms of Cooperation among Urban and Rural Dwellers: Experimental Evidence from Russia[END_REF]. [START_REF] Anderson | Do Non-Strategic Sanctions Obey the Law of Demand? The Demand for Punishment in the Voluntary Contribution Mechanism[END_REF] show that while punishing behavior follows the law of demand, subjects will punish even if there is a higher cost to the punisher than to the target. While Fehr and Gächter 1 (2002) show that even selfish subjects have reasons to cooperate because they anticipate punishment from altruistic punishers, [START_REF] Hopfensitz | The Importance of Emotions for the Effectiveness of Social Punishment[END_REF] emphasise the importance of social emotions such as shame and guilt in the punished as an essential component for the successful enforcement of cooperation. Such emotion-based punishment underscores the importance of intentions [START_REF] Rabin | Incorporating Fairness into Game Theory and Economics[END_REF][START_REF] Dufwenberg | A Theory of Sequential Reciprocity[END_REF]. The importance of emotions in decision-making has also been established by neuroscientists [START_REF] Damasio | Descartes' Error -Emotion, Reason and the Human Brain[END_REF]) and research in neuroeconomics has started to provide some evidence of a neural basis for altruistic punishment [START_REF] Quervain | The Neural Basis of Altruistic Punishment[END_REF].

A second non-strategic reason to punish group members is inequality aversion [START_REF] Fehr | A Theory of Fairness, Competition and Cooperation[END_REF][START_REF] Falk | Driving Forces Behind Informal Sanctions[END_REF]Bolton and Ockenfels, 1999). 3 In economics, the relationship between punishment and inequality aversion has been far less scrutinized than that with emotions. 4 In contrast, the fitness differential theory in evolutionary psychology posits that punishment aims to reduce the payoff advantage of free-riders [START_REF] Price | Punitive Sentiment as an Anti-Free Rider Psychological Device[END_REF]. Individuals with distributional concerns who suffer from disadvantageous inequality may be willing to pay to punish defectors in order to reduce earnings inequality, if the cost they bear themselves is smaller than the impact of sanctions on the target's payoff. This suggests that we should take a closer look at the influence of the feelings caused by inequality on punishment.

The contribution of our paper is threefold. We first examine the relative importance of inequality aversion and emotions among the non-strategic motives for punishment in a public good game.

If motivated by emotions, an individual may be willing to undertake costly punishment even when punishment does not affect the distribution of earnings. In contrast, if only motivated by inequality aversion, individuals will not punish when punishment does not change the distribution of earnings. Our experiment consists of two treatments of a public good game, implemented with a stranger matching protocol. In the Unequal Cost Treatment, punishment affects payoff inequality between the punisher and the target (replicating the design of Fehr and Gächter, 2000), and in the Equal Cost Treatment it does not affect the current-period level of inequality. In the second treatment the ratio between the cost of one punishment point to the punisher and the cost of this point to the target equals one. On the contrary, in the Unequal Cost Treatment, this ratio should be greater than one in most circumstances. 5 The strong prediction is that subjects who only care about their own payoff and earnings inequality should then not punish in the Equal Cost Treatment. This prediction is based on [START_REF] Fehr | A Theory of Fairness, Competition and Cooperation[END_REF] model. A weaker hypothesis allows several different motives to play a role; the comparison between treatments allows us to identify the importance of the concern for inequality in punishment behavior.

Second, we measure the importance of strategic motives in punishment by repeating the Equal Cost Treatment in a partner-matching protocol in which subjects repeatedly play the public good game with the same group. Comparing behavior across the two matching protocols allows us to separate strategic from non-strategic punishment behavior, and to differentiate between related and unrelated individuals. While any immediate effect of punishment on inequality is ruled out in this treatment, the design does help us to see whether the demand for punishment is partly explained by forward-looking distributional considerations (i.e. the desire to reduce the payoff advantage accruing to free-riders in the future).

The third contribution of this paper is to examine the indirect relationships between inequality and punishment. We also investigate the impact of punishment on the evolution of inequality.

Will punishment reduce the level of inequality over time by inducing free-riders to increase their contributions after being punished?

In contrast with most papers on punishment, we focus on the distance between the punisher's and the target's payoffs, rather than on the distance between the target's payoff and the average group payoff. We are close to [START_REF] Falk | Driving Forces Behind Informal Sanctions[END_REF] who analyze the importance of fairness and spite in punishment in three-player one-shot Prisoner's Dilemma games. Spiteful punishment is carried out by subjects who value the target's payoff negatively, irrespective of the distribution of pre-punishment payoffs. They conclude that punishment is not primarily driven by the willingness to change payoff shares, but by the desire to retaliate and harm those who behave unfairly. Here we consider another type of cooperation game, a four-person public-good experiment, in which the set of actions is larger.

Our first finding is that our experiment rejects the strong assumption that punishment is only motivated by inequality aversion, but accepts the weaker assumption of punishers paying some attention to the distribution of payoffs. Subjects do punish even in the Equal Cost Treatment.

However, the comparison across treatments indicates that a significant number of subjects refrain from punishing when they cannot affect earnings inequality by doing so. The second finding shows the importance of negative emotions. If we control for selection bias, as some subjects do not punish, and the level of inequality between the punisher and the punished, we find that cooperators punish more heavily in the Equal than in the Unequal Cost Treatment. This indicates that they are willing to impose the same payoff reduction as in the Unequal Cost Treatment, since the monetary impact of each punishment point on the target is smaller than in the Unequal Cost Treatment. This finding is consistent with negative emotions: feelings of anger provoke greater expenditure in punishment since the punishers focus more on the consequences of the sanction on the target than on their personal cost of punishing.

A third finding is that although subjects are primarily motivated by negative emotions, these emotions arouse from the observation of inter-individual inequality in contributions and earnings.

Fourth, people do not punish more in the Equal Cost Treatment played under the partnermatching protocol. We observe however that the efficiency of punishment is greater in fixed groups, underscoring the importance of strategic motives with repeated interactions. Last, punishment reduces inequality over time, by inciting free-riders to increase their effort in following periods, which reduces the future inequality in payoffs to cooperators. We therefore cannot reject that in repeated interactions subjects punish in order to reduce inequality over time.

The remainder of the paper is organized as follows. Section 2 details our experimental design and the procedures, and Section 3 presents the theoretical predictions of the model, with either purely selfish agents or in the presence of agents with distributional concerns. Section 4 displays the results and Section 5 concludes.

II. EXPERIMENTAL DESIGN AND PROCEDURES

Design. The experiment is based on a public good game, and involves groups of four subjects. It consists of 30 periods, divided into three segments. In the first ten periods and the last ten periods, subjects play a standard public good game. This no-punishment condition serves as a benchmark for the condition with punishment opportunities which occurs from periods 11 to 20.

At the beginning of each period, each group member is endowed with 20 units. Each member simultaneously selects a fraction of her endowment to contribute to a group account, while keeping the remainder in her private account. All funds in the group account pay a positive return to each member. The parameters are chosen so that full free-riding is a dominant strategy whereas full contribution to the public good corresponds to the social optimum.

First consider the condition without punishment. In periods 1-10 and 21-30, each subject i chooses a fraction g i of his endowment as a contribution to the public good. All group members' decisions regarding g i are made simultaneously. The marginal per capita return from a contribution to the group account is 0.4. Subject i 's payoff is given by: 4 1

20

.4

ii j g π = =-+ ∑ j g (1)
The group members are informed of both the amount of the group contribution and their individual payoff. Now consider the condition with punishment. Each period 11-20 of both treatments consists of a two-stage game. The first stage is identical to that in the previous condition: each group member receives an endowment and has to decide how much to invest in the group account. In the second stage, each subject, after being informed about each other group member's contribution, can assign 0 to 10 punishment points to any of the other three group members. Assigning points is costly. The main difference between the Unequal and Equal Cost Treatments lies precisely in the monetary consequences of punishment points. The schedule of costs in these two treatments is given in Table 1.

In the Unequal Cost Treatment, each punishment point received from any other subject reduces first-stage earnings by 10%, up to a maximum of 100%. This treatment intends to replicate that in Fehr and Gächter (2000). As usual, contributions are listed in random order and with a different identification number on the screen each period so that it is impossible to target another subject for punishment for more than one period. This rules out motivations such as revenge. Subject i's earnings per period are now given by:

() ∑ ∑ ∑ ≠ ≠ = -       - *       * + - i k ik i k ki n k k i P C P g g 10 10 , 0 max 4 . 0 20 1 (2)
where is the number of points assigned by to , and

ik P i k ( ) ik P C
is the cost to i of assigning the points to k. The design of punishment is such that if a subject who plays the Nash equilibrium is punished, the payoff inequality is necessarily reduced between the punisher and the target. In most circumstances it guarantees a reduction of payoff inequality. 6 Losses are possible if the cost of punishing others exceeds the individual's net income but they are extremely unlikely (this situation never occurred in our experiment).

In the Equal Cost Treatment, each point given by a punisher has the same monetary cost for himself and the target: the cost of being punished always equals the cost of punishing. Suppose that subject i assigns say 3 punishment points to subject j; the first-stage earnings of both subject i and subject j are reduced by 4 units, as described in Table 1. Subject i's earnings per period in this treatment are given by:

( ) (

1 20 0.4 n ik k i kk i k i gg C P C =≠ ≠  -+ * - -   ∑∑ ∑ ) i k P (3)
Procedures. The experiment was computerized using REGATE software [START_REF] Zeiliger | A Presentation of Regate, Internet Based Software for Experimental Economics[END_REF]. We ran eight sessions in total under the stranger matching protocol, split 50:50 between the Equal and Unequal Cost Treatments. In total 72 subjects participated in each treatment. We also ran two sessions with the partner matching protocol under the Equal Cost Treatment, with 12 participants each. Six sessions were conducted in the experimental laboratory of the Groupe d'Analyse et de Theorie Economique (GATE) at the University of Lyon, and four sessions were organized in the LABoratory of EXperimental Economics at the University of Rennes, France.

In total, 168 subjects were recruited from undergraduate classes in business and engineering schools in Lyon and in various departments in Rennes. We did not recruit any economics students, and none of the subjects had any experience of this particular type of experiment. No subject participated in more than one session.

Upon arrival, the subjects drew a label from a bag, indicating the name of their computer. The instructions (see the Appendix) were distributed and read aloud. The subjects then filled out a questionnaire that allowed us to check their understanding of the rules of the game. Questions were answered in private. The program then matched subjects randomly and anonymously.

Under the stranger matching protocol, groups were reshuffled after each period, whereas under the partner matching protocol, the composition of groups remained unchanged over time. During each ten-period segment, subjects did not know if the experiment would extend beyond the current segment.

On average sessions lasted for 90 minutes, including reading instructions and payment. Each unit was convertible to Euro at 100 units = 2 Euros. Each participant received €16.40 on average, including a show-up fee of €3.

III. THEORETICAL PREDICTIONS

If players are selfish, the unique subgame-perfect equilibrium of both the Unequal and Equal

Cost Treatments is to contribute nothing in each period and never to punish. Punishment is not credible in either treatment. Therefore, complete free-riding is a dominant strategy in all periods and all treatments.

Relaxing the selfishness assumption, consider the predictions of [START_REF] Fehr | A Theory of Fairness, Competition and Cooperation[END_REF] inequality-aversion theory. Here individual utility depends not only on ones own payoffs but also on the equality of the income distribution. Individuals are inequality averse if they incur disutility both from being worse off in material terms than others (disadvantageous inequality)

and from being better off than others (advantageous inequality). This disutility is self-centered in the sense that subject i compares herself to each of the other group members but does not care about inequalities between the other members. Moreover subjects are assumed to be more sensitive to disadvantageous inequality, as shown by the inequality-aversion term α i , α i >0, than advantageous inequality, given by β i with 0 ≤ β i <1 such that α i >β i . The utility function of player i∈{1,…,n} is:

11 ( ) max( ,0)- max( ,0) 11 ii i j i i i j ji ji Ux x x x x x nn α β ≠≠ =- - - -- ∑∑ (4)
The first term of equation ( 4) represents the pecuniary payoff of subject i. The second and the third terms measure the utility loss from disadvantageous and advantageous inequality respectively. Following Fehr and Schmidt, we assume that the utility function is linear in both inequality aversion and x i . This implies that the marginal rate of substitution between monetary income and inequality is constant.

To solve the standard public good game, we first substitute equation (1) (the monetary payoffs x i and x j ) into the utility function (4). Without any chance of punishment, if subjects are sufficiently averse to disadvantageous inequality, and if advantageous inequality aversion is sufficiently low relative to marginal earnings (i.e. if 1-α > β i ), then not contributing is a Nash equilibrium. When subjects cannot punish each other, cooperation cannot be enforced, even if they are inequality averse. The only way for inequality-averse subjects to reduce such inequality is by reducing their own contributions (the formal proof is provided in [START_REF] Fehr | A Theory of Fairness, Competition and Cooperation[END_REF]. Now consider punishment in the Unequal Cost Treatment. We substitute the monetary payoffs of i and j from equation (2) into equation ( 4). In the second stage, punishment strategies are credible since no enforcer gains by not punishing. If he does not punish, he reduces his expenditure but suffers from both disadvantageous inequality relative to the defector and advantageous inequality relative to the enforcers. Under sufficient conditions, subjects will always choose to punish. Anticipating credible punishment in the second stage, no-one free rides in the first stage. Anyone deviating by free riding in stage 1 will be punished by the other group members in stage 2. Here, disutility from inequality can be reduced since players can affect the payoff of others by punishing them (see the formal proof in [START_REF] Fehr | A Theory of Fairness, Competition and Cooperation[END_REF]. Do these predictions hold when the cost of punishment is the same for the punisher and the target, as in the Equal Cost Treatment? Inequality-aversion predicts no punishment and no cooperation in this treatment. If a cooperator punishes a defector, whereas the other subjects play the equilibrium in the second stage, she incurs the cost of the punishment but does not change the earnings gap between herself and the target. In addition, she suffers from disadvantageous inequality relative to the cooperators who do not punish.

Last consider the case of subjects who are neither selfish nor inequality averse, but driven by emotions. They may punish a norm violator even though punishment will not affect the level of inequality between themselves and the target. With a sufficient number of subjects driven by anger or disapproval, we may observe no difference in punishment between the Equal and Unequal Cost Treatments.

V. EXPERIMENTAL RESULTS

We first examine the impact of the various punishments on contribution decisions. We then analyze punishment behavior to identify inequality considerations and negative emotions, before studying strategic punishment. We then focus on inter-personal comparisons as a source of norms, and last show the effect of punishment on the evolution of inequality over time.

Contribution Decisions

We look at contributions to the public good by period and by treatment to see whether behavior changes when punishment is possible and, if so, whether it is influenced by the treatment and the matching protocol. Figure 1 displays the evolution of average contributions over time in both the Unequal and Equal Cost Treatments, under both the stranger and partner matching protocols.

In all of the treatments the data exhibit the same pattern as in Fehr and Gächter (2000): in periods 1-10, subjects initially contribute more than the Nash equilibrium level, but progressively reduce their contributions; in periods 11-20 the introduction of the punishment opportunity entails an increase in average contributions; last, in periods 21-30, average contributions drop off sharply when punishment opportunities are withdrawn.

Result 1: In all treatments, punishment causes an increase in average contributions. Support for result 1: In the Equal Cost Treatment under the partner matching protocol, a Wilcoxon matched pair test rejects the null hypothesis that contributions are identical between periods 11-20 (punishment) and the pooled data from periods 1-10 and 21-30 (no punishment) (p=.027). 7 In the Unequal cost treatment under the stranger protocol, an analogous Wilcoxon matched pair test also rejects this null hypothesis (p=0.0679). The same test for the Equal cost treatment with stranger matching also reveals a significant difference between periods 11-20 and the pooled data from periods 1-10 and 21-30 (p<0.1).

In the stranger matching protocol, a Mann-Whitney rank sum test accepts the null hypothesis that there is no difference in contributions in periods 11-20 between the Unequal and Equal Cost Treatments (p=.309). The same test comparing contributions in the stranger and partner conditions for the Equal Cost Treatment reveals that subjects contribute significantly more under repeated interactions (p<0.05).

Punishment behavior

We next examine the distribution of punishment points by subjects to other group members to determine whether behavior differs according to the treatment and the matching protocol. Table 2 shows the evolution of the average number of points distributed over time. In all treatments, subjects use costly punishment, even in the one-shot game, but the average number of punishment points decreases over time. This is consistent with results from previous work. We also show that subjects punish even in the Equal Cost Treatment.

Result 2: Subjects punish even when they cannot alter the distribution of payoffs. Support for result 2. In any period of the stranger condition, 36.4% of subjects distribute at least one punishment point in the Unequal Cost Treatment; the analogous figure in the Equal Cost Treatment is 32.2%, and in the partner condition of the Equal Cost Treatment it is 37.1%. In the stranger condition, a Mann-Whitney rank sum test accepts the null hypothesis that there is no difference in the proportion of punishers between the Unequal and Equal Cost Treatments (p=.248). The same conclusion results from the comparison between the stranger and partner conditions in the Equal Cost Treatment (p=.199).

These statistics confirm that subjects use punishment even when it has no effect on inequality. However, they do not take into account the amount contributed in groups. Further, it is important to dissociate the decision to punish from the intensity of punishment: if partly motivated by inequality-aversion, some subjects may abstain from punishing, which introduces a potential selection bias in the estimation of the number of punishment points.

Result 2': Conditional on the willingness to punish, subjects who punish defectors do so more intensely in the Equal than in the Unequal Cost Treatment.

Support for result 2':

To analyze the sensitivity of the number of punishment points distributed in periods 11 to 20 to the experimental conditions, we estimate a model including correction for selection bias. As the treatment may have different effects on the decision to punish and the intensity of punishment, we use a two-step estimation procedure. We consider punishment probability using a random-effects Probit model; we then explain the number of points, conditional on the punishment decision, by a Generalized Least Squares model corrected for selection bias via the inverse of the Mills ratio (the "IMR" variable). Each regression measures the influence of the Equal Cost relative to the Unequal Cost Treatment in the stranger matching protocol. The exogenous variables in the selection equation include the positive and negative deviations between the target's contribution and both the punisher's and average group contributions 8 , as well as a period variable. The GLS regression includes the same variables except for the period variable, which allows us to identify the model.

The estimation results are shown in Table 3. Column (1) presents the selection results and column (2) the marginal effects. Column (3) shows the results of the GLS estimation for the whole population. The last two columns give the results of the GLS estimations for the subsamples of observations in which the subject punishes a group member who contributes less than himself (column (4)) or more than himself (column ( 5)).

Controlling for selection bias and for relative contribution levels, the regressions show that, first, subjects are significantly less likely to punish in the Equal than in the Unequal Cost Treatment and, second, that those who decide to punish do so more intensely in the Equal Cost treatment. This shows that inequality aversion does play some role, since some subjects do not punish when payoff shares cannot be altered (see regression (1)). However, the marginal effect of the treatment is small: the Equal Cost Treatment reduces the probability of punishment by 4% (see column (2)).

We can also see the strength of negative emotions. The regression in column (3) shows that, conditional on the willingness to punish, subjects are willing to pay more to increase the harm imposed on targets in the Equal Cost Treatment. This is because the monetary consequence of each punishment point on the target is lower in the Equal than in the Unequal Cost Treatment. Indeed, on average the monetary consequence of a punishment point is 2.7 times lower in the Equal than in the Unequal Cost Treatment. This is consistent with the results obtained by [START_REF] Falk | Driving Forces Behind Informal Sanctions[END_REF] in PD games where cooperators increase their punishment expenditures if the impact of punishment on the targets is reduced. Therefore, while [START_REF] Anderson | Do Non-Strategic Sanctions Obey the Law of Demand? The Demand for Punishment in the Voluntary Contribution Mechanism[END_REF] have shown that punishment behavior obeys the law of demand, individuals take into account not only the cost to themselves but also the cost imposed on the target. They are ready to pay more if the "quality" of punishment is higher. Negative emotions lead them to pay a higher price in the Equal Cost Treatment to increase the consequences of their sanction on the targets. Interestingly, this is only true for the subjects who punish those who contribute less than themselves (regression (4)). In contrast, the subjects who punish those who cooperate more and are not inequality averse, by definition, are not willing to pay more to punish in the Equal than in the Unequal Cost Treatment (regression (5)). An interpretation is that the intensity of their negative emotions is lower than that of the cooperators vis-à-vis free-riders.

Again this result is consistent with that observed in PD games.

We now consider the strategic motive for punishment. To isolate this motive, we measure the impact of the protocol on the decision to distribute punishment points which do not affect the distribution of payoffs. We again use a two-step procedure to account for potential selection bias.

In Table 4, column (1) presents the results of a random-effects Probit model analyzing the probability of punishment and column (2) shows the marginal effects. Column (3) shows the GLS estimation of the number of punishment points corrected for selection. Column (4) shows the same model estimated on the sub-sample of observations in which a subject punishes those who contribute less than himself. 9

In Table 4 the nature of the interaction has no significant effect on either the decision to punish or the intensity of punishment. This suggests that pressure is exerted as strongly between unrelated individuals (the stranger condition) as between related individuals (the partner condition).

Result 3: The strategic motive for punishment is less important than non-strategic motives.

As in PD games [START_REF] Falk | Driving Forces Behind Informal Sanctions[END_REF], people punish mostly to harm targets in retaliation for the negative emotions they have aroused. The reduction of payoff differentials is a secondary motive and when sanctions cannot affect the payoff shares there is no difference between the one-shot and the repeated game.

Inequality and the arousal of emotions

If people allocate punishment points even when they do not affect the earnings gap with the punished, this does not necessarily mean that they do not care about the level of inequality. On the contrary, the observation of inter-individual differences in contributions brings about negative emotions.

Result 4: Inter-individual comparisons cause negative emotions and punishment. Support for result 4. Figure 2 displays the distribution of punishment points as a function of the level of inequality between the punisher and the target, by treatment and protocol. This difference represents the absolute level of inequality in both contributions and earnings at the end of the first stage of each period. 10 The relative importance of each category of deviation is indicated above each bar.

This figure clearly shows that punishers react strongly to the observed level of inequality under all conditions. Subjects who contribute less and earn more are punished much heavily than those who earn almost the same payoff after the first stage. For example, a subject who earns between 14 and 20 units more than the punisher receives on average 1.9 points of punishment from this punisher whereas he receives only .07 point if he contributes and earns almost the same amount as the punisher (within the [-2,2] interval). This is formalised in the econometric analysis. The regressions reported in Table 3 show that the distribution of punishment points is less likely when an individual contributes more and earns less than the potential punisher. In contrast, disadvantageous inequality between the punisher and the target increases both the probability of punishment and the number of punishment points distributed. The regressions reported in Table 4 confirm that the intensity of punishment is influenced by inter-individual deviations in contributions and earnings; they also show that individuals are not punished for deviations from average group behavior. This confirms that inter-individual inequality gives rise to negative emotions and retaliation. Inter-individual comparisons are more relevant than comparisons between the individual and average behavior and earnings within the group.

We also see some subjects who punish cooperators who are already suffering from disadvantageous inter-personal inequality. In this sub-sample (see regression (5) in Table 3), the number of punishment points distributed increases in the deviation of the target's contribution from the average and not from the punisher's contribution. This suggests that free-riders' negative emotions are triggered by the fact that cooperators raise the social norm, and not by inter-individual comparisons. This is consistent with [START_REF] Gächter | Norms of Cooperation among Urban and Rural Dwellers: Experimental Evidence from Russia[END_REF].

Overall, these results show that while negative emotions seem to be the main reason for punishment, rather than inequality reduction, inter-individual comparisons and inequality are an important source of the negative emotions that trigger the desire to punish. more effective under the Unequal than under the Equal Cost Treatment. As we saw earlier, this can be explained by the smaller monetary consequences of each punishment point on the target in the Equal Cost Treatment, which is anticipated by punishers who punish a given deviation more heavily. Finally, the regressions show that subjects react more to punishment in the partnermatching protocol, underlining the importance of strategic motives for contributing in repeated interactions.

How do these results translate into payoffs and inequality over time? Table 6 displays the evolution of average payoffs and Gini coefficients over time with and without punishment. To avoid possible biases due to restart and end-game effects, we do not compare directly the levels of inequality in the first and in the last period. Instead, we consider pooled data over three periods at the beginning and at the end of each set of rounds played under the same treatment.

At the aggregate level, Wilcoxon tests do not reject the null hypothesis that the Gini index is similar for treatments with and without punishment played under a stranger matching protocol.

Considering disaggregated data by sets of periods, Table 6 shows that inequality declines over time within each set of periods for each treatment.

We also observe that when punishment is not possible, free-riding leads to more equal payoffs but with falling earnings. On the contrary, punishment produces falling inequality along with a slight tendency to rising earnings (except in the Equal Cost Treatment under the stranger matching protocol where earnings are flat). 12 For example, in the Unequal Cost Treatment without punishment, the average payoff drops from 23.15 to 20.8, whereas with punishment average earnings rise from 20.70 at the beginning of the set of periods to 23.06 at the end of the periods.

V. DISCUSSION AND CONCLUSION

Previous research has shown that punishment is an important mechanism underlying cooperation among unrelated individuals, but the reasons why individuals are willing to pay to punish others remain unclear. In this paper we have looked at the influence of inequality aversion on the punishment decision, compared to that of negative emotions and strategic motives. We have considered a number of treatments in a public good game in which we introduce punishment opportunities with different consequences on the distribution of payoffs. Our econometric analysis allows us to separate the decision to punish from the intensity of punishment, so that we can control for possible selection bias.

We reject the strong hypothesis that individuals only punish to reduce inequality between themselves and the targets, but support the weak hypothesis of some influence of inequality aversion on punishment. However, the marginal effect of the nature of the punishment on the decision to punish shows that punishment results more from negative emotions than from inequality aversion or strategic motives (i.e. increasing the target's contribution in future periods). Introducing punishment that does not affect the distribution of payoff shares reduces the probability of punishment by only 4%. These results are remarkably consistent with those obtained in one-shot PD games by [START_REF] Falk | Driving Forces Behind Informal Sanctions[END_REF].

While punishers are not primarily motivated by inequality-aversion, nonetheless, in all treatments, inter-individual comparisons are used as a social norm to decide whom to punish.

Negative emotions are primarily triggered by the observation of inequality in contributions and earnings more than by the lack of adherence to the social norm of contributing. Thus, instead of contrasting inequality aversion and negative emotions, further research is needed to understand 20 how inequality may give rise to negative emotions. Another implication is that we should concentrate more on inter-individual comparisons than on comparisons between individual and average behavior in groups.

Finally, we also show that punishment within a group reduces inequality over time. This suggests that, when groups are fixed, team members might punish strategically not only to increase their own payoff in future periods, but also to reduce earnings inequality over time, even though punishment leaves current-period inequality unaffected. This impact of punishment on inequality over time is also seen in groups of unrelated individuals. We find evidence that freeriding progressively gives rise to reduced inequality coupled with low earnings; in contrast, the disciplinary effect of punishment brings about falling inequality associated with higher earnings.

Fig.1. Average contributions over time

For each ECU that you keep for yourself you earn an income of 1 ECU. For every ECU you contribute to the project instead, the total contribution rises by one ECU. Your income from the project will rise by 40% (1) = 0.4 ECU. The income of the other group members will also rise by 0.4 ECU each, so that the total income of the group from the project rises by 1.6 ECU. Your contribution to the project therefore also increases the income of the other group members. On the other hand you will earn money from each ECU contributed by other members to the project. For each ECU contributed by any member you earn 40% (1) = 0.4 ECU.

At the end of each period, groups randomly reshuffled.

If you have any question on these instructions, please raise your hand. We will answer your questions in private.

INSTRUCTIONS FOR THE FOLLOWING 10 PERIODS

[These instructions were distributed only at the end of the first 10 periods]

As before, groups are reshuffled randomly after each period. Each period consists of two stages.

In the first stage you have to decide how many ECU you would like to contribute to a project. In the second stage you are informed about the contributions of each of the three other group members to the project. You can decide whether or by how much to reduce their earnings from the first stage by distributing points to them. The following sections describe the activity in detail.

The first stage is identical to the first ten periods.

At the beginning of each period each participant receives 20 ECU.

Each of the four subjects simultaneously decides how much of his or her endowment he/she will contribute to the project. After having decided how much of the 20 ECU you want to contribute to the project, by choosing a number between 0 and 20, you must press the OK button Once you have done this, you can no longer change your decision for the current period..

At the end of this first stage, you will be individually informed of the amount contributed and of your income in the first stage which consists of two parts: the amount of your endowment which you have kept for yourself (i.e. 20 -your contribution to the project), the income from the project: this income represents 40% of the total contribution of all four group members to the project Your income of the first stage in each period is therefore:

(20-your contribution to the project)+ 40% (total contributions to the project)

The income of each group member from the project is calculated in the same way, this means that each group member receives the same income from the project. Similarly, your income can be modified if the other group members wish to do so.

You are first informed how much each group member contributed to the project in the first stage. Please note that the order in which the contribution decisions of the three other subjects appear on your screen changes randomly each period (i.e. the first number which appears on your screen does not always correspond to the same subject).

You must decide how many points to give to each of the other three group members to reduce their income or leave it unchanged. You must enter a value for each subject, between 0 and 10 points. If you do not wish to change the income of a specific subject, then you must enter 0.

If you distribute points, you bear a cost in ECU which depends on the number of points you distribute to each subject. The more points you give to any subject, the higher your costs. Your total costs are equal to the sum of the costs of distributing points to each of the other three group members. The following table illustrates the relation between points distributed to a subject and the cost of doing so in ECU. Suppose for example that you give 2 points to one subject. This costs you 2 ECU. If you give 9 points to another subject, this costs you an additional 25 ECU; if you give the last subject no points, this has no cost for you. In this example, your total costs of distributing points would be 27 ECU (2 + 25 + 0). These costs will be displayed on your screen. As long as you have not pressed the OK button you can alter your decisions.

Number of points distributed to a subject

Your final income in ECU in each period for the two stage is therefore calculated as follows:

= (income from the 1st stage) -cost of the points you have received -cost to you of points you distribute Your income in ECU at the end of the second stage can be negative, if the costs of the points you distribute exceeds your income from the first stage. You can however avoid such losses with certainty through you own decisions.

* * *

If you have any question about these instructions, please raise your hand. We will answer your questions in private.

INSTRUCTIONS FOR THE LAST 10 PERIODS

[These instructions were distributed only at the end of the 20th period]

In the next ten periods, each period will follow the same rules as periods 1-10. The groups are reshuffled randomly after each period.

. * * * 34 35 NOTES 1 In the first stage of the game designed by Fehr and Gächter (2000), subjects contribute to a public good; in the second stage, after being informed about their individual contributions, the subjects can impose costly punishment on their team members. Contrary to the unique sub-game perfect Nash equilibrium of this game, subjects do punish their teammates whose level of contribution is lower than the average. The targets increase their contributions in reaction to punishment and the groups converge to the optimum of full cooperation. 2 The results of Fehr and Gächter (2000) on punishment have inspired many other studies that confirm the existence of peer pressure in groups and examine many variants of punishment. For example, [START_REF] Carpenter | Punishing Free-Riders: How Group Size Affects Mutual Monitoring and the Provision of Public Goods[END_REF] shows the influence of the group size on the efficiency of punishment. He also demonstrates that punishment is more efficient when agents are allowed to punish only a fraction of the group members. Bochet, Page, and Putterman (Forthcoming) show however that the efficiency of punishment on cooperation is lower than when communication is allowed. For their part, [START_REF] Fehr | Detrimental Effects of Sanctions on Human Altruism[END_REF] identify a detrimental effect of sanctions; the crowding-out of norm-based motivations by punishment is also suggested by [START_REF] Houser | When Punishment Fails: Research on Sanctions, Intentions and Non-Cooperation[END_REF].

3 [START_REF] Fehr | A Theory of Fairness, Competition and Cooperation[END_REF] develop a model in which individuals exhibit inequality aversion if they dislike being better or worse than others. [START_REF] Bolton | ERC: A Theory of Equity, Reciprocity, and Competition[END_REF] propose another model of inequality aversion in which people compare their payoff to the average payoff of the group. Whereas the latter assumes that subjects like being as close as possible to the average payoff, Fehr and Schmidt assume that subjects dislike payoff differences compared to any other individual. [START_REF] Engelman | Inequality Aversion, Efficiency and Maximin Preferences in Simple Distribution Experiments[END_REF] designed an experiment to compare the relative performance of these two theories and conclude in favor of Fehr and Schmidt.

Table 1 .

 1 Levels of punishment and associated costs

	Unequal Cost Treatment
	Punishment Points	0	1	2	3	4	5	6	7	8	9	10
	Cost to the punisher in units	0	1	2	4	6	9	12	16	20	25	30
	Cost to the target in % of the	0	10	20	30	40	50	60	70	80	90 100
	target's earnings from the 1 st stage
	Equal Cost Treatment
	Punishment Points	0	1	2	3	4	5	6	7	8	9	10
	Cost to the punisher in units	0	1	2	4	6	9	12	16	20	25	30
	Cost to the target in units	0	1	2	4	6	9	12	16	20	25	30

Table 3 .

 3 Punishment (probability and intensity) in the Stranger Matching protocol errors in parentheses. *** statistically significant at the .01 level; ** at the .05 level; * at the .10 level. Each individual appears 30 times (1 observation for each pair within the group over 10 periods)

			Variables	Random effects Probit model	GLS models
						(1)	(2)	(3)	(4)	(5)
	Equal Cost	-.416***	-.036***	.488***	.493***	-.099
	Treatment	(.140)		(.121)	(.134)	(.199)
	Negative difference	.022	.002	-.021	-.023	.007
	from the average	(.023)		(.187)	(.045)	(.038)
	Positive difference	-.227***	-.019***	-.037*	-.040*	.843***
	from the average	(.020)		(.023)	(.023)	(.154)
	Average group	-.032**	-.003**	-.015	-.029**	-.003
	contribution	(.013)		(.013)	(.014)	(.034)
	Negative difference	.361***	.031***	.112***	.101***
	from the punisher	(.016)		(.029)	(.030)
	Positive difference	-.042**	-.004**	.085***	.046
	from the punisher	(.018)		(.024)	(.032)
	Period			-.114***	-.010***
						(.013)
	Constant	.180		1.224***	1.519***	.205
						(.276)		(.310)	(.327)	(.897)
	ρ					.496 (.040)
	IMR						-.210*	-.308**	.322
								(.127)	(.132)	(.415)
	Nb observations	4320		668	597	59
	Log Likelihood	-1119.615
	R 2							.258	.261	.551
	Wald	χ	2	711.29	238.77	242.54	64.17
	p>	χ	2			.0000		.0000	.0000	.0000
	Note: standard

Table 4 .

 4 Punishment (probability and intensity) in the Equal Cost Treatment (pooled data from errors in parentheses. *** statistically significant at the .01 level; ** at the .05 level; * at the .10 level.

	both protocols)
				Variables	Random effects Probit model	GLS models
						(1)	(2)	(3)	(4)
	Partner matching	.072	.005	.079	.162
	protocol	(.239)	(.206)	(.220)
	Negative difference	.002	.0001	-.014	-.0202
	from the average	(.032)	(.044)	(.059)
	Positive difference	-.233***	-.016***	-.005	-.007
	from the average	(.024)	(.039)	(.039)
	Average group	-.003	-.0002	-.029*	-.042**
	contribution	(.015)	(.018)	(.019)
	Negative difference	.318***	.022***	.112**	.100**
	from the punisher	(.018)	(.046)	(.046)
	Positive difference	-.140***	-.009***	.113**
	from the punisher	(.032)	(.058)
	Period			-.094***	-.006***
						(.016)
	Constant	-.3821	1.794***	2.091***
						(.338)	(.519)	(.534)
	ρ					.430 (.050)
	IMR					-.29793	-.400*
							(.221)	(.227)
	Nb observations	2880	410	384
	Log Likelihood	-707.945
	R 2						.254	.252
	Wald	χ	2	461.93	161.12	161.70
	p>	χ	2			.000	.000	.000
	Note: standard

Table 5 .

 5 The change in contributions between periods t and t+1 errors in parentheses. *** statistically significant at the .01 level; ** at the .05 level: * at the .10 level.

	Dependent variable:	Targets who contributed less	Targets who contributed more
	Change in		than the average		than the average
	contribution	Stranger	Equal	Pooled data Stranger	Equal	Pooled data
	between t and t+1
	Points received in	.894***	.506***	.833***	.653*	.220	.631
	period t		(.126)	(.104)	(.125)	(.388)	(.523)	(.414)
	Points received *	-.412***		-.376***	-.448		-.441
	Equal Cost Treatm.	(.111)		(.116)	(.597)		(.639)
	Points received *		.251*	.238*		1.660	1.630
	partner protocol		(.146)	(.130)		(1.823)	(1.742)
	Deviation from the	-.314***	-.303***	-.349***	-.770***	-.690***	-.713***
	average		(.060)	(.077)	(.055)	(.054)	(.077)	(.054)
	Constant	-.522**	-.398	-.492**	.277	-.214	-.002
				(.215)	(.317)	(.211)	(.288)	(.376)	(.280)
	Observations	616	380	692	634	434	744
	R 2			.308	.276	.322	.207	.150	.166
	Wald	χ	2	287.68	157.63	352.27	203.82	82.49	174.69
	p> 2 χ		.000	.000	.000	.000	.000	.000
	Note: standard

Table 6 .

 6 Changes over time in average payoffs and Gini coefficients by treatment

		All periods	Periods	Periods
				without punishment	with punishment
	Groups of periods	Without punishment	With punishment	21-23 1-3 and	28-30 8-10 and	11-13	18-20
	Unequal Cost Strangers
	Mean Payoff	21.81	22.51	23.15	20.80	20.70	23.06
	Gini	.10	.09	.12	.06	.13	.08
	Equal Cost Strangers
	Mean Payoff	22.09	21.97	23.37	21.17	21.94	20.93
	Gini	.10	.10	.11	.08	.13	.10
	Equal Cost Partners
	Mean Payoff	22.22	25.66	24.02	20.67	24.08	25.72
	Gini	.10	.08	.11	.06	.09	.08

  At the beginning of the second stage, your screen shows you how much each of your group members contributed to the project. You have now the possibility to reduce or leave unchanged the income of each group member by distributing points. You can distribute points to any member of your group to indicate your disapproval. Each point you distribute to a particular player lowers his or her payment according to the following schedule:Suppose for example that you give two points to one subject. This will reduce his first-stage income by 2 ECU; if you give 9 points to another subject, this will reduce his first-stage income by 25 ECU; if you give the last subject no points, this does not change his income. The amount of points you distribute to each subject determines therefore how much you reduce his income from the first-stage.

	Number of points received from the	0	1	2	3	4	5	6	7	8	9	10
	same subject
	Cost deduced from first-stage income of	0	1	2	4	6	9	12	16	20	25	30
	the subject who receives the points
				33

Most papers which study the importance of inequality in public good games have focused on endowment heterogeneity. In contrast, our paper examines the relationship between punishment and inequality aversion when inequality arises endogenously from the subject's behavior.

In our experiment, the cost to the target was higher than the cost to the punisher in all the cases where a subject punished another subject in the Unequal Cost Treatment. On average, the ratio is 2.74.

Alternatively, we could have imposed a fixed cost ratio between the punisher and the target higher than 1. We rejected this option in order to replicate the design in use inFehr and Gächter (2000).

In all non-parametric statistical tests reported in this paper, the unit of observation is the group in the data collected under the partner matching protocol (N=6) and the session for the data collected under the stranger matching protocol (N=8).

The "negative deviation from punisher" variable is the absolute value of the difference between subject j 's contribution and the contribution of subject i. This variable is set equal to zero if the deviation is positive. The other deviation variables are constructed analogously.

The number of observations in which subjects punish individuals who contribute more than themselves is not sufficient to allow for a separate estimation of the model on this sub-sample.

It should be noted that in standard linear public good games, the difference in contributions between two players is similar to the difference between their first-stage payoffs.

Here we consider the deviation from the average since a punished subject is only aware of the total number of points he received from the other group members and does not know who punished him.

In addition, when punishment is possible, we observe that average first-stage payoffs (i.e. before punishment decisions are made) are significantly higher than in treatments without punishment. The figures are 25.37 in the Unequal Cost Treatment, 24.22 in the Equal Cost Treatment under the stranger matching protocol and 27.83 under the partner matching protocol.

This paper has benefited from useful comments from participants at the 2005 paper, and to the MiRe -DREES

Consequences of punishment on the level of inequality

How does punishment change the level of inequality in groups over time? We suspect that even if punishment does not directly affect inequality, punishment in one period may reduce inequality in further periods if it affects future contributions by the punished.

Result 5: Under all conditions, individuals who contributed less than average in period t increase their contributions more in period t+1 the more they were punished in t. As a result of both the credible threat of punishment and its actual impact, payoff inequality decreases over time.

Support for result 5: Table 5 displays the estimates of random-effects GLS models analyzing the determinants of the change in contribution between periods t and t+1. These are estimated separately for those who contributed less than average and more than average (on the left and right sides of the Table respectively). The first columns correspond to the pooled data from the Equal and Unequal Cost Treatments under the stranger matching protocol; the second columns correspond to the pooled data from the Equal Cost Treatment under both matching protocols; and the third columns pool all data together. The explanatory variables include the punishment received in period t and the deviation from the average contribution of other group members. 11 We also interact the punishment points received with the treatment and with the protocol. These interactions show whether punishment is more or less effective under the Equal Cost treatment, and under the partner matching protocol.

Table 5 reveals a significant negative relationship between deviation from the average contribution and the subsequent change in contributions. In contrast, punishment boosts the contributions of those who contributed less than average in all treatments. Those who contributed more than average are unaffected. The Table also shows that a punishment point is

Fig.2. Punishment intensity as a function of payoff and contribution inequality between the punisher and the target (the percentages at the top of the bars represent the relative frequency of each category of deviation in the various treatments)

Appendix -Instructions of the Equal Cost Treatment [other instructions are available upon request]

You are now taking part in an economic experiment during which you can earn money. Your earnings depend on your decisions and the decisions of others. It is therefore very important that you read these instructions with care.

The instructions we have distributed to you are solely for your private information. It is forbidden to communicate with the other participants during the experiment. If you violate this rule, we shall have to exclude you from the experiment and from all payments.

During the experiment your entire earnings will be calculated in ECU (Experimental Currency Units). At the end of the experiment the total amount of ECU you have earned will be converted to Euros and paid in cash according to the following rules:

Your final payoff in ECU consists of the sum of your payoffs in each period.

This final payoff in ECU will be converted into Euros at the rate: 100 ECU = 2 Euros.

In addition, you will be given a show up fee of 3 Euros.

The experiment is divided into successive periods. In each period the participants are divided into groups of four. You will therefore be in a group with 3 other participants. In each new period, the composition of your group will change: your probability of interacting more than one time with the same three other people is very low. You will not be informed of the identity of the other group members.

INSTRUCTIONS FOR THE FIRST 10 PERIODS

The four subjects belonging to a group can participate in a project, by contributing to a group account that will be shared among them. This amount of this group account is determined by the individual contributions of the four members of the group.

At the beginning of each period each participant receives 20 ECU. In the following we will call this his or her endowment.

Each of the four subjects simultaneously decides how much of his or her endowment he/she will contribute to the project. After having decided how much of the 20 ECU you want to contribute to the project, by choosing a number between 0 and 20, you must press the OK button Once you have done this, you can no longer change your decision for the current period.

After all members of your group have made their decision your screen will show you the total amount of ECU contributed to the project by the four group members (including your own contribution). This screen shows you how many ECU you have earned for this period.

Your income consist of two parts: the amount of your endowment which you have kept for yourself (i.e. 20 -your contribution to the project), the income from the project: this income represents 40% of the total contribution of all four group members to the project (the total includes your own contribution)

Your income in ECU in each period is therefore:

(20-your contribution to the project) + 40%(total contributions to the project)

The income of each group member from the project is calculated in the same way, this means that each group member receives the same income from the project.

For example, suppose the total of the contributions of all group members is 60 ECU. In this case each member of the group receives an income from the project of 40% (60) = 24 ECU. If the total contribution to the project is 9 ECU, then each member of the group receives an income of 40% (9) = 3.6 ECU from the project.


¸ois Maucourant

UNIQUE ERGODICITY OF ASYNCHRONOUS ROTATIONS, AND APPLICATION

Keywords:

The main result of this paper is an analogue for a continuous family of tori of Kronecker-Weyl's unique ergodicity of irrational rotations. We show that the notion corresponding in this setup to irrationality, namely asynchronicity, is satisfied in some homogeneous dynamical systems. This is used to prove the ergodicity of naturals lifts of invariant measures.

Résumé. Nous étudions sur une famille continue de tores les rotations dites asynchrones, analogues aux rotations irrationnelles sur les tores classiques. Le résultat principal est l'unique ergodicité de ces rotations sur un monoïde adapté. Nous prouvons que la condition d'asynchronicité est vérifiée dans une famille d'exemples issue de la dynamique homogène, ce qui nous permet de déduire l'ergodicité de relevés de certaines transformations dans des fibrés en tores.

1. Introduction 1.1. Motivations. The original motivation of this study is the inquiry of ergodic properties of torus extension of homogeneous dynamical systems. Such dynamics have drawn some attention recently -see for example [START_REF] Elkies | Gaps in √ n mod 1 and ergodic theory[END_REF], [START_REF] Marklof | The Boltzmann-Grad limit of the periodic Lorentz gas[END_REF], [START_REF] Strömbergsson | An effective Ratner equidistribution result for SL(2, R) ⋉ R 2[END_REF] for unipotent actions, and [START_REF] Arnoux | The scenery flow for geometric structures on the torus: the linear setting[END_REF] and [START_REF] Shapira | A solution to a problem of Cassels and Diophantine properties of cubic numbers[END_REF] for diagonal actions.

As an informal example, consider a diagonal element a ∈ SL(d, R) = G 0 (d ≥ 2) with positive diagonal entries, acting by left multiplication on the homogeneous space SL(d, R)/SL(d, Z) = G 0 /Γ 0 , and let µ be a probability invariant by a and ergodic on G 0 /Γ 0 . The main interesting cases for our purpose occur when the measure µ is not algebraic. This dynamical system is a factor of the action of (a, α), where α ∈ R d is arbitrary, by left multiplication on (SL(d, R) ⋉ R d )/(SL(d, Z) ⋉ Z d ) = G/Γ. This latter space is a torus bundle above G 0 /Γ 0 . Amongst the possible (a, α)-invariant measures that projects onto µ, there is a particular one, denoted by λ, which decomposes into the Haar measure of tori on each fiber. It is natural to ask about its ergodicity with respect to (a, α).

Following the classical Hopf argument (see [START_REF] Coudène | The Hopf argument[END_REF]), one is naturally led to inquire about ergodic properties of the strong stable foliation of (a, α). It turns out that this foliation contains the orbits of another action, namely the multiplication by (e, β) on G/Γ, where β ∈ R d is an eigenvector for a associated to an eigenvalue < 1. This action is an unipotent action, but since it is "vertical" (in the sense trivial in the factor G 0 /Γ 0 ), Ratner's theory yield in this case no more information than Kronecker-Weyl's uniform distribution on the torus.

To visualize this action of (e, β) on each fiber, one may think of it as a rotation by a fixed vector β on a varying torus depending on the base-point. Here, we will prefer to think of it as the rotation by a varying vector f β (x) depending on the base-point x, on a fixed torus T d . One may hope in this situation that f β (x) ∈ T d is irrational for almost every x. It turns out that under appropriate assumptions, the rotations defined by f β satisfy a stronger property, namely asynchronicity.

As we will see shortly, such rotations on torus bundle above a measured space like (G 0 /Γ 0 , µ) enjoy strong ergodic properties, enabling us in this setting to prove a unique ergodicity result. In some sense, this can be considered as a weak analogue of Furstenberg's unique ergodicity of horocyclic flow.

Finally, we will return to the question of the ergodicity of λ with respect to (a, α), and related mixing properties. These kind of fiber-wise system were also investigated independently by Damien Thomine [START_REF] Thomine | Keplerian shear in ergodic theory[END_REF], using another point of view.

1.2. Asynchronous rotations. Let (I, B(I), L) be a standard probability space without atoms, and let K be a torus T d = (R/Z) d , for some integer d ≥ 1. We think of a measurable map f : I → K as the data, for each x ∈ I, of a rotation adding the angle f (x) in a torus indexed by x. Despite what the above motivational example might suggest, in this abstract setting, the case d = 1 of rotations on a family of circles above a probability space is already interesting, and contains most of the difficulties.

The set of such measurable maps {f : I → K}, where we identify two maps if they coincide L-almost everywhere, is naturally an abelian group under pointwise addition of functions. We denote by (K I , +) this group, by a slight abuse of notation. We would like to study the translation by f in K I , but as it lacks a nice topology, we consider a compactification of the group (K I , +), which will be a monoid, as follows.

Let M L be the space of probability measures on I ×K which project to L on the first factor. To an element f ∈ K I , we can associate the probability measure D f on I × K, supported by its graph, which is the pushforward of L by the map x ∈ I → (x, f (x)) ∈ I × K. This defines an embedding of K I into M L . It is not hard to see that the group law + on K I correspond to a fiber-wise convolution product * on M L , which turns (M L , * ) into an abelian monoid, with neutral D 0 , where 0 : I → K is the zero map.

The space M L is equipped naturally with a weak-* topology, for which it is a compact metric space. A tricky fact is that the convolution product (µ, ν) → µ * ν is not continuous of the two variables, but is of each variable separately. A more detailed description of these objects, and explanations of the implied claims, are given in Section 2.

There is a particular element in M L , the measure λ = L ⊗ Haar K . It satisfies the relation: ∀µ ∈ M L , µ * λ = λ.

We are interested in studying the dynamics of translation D f * on the monoid M L . Unsurprisingly, we now need a kind of irrationality condition.

Definition. The angle map f : I → K is said to be asynchronous if the image measure f * L gives zero mass to any translate of any proper closed subgroup of K. Equivalently, for any non-trivial character χ ∈ K, (χ • f ) * L has no atoms.

Intuitively, for d = 1, this means that one looks at an action by rotation on a family of circles indexed by x ∈ I, by angles f (x), which are different from one another if picked randomly following the probability L. For d ≥ 1, it means that for almost every couple (x, y), f (x) and f (y) do not belong to the same coset modulo any closed, strict subgroup of K.

Theorem 1. The following are equivalent.

(1) The angle map f : I → K is asynchronous.

(2) The closure {D nf } n∈Z contains λ.

(3) The convolution action of D f on M L is uniquely ergodic. If these are true, the only invariant probability measure is the Dirac measure δ λ .

The fact that the invariant measure is a Dirac measure implies (see Proposition 3.3) that there exists a subset of the integers E ⊂ Z, of natural density 1, such that for any µ ∈ M L , lim n→±∞, n∈E

D nf * µ = λ.
The question whether λ is an attracting point of the dynamic, that is if lim

n→±∞ D nf * µ = λ,
or if this fails along some subsequence of zero density, is more delicate, and its answer depends on f .

If I = [0, 1] equipped with Lebesgue measure, d = 1, and f is a C 2 map with non-vanishing derivative, then λ is an attracting point (Proposition 3.1), and there is no exceptional subsequence. The C 2 regularity condition is not optimal, as Thomine obtained similar results for C 1 maps [START_REF] Thomine | Keplerian shear in ergodic theory[END_REF].

However, for an angle map f which is only measurable, the convolution action of D f might behave more like an intermittent map with the neutral fixed point λ. An example of this phenomenon is the following. Again, let I = [0, 1] endowed with the Lebesgue measure L. Let ν be the (probability) Hausdorff measure of dimension log 2/ log 3 on the usual Cantor set C, viewed as a subset of K = T 1 by identifying 0 and 1. Define

f : [0, 1] → T 1 , by f (x) = inf{t ≥ 0 : ν(0, t) ≥ x} mod 1.
Then f * L = ν, and f (I) ⊂ C. Alternatively, f can be defined as the reciprocal, outside of dyadic rationals, of the usual devil's staircase, modulo 1. Since ν does not have any atom, f is an asynchronous angle map. We claim that the sequence (D 3 k f ) k≥1 does not intersect a fixed neighborhood of λ. Indeed, since C is invariant by the multiplication ×3 on the circle, the graph of 3 k f is contained in I × C, so the measure D 3 k f is supported on I × C, a proper compact subset of I × K. This forbids D 3 k f to be close to λ. Still, by Theorem 1, subsequences like (D 3 k f ) k≥0 are scarce, as the points (D nf ) n∈Z tend to λ for a subset of Z of density one. 1.3. Main example, and a related ergodicity result. As hinted in the motivational paragraph, asynchronous rotations occur naturally in the context of homogeneous dynamics on torus bundle.

More precisely, let G 0 be a connected, semisimple algebraic linear group defined over Q, G 0 = G 0 (R) the group of its R-points and Γ 0 = G 0 (Z) its integer points. By the Borel -Harish-Chandra Theorem, Γ 0 is a lattice in G 0 . We will consider invariant measures on G 0 /Γ 0 under some elements a ∈ G 0 , under the following assumptions.

Definition. An element a ∈ G 0 -{e} is said triangularizable with positive eigenvalues if for every finite dimensional representation α of G 0 defined over Q, α(a) has only real, positive eigenvalues.

It is the case, for example, when G 0 is the real split form of G 0 , meaning the real rank equals the complex rank, and if a is the exponential of a non-zero element of a Cartan subalgebra. It also happens when a is unipotent, but this case is less interesting for our purpose, since by Ratner's Theory, a-invariant ergodic measures are algebraic. This hypothesis implicitly rules out the case where G 0 is the real compact form of G 0 , as it cannot contain such element a.

Definition. Let µ be a probability measure on G 0 /Γ 0 , invariant and ergodic under the action of a. Such a measure is said to be nonconcentrated if for every H ⊂ G 0 closed algebraic, strict subgroup containing a, and every x ∈ G 0 /Γ 0 such that HxΓ 0 is closed, then µ(HxΓ 0 ) < 1.

We now consider a fiber bundle over the probability space (G 0 /Γ 0 , µ), whose fibers are tori.

Let ρ : G 0 → GL(V ) a representation defined over Q on a finitedimensional space V = R d endowed with the Z-structure Z d . We will always assume that d ≥ 2, and that ρ is irreducible over Q. The semidirect product G = G 0 ⋉ ρ V is endowed with the group law

∀(g, v) ∈ G, ∀(h, w) ∈ G, (g, v)(h, w) = (gh, v + ρ(g)w).
Up to replacing Γ 0 with a subgroup of finite index in a way such that

ρ(Γ) ⊂ GL(d, Z), the set Γ = Γ 0 ⋉ ρ Z d is a subgroup of G, and the map π : G/Γ → G 0 /Γ 0 ,
is a torus bundle. Indeed, the lattice of {e}×R d (for the action of multiplication on the left) stabilizing a point (x, v)Γ is precisely {e}×ρ(x)Z d , thus the fiber of π over xΓ 0 is the torus R d /ρ(x)Z d . It will be convenient to have measurable coordinates where this fiber bundle is a direct product.

Let I ⊂ G 0 be a measurable fundamental domain for the action of Γ 0 , and put L the restriction to I of the Γ 0 -invariant lift of µ. As previously, we denote by K the d-dimensional torus

T d = R d /Z d . The map I × K → G/Γ, (x, v) I×K → (x, ρ(x)v)Γ,
is a measurable bijection, that we will use as an identification between I × K and G/Γ, the subscript I × K indicating the coordinates we are using. Likewise, we will identify G 0 /Γ 0 with I and µ with L.

For β ∈ V , the action of (e, β) by multiplication on G/Γ on the left, can be read in the I × K coordinates as the map

(x, v) I×K → (x, v + ρ(x) -1 β) I×K ,
i.e. it is a rotation by an angle map f β : I → K, with

f β (x) = ρ(x) -1 β mod Z d .
We prove: Theorem 2. Assume that a ∈ G 0 is triangularizable with positive eigenvalues, that µ is an a-invariant, non-concentrated, ergodic probability on G 0 /Γ 0 , that ρ is irreducible over Q, and dim(V ) > 1. Assume also that β ∈ V -{0} is an eigenvector for ρ(a). Then the angle map

f β : I → K x → ρ(x) -1 β mod Z d , is asynchronous.
Using the identification of G/Γ with I × K, we still denote by λ the measure on G/Γ such that π * λ = µ, whose disintegration along each fiber of µ are the Haar measures on each tori.

Direct application of Theorem 1 gives:

Corollary 1.1. We assume the same hypotheses as Theorem 2. Let M µ be the set of probabilities on G/Γ projecting onto µ. The action on M µ induced by the left multiplication by (e, β) on G/Γ is uniquely ergodic, with invariant measure δ λ .

Now choose any α ∈ V . The action of left multiplication by (a, α) on G/Γ admits the action of a on G 0 /Γ 0 as a factor. A natural question is if the measure λ, which is invariant, is ergodic with respect to this action.

If β is any eigenvector of ρ(a), multiplication by (e, β) is in some sense moving in some part of the stable, unstable or neutral direction (depending on the eigenvalue) of the action of (a, α). Theorem 2 and Hopf's argument allows us to prove the following ergodicity result: Theorem 3. Assume that a ∈ G 0 is triangularizable with positive eigenvalues, that µ is an a-invariant, non-concentrated, ergodic probability on G 0 /Γ 0 , and that ρ is irreducible over Q, of dimension > 1.

Choose α ∈ V , then the action by left multiplication by (a, α) on G/Γ is ergodic with respect to the invariant measure λ. If we assume moreover that ρ(a) is not unipotent, then the action of (a, α) on (G/Γ, λ) is weakly mixing if and only if the action of a on (G 0 /Γ 0 , µ) is weakly mixing, and the same property holds for strong mixing.

1.4. Plan of the paper. In Section 2, we collect some facts about the topology of M L .

In Section 3, after dealing with the toy example where f is a C 2 map, we prove Theorem 1.

In Section 4, we prove that in the algebraic setting, the smallest algebraic subgroup of G 0 containing the elements γ ∈ Γ 0 induced by Poincaré recurrence of tha a-action, is G 0 itself. This result (Theorem 4), which is the main ingredient of the proof of Theorem 2, relies crucially on the non-concentration of µ.

In Section 5, we prove the asynchronicity of the rotation obtained by the construction in homogeneous dynamics (Theorem 2).

In Section 6, we prove the ergodicity of the extension of the action of a on G 0 /Γ 0 (Theorem 3). This mainly relies on the following easy observation: if an angle map f : I → K is asynchronous, then for almost every x, the action of translation by f (x) on the torus K is (uniquely) ergodic. 1.5. Acknowledgements. I would like to thank to Jean-Pierre Conze, Serge Cantat, Sébastien Gouëzel, Barbara Schapira and Damien Thomine for their feedback and comments on the subject.

2. The space M L 2.1. Topology of M L . We recall that (I, B(I)) be a standard measurable space means that is I can be endowed with a complete, separable distance d I , such that B(I) is the σ-algebra of its Borel sets. The facts about standard probability spaces we will use are summarized in [START_REF] Aaronson | An introduction to infinite ergodic theory[END_REF]Chapter 1.1]. Choosing such a distance on I defines a topology on the space of probability measures on I × K, and hence on M L .

Although the weak-* topology of the space of measures on I × K depends in a strong way on the choice of topology on I, it turns out that:

Lemma 2.1. The topology induced on M L does not depends on the choice of topology on I.

Proof. Let I 1 , I 2 two complete, separable metric space endowed with probabilities L i , with a map ϕ : I 1 → I 2 an isomorphism such that ϕ * L 1 = L 2 . The topologies induced on measures on I i ×K are generated by the open sets:

U i (F, δ, µ) = ν : I i ×K F dµ - I i ×K F dν < δ ,
where F : I i × K → C is continuous with compact support for the relevant topology on I i × K.

Denote by φ the map

I 1 × K → I 2 × K, φ(x, y) = (ϕ(x), y). To show that φ * : M L 1 → M L 2 is
a homeomorphism, it is sufficient by symmetry to show its continuity.

We fix a neighborhood U 2 (F, ǫ, φ * µ), and wish to show that its preimage contains some neighborhood of the initial point U 1 (G, δ, φ * µ), for some G, δ.

The map ϕ from I 1 to I 2 is measurable. By Lusin's Theorem, for every δ > 0, there is a compact set J ⊂ I 1 , such that L 1 (J) > 1δ, on which ϕ is continuous. Let F : I 2 × K → C be continuous, by Tietze-Urysohn's Theorem, there exists a continuous function G δ :

I 1 ×K → C which extends the continuous map F • φ : J × K → C. Moreover, since F is bounded, G δ can be chosen such that G δ ∞ = F ∞ . If ν ∈ M L 1 , I 2 ×K F d φ * µ - I 2 ×K F d φ * ν ≤ 2δ F ∞ + J×K G δ dµ - J×K G δ dν , because µ((I 2 -ϕJ) × K) = L 2 (I 2 -ϕJ) = L 1 (I 1 -J) < δ
and the same holds for ν. If we choose δ > 0 such that 2δ F ∞ < ǫ/2, we have:

I 2 ×K F d φ * µ - I 2 ×K F d φ * ν ≤ ǫ/2 + I 1 ×K G δ dµ - I 1 ×K G δ dν ,
and therefore, provided that δ < ǫ/2,

M L 1 ∩ U 1 (G δ , δ, µ) ⊂ φ-1 * (U 2 (F, ǫ, φ * µ) ∩ M L 2 ).
As any neighborhood of ϕ * µ contains finite intersections of sets of the form U 2 (F, ǫ, ϕ * µ), this implies that φ * :

M L 1 → M L 2 is continuous, as required.
A corollary of this discussion is that we can assume for example that I = [0, 1] and L is the Lebesgue measure on this interval, endowed with its usual topology. Since in this case, the set of probability measures on I × K is a compact, separable metric space, it follows that M L is also compact, separable and metric.

2.2. Graphs and measures. For a measurable map g : I → K, we define the graph measure D g , of g as the direct image of L by the map x ∈ I → (x, g(x)) ∈ I × K. Two measurables maps I → K define the same graph measure if and only if they are equal L-almost everywhere.

Let G be the set of graph measures, this is a subset of M L .

Disintegration along L.

Any µ ∈ M L can be disintegrated as a family of measures (µ x ) x∈I , such that for any continuous test-function with compact support F :

I × K → C, µ(F ) = I K F (x, y)dµ x (y) dL(x).
Moreover, the map x → µ x is measurable, and uniquely defined modulo zero sets. See e.g. [8, Th. 5.8].

2.4. Convolution product. For µ 1 , µ 2 two measures in M L , we define the fiberwise convolution product of µ 1 , µ 2 by

µ 1 * µ 2 (F ) = I K 2 F (x, y + z)dµ x 1 (y)dµ x 2 (z) dL(x)
, where F : I × K → C a continuous test-function with compact support. Equivalently, (µ * ν) x is the usual convolution product of µ x and ν x .

The following Lemma, whose proof is left to the reader, summarizes elementary properties of this fiberwise convolution product.

Lemma 2.2. The following holds.

(1)

∀(f, g) ∈ (K I ) 2 , D f +g = D f * D g . (2) D 0 is the neutral element of the commutative monoid (M L , * ),
where 0 : I → K is the map almost everywhere zero. (3) The set of invertible elements for * is G, the set of graph measures.

Remark that, if f : [0, 1] → T 1 , f (x) = x mod 1, then one can check by hand (or see e.g. Proposition 3.1) that D nf tends to λ as n → ±∞, but D nf * D -nf = D 0 = λ * λ, so the fiberwise convolution product is not continuous. However, we have:

Lemma 2.3. For any ν ∈ M L , the convolution map * ν : M L → M L , µ → µ * ν, is continuous.
Proof. It is sufficient to check that the preimage by * ν of any set of the form U(F, ǫ, µ * ν), for any F , ǫ, µ, contains a set of the form U(G, δ, µ) for some G and some δ > 0. Let U(F, ǫ, µ * ν) be such a neighborhood of µ * ν, and let δ such that δ(4 F ∞ + 1) < 1. As the map x → ν x is measurable, again by Lusin's Theorem, it is continuous on a set E of measure 1δ. Define

H(x, y) = K F (x, y + z)dν x (z).
It follows from the continuity of F and the continuity of x → ν x that this is a continuous map on E × K, and moreover bounded by F ∞ .

Thus it can be extended to a bounded continuous map, say G, on I ×K, still bounded by F ∞ . Notice that for any η ∈ M L ,

E×K F d(η * ν) = E K K F (x, y + z)dν x (z) dη x (y)dL(x) = E×K Gdη. Let η ∈ U(G, δ, µ), then I×K F d(η * ν) - I×K F d(µ * ν) ≤ 2δ F ∞ + E×K F d(η * ν) - E×K F d(µ * ν) ≤ 2δ F ∞ + E×K Gdη - E×K Gdµ ≤ 4δ F ∞ + δ.
By the choice of δ,this implies that * ν (U(G, δ, µ)) ⊂ U(F, ǫ, µ * ν), as announced.

Asynchronous maps

3.1. A simple example. As stated in the introduction, if f has enough regularity properties, it turns out that λ is the limit point of the dynamic of * D f on M L . This result will not be used in the sequel.

Proposition 3.1. Assume f : [0, 1] → R/Z is a C 2 -map, such that f ′ does not vanish. Then for all µ ∈ M L , D kf * µ → λ when k → ±∞.
Proof. By continuity of * µ, it is sufficient to check that D kf tends to λ as k tends to infinity. To do so, compute the Fourier-Stieltjes coefficients

Dkf (n, m) = [0,1]
e 2iπ(nx+mkf (x)) dx.

If m = 0, this coefficient is 1 or 0, depending on whether n = 0 or not. If m = 0, we can write

Dkf (n, m) = [0,1] e 2iπnx 2iπmk f ′ (x) ∂ ∂x e 2iπmkf (x) dx,
and integration by parts immediately shows that Dkf (n, m) = O( 1 k ) when k → ±∞ with n, m fixed.

Proof of 1) ⇒ 2)

. Assume that f is asynchronous. We may, and will, assume that I = T 1 endowed with its Haar probability measure L. The space I × K is then a (d + 1)-dimensional torus, and to check that some D nf is close to λ, it is sufficient to show that for a finite set of the non-trivial Fourier-Stieljes coefficients of D nf are close to zero. Lemma 3.2. For any nontrivial character χ 0 of I × K = T d+1 , we have

lim N →+∞ 1 N N -1 n=0 D nf (χ 0 ) 2 = 0.
Proof. For an integer k ∈ Z, let e k be the character of T 1 , e k (x) = e 2iπkx . Any character χ 0 of I × K = T n+1 can be written uniquely as a product χ 0 (x, y) = e k (x)χ(y), for some k ∈ Z and every x ∈ I, y ∈ K. We have

1 N N -1 n=0 D nf (χ 0 ) 2 = I 2 e 2iπk(x-x ′ ) 1 N N -1 n=0 χ(f (x) -f (x ′ )) n dL 2 (x, x ′ ).
If χ = 1, then k = 0 since χ 0 = 1. In this case, we have

1 N N -1 n=0 D nf (χ 0 ) 2 = 0,
so the statement is trivial.

If χ = 1, then by assumption χ(f (x)f (x ′ )) = 1 for almost every (x, x ′ ), so lim

N →+∞ 1 N N -1 n=0 χ(f (x) -f (x ′ )) n = 0.
Therefore, Lebesgue's dominated convergence Theorem applies and we obtain the desired result.

Let F ⊂ Td+1 -{1} be a finite subset of nontrivial characters, and ǫ > 0. By the previous Lemma, we have lim

N →+∞ 1 N N -1 n=0 χ 0 ∈F D nf (χ 0 ) 2 = 0,
so there exists n ≥ 0 such that for all χ 0 ∈ F , D nf (χ 0 ) ≤ ǫ, meaning that D nf is close to λ.

3.3.

Proof of 2) ⇒ 3). Let us check that λ ∈ {D nf } n∈Z implies that * D f is uniquely ergodic. Let (n i ) i≥1 be a sequence such that D n i f converges weakly to λ when i → +∞. Let µ ∈ M L , then by one-sided continuity of convolution,

µ * D n i f → i→+∞ µ * λ = λ.
Let m be any invariant measure on M L , F : M L → R be a continuous function. Then

M L F (µ)dm(µ) = M L F (µ * D n i f )dm(µ),
by invariance of m. The Lebesgue dominated convergence Theorem implies

M L F (µ)dm(µ) → M L F (λ)dm(µ) = F (λ),
which means that m is the Dirac measure at λ, as required.

3.4. Proof of 3) ⇒ 1). Assume that the convolution action of D f is uniquely ergodic. As λ is a fixed point, the invariant measure is necessarily δ λ , and thus as the invariant measure is a Dirac mass, there exist a subsequence n i → +∞ such that D n i f tends to λ as i → +∞.

Let χ be any non-trivial character of K. Assume that (χ • f ) * L has an atom. In this case, there would be a set E ⊂ I of positive measure on which χ • f is a constant, say c. Thus χ • (n i f ) is a constant on E, namely c n i , and (χ • (n i f )) * L will have an atom of mass L(E). Note that (χ

• (n i f )) * L = (χ • π K ) * D n i f . But (χ • π K ) * D n i f tends to (χ • π K ) * λ
, namely the Lebesgue measure on T 1 , which cannot be a limit of measures having a atom of fixed mass. This is a contradiction.

D nf * µ = λ.
Proof. We consider the measure on M L ,

ν N = 1 2N + 1 |k|≤N δ D kf .
As any weak limit of ν N is D f * -invariant and M L is compact, by unique ergodicity of (M L , D f * ), ν N converges to δ λ when N goes to +∞. This implies that for any neighborhood U of λ, the proportion of {D kf } |k|≤N outside U goes to zero as N → +∞. Let (U m ) m≥1 be a decreasing basis of neigborhood of λ, and

E m = {k ∈ Z : D kf ∈ U m }.
Let N m be an integer such that for all N ≥ N m ,

P N (E m ) ≥ 1 - 1 m ,
where P N is the uniform probability on [-N, N]. We can modify the sequence (N m ) m≥0 to be strictly increasing, and choose N 1 = -1. Let E be the subset

E = m≥1 E m ∩ {k ∈ Z : N m < |k| ≤ N m+1 }.
Notice that since the sets E m are decreasing with m,

if n ≤ N m+1 , E ∩ [-n, n] ⊂ E m .
Thus, for n such that N m < n ≤ N m+1 , we have

P n (E) ≥ 1 - 1 m .
This proves that E is a set of natural density one. By construction, we have lim n→±∞, n∈E

D nf = λ.
By continuity of the convolution with µ, the latter limit holds for the sequence D nf * µ with the same set E.

On the smallest algebraic group containing return elements

The following Theorem, which will be a crucial ingredient of the proof of Theorem 2, might be of independent interest. Theorem 4. Let G 0 be the group of real points of an algebraic group G 0 defined over Q, without nontrivial Q-characters, Γ 0 = G 0 (Z) be its integer points, a ∈ G 0 be triangularizable with positive eigenvalues, and µ an a-invariant measure on G 0 /Γ 0 . We assume that the measure µ is ergodic and non-concentrated. Let I ⊂ G 0 be a fundamental domain for Γ 0 , and denote by L the lift to I of µ. Let E ⊂ I be a subset of positive L-measure. Define

P E = γ ∈ Γ 0 : L ∪ k∈Z a k Eγ ∩ E > 0 ,
the set of elements of Γ 0 associated to return times in E. Then the smallest algebraic subgroup of G 0 containing P E is G 0 .

To prove this, let H be the smallest algebraic subgroup of G 0 containing P E . Our aim is to show that H = G 0 . This will be done in the following sequence of Lemmata. Notice that H is defined over Q, since P E consists of integer points. We claim that the the non-trivial Q-characters of H are of order 2. Indeed, if c is such a character defined over Q, the image by c of the subgroup generated by P E ∩ H 0 ⊂ Γ 0 = G(Z) consists of rational with bounded denominators, and is a multiplicative subgroup, so c(P E ) ⊂ {-1, +1}. Therefore, P E is contained in Ker(c 2 ), an algebraic group defined over Q. By definition of H, H ⊂ Ker(c 2 ), so H = Ker(c 2 ) as required. In particular, H/(H ∩Γ 0 ) is of finite volume, by the Theorem of Borel and Harish-Chandra [START_REF] Borel | Introduction aux groupes arithmétiques[END_REF]Corollaire 13.2]. By [START_REF] Borel | Introduction aux groupes arithmétiques[END_REF]Proposition 8.1], this also implies that H 0 Γ 0 is a closed subset of G 0 /Γ 0 , where H 0 is the connected component of the identity of H, in the Zariski topology (a subgroup of finite index). This implies that HΓ 0 is closed.

Reduction step.

Lemma 4.2. To prove Theorem 4, we can (and will) assume that for all k ∈ Z and γ ∈ Γ 0 such that a k Eγ ∩ E = ∅, then γ ∈ P E .

Proof. Consider the subset

F = E - (k,γ)∈Z×Γ 0 s.t. L(a k Eγ∩E)=0 a k Eγ.
Clearly, F is a subset of E of the same measure, and P F = P E . So it is sufficient to prove the statement of Theorem 4 for F instead of E, and F satisfies the above property.

k ∈ Γ 0 such that a n k xγ k → x and a n k xγ k ∈ E. Fix such an x ∈ E. By Lemma 4.2, we known that γ k ∈ P E ⊂ H. It follows that α(γ k )D = D, so since a n k xγ k → x, we have (1) lim k→+∞ α(a) n k α(x)D = α(x)D.
By assumption, a is triangularizable with positive eigenvalues, so α(a) has only positive, real eigenvalues. We claim that (1) implies that α(x)D is contained in one of its eigenspaces.

Let

α(a) = δ + η,
be the Jordan-Chevalley decomposition of α(a), that is: δ and η commutes, η is nilpotent, δ diagonalizable (with positive, real eigenvalues).

If p is the nilpotent index of η, for k such that

n k > p, α(a) n k = p-1 i=0 n k i δ n k -i η i .
Let v ∈ α(x)D -{0}, and v = θ v θ be its decomposition along the eigenspaces of δ corresponding to the eigenvalues {θ} of δ. Then

α(a) n k (v) = θ p-1 i=0 n k i θ n k -i η i (v θ ) .
As a function of n k , this is a combination of polynomials and powers of eigenvalues. If θ 0 is the highest eigenvalue θ for which v θ = 0, and i 0 is the largest i for which η i (v θ 0 ) = 0, then we have the asymptotic as k → +∞,

α(a) n k (v) ∼ n k i 0 θ n k -i 0 η i 0 (v θ 0 ).
However, we know that projectively, α(a) n k α(x)D → α(x)D, so v is colinear to η i 0 (v θ 0 ). As η preserves the eigenspace of δ associated to θ 0 ,

α(a)η i 0 (v θ 0 ) = θ 0 η i 0 (v θ 0 ) + η i 0 +1 (v θ 0 ) = θ 0 η i 0 (v θ 0 ),
because by definition of i 0 , η i 0 +1 (v θ 0 ) = 0. This shows that η i 0 (v θ 0 ) is an eigenvector, and so is v.

We have proved that α(x)D is contained in an eigenspace of α(a). So D is stabilized by α(x -1 ax), meaning that a ∈ xHx -1 , as required. Proof. By ergodicity of a with respect to µ, for µ-almost every xΓ 0 , a Z xΓ 0 is dense in the support of µ. By the previous Lemma, we have also for almost every x ∈ E, a ∈ xHx -1 , so

a Z xΓ 0 ⊂ xHΓ 0 .
Consider a typical x ∈ E satisfying both of these properties. By Lemma 4.1, xHΓ 0 is a closed set. By density of the a-orbit of x in the support of µ, this implies that supp(µ) ⊂ xHΓ 0 , so

µ(xHΓ 0 ) = µ (xHx -1 )xΓ 0 = 1.
By assumption, µ is non-concentrated, so H = G 0 necessarily.

Proof of Theorem 2

The proof is by contradiction. We assume that f β is not asynchronous. By translate of a Q-subspace of V = R d , we mean a set T ⊂ V of the form T = v + W , where v ∈ V , and W ⊂ V a subspace of V defined over Q; in particular, there is no rationality assumption on v, and T itself does not have to be defined over Q. The Q-subspace W is called the direction of T .

If non-empty, the intersection of two translates of Q-subspaces is again a translate of Q-subspace. This property allows us to define, for a set E ⊂ I of positive L-measure, the set T E which is the smallest translate of Q-subspace containing {ρ(x) -1 β} x∈E . We denote by W E its direction. Lemma 5.1. There exist E ⊂ I of positive measure such that W E = V .

Proof. Since f β is not asynchronous, so for some non-trivial character χ ∈ K, χ • f β is constant on a set F of positive L-measure. There exists n ∈ Z d -{0}, such that

χ(v) = e 2iπ n,v .
So the set { n, ρ(x) -1 β } x∈F is contained in a countable set of the form c + Z, for some c ∈ R. This implies that at least one of the sets We now fix E ⊂ I a set of positive measure, such that W E is of minimal possible dimension (it exists). By Lemma 5.1, W E = V . Like in Theorem 4, we define

F m = {x ∈ F : n, ρ(x) -1 β = c + m},
P E = γ ∈ Γ 0 : L ∪ k∈Z a k Eγ ∩ E > 0 . Lemma 5.2. For all γ ∈ P E , ρ(γ)W E = W E .
Proof. Recall that β is an eigenvector for ρ(a), denote by κ the corresponding eigenvalue. By definition of P E , there exists k ∈ Z with L a k Eγ ∩ E > 0. Let F = a k Eγ ∩ E, then for all x ∈ F , there exist y ∈ E such that x = a k yγ. We have:

ρ(x) -1 β = ρ(γ -1 y -1 a -k )β = κ -k ρ(γ) -1 ρ(y) -1 β ∈ κ -k ρ(γ) -1 T E .
Note that since ρ(γ) -1 is a matrix with integer coefficients, κ -k ρ(γ -1 )T E is also the translate of a Q-subspace, containing {ρ(x) -1 β} x∈F . By definition of T F , this means that

T F ⊂ κ -k ρ(γ) -1 T E .
Since F ⊂ E, T F ⊂ T E , and because E was chosen such that T E is of minimal possible dimension, we have T E = T F , so from the rank-nullity Theorem, κ -k ρ(γ) -1 T E = T E . This implies equality for the directions, ρ(γ) -1 W E = W E , and multiplication by ρ(γ) concludes the proof. Lemma 5.3. We have W E = {0}, that is, the map x → ρ(x) -1 β is constant on E.

Proof. The subgroup Stab G 0 (W E ) = {g ∈ G 0 : ρ(g)W E = W E } is an algebraic subgroup containing P E , by Lemma 5.2. By Theorem 4, this group is G 0 . Since ρ is irreducible over Q and W E is defined over Q, W E = {0}, or W E = V . But the latter cannot happen, because of the choice of E.

From now on, we fix some x 0 ∈ E. By the previous Lemma, T E is the point ρ(x 0 ) -1 β. Lemma 5.4. For any γ

∈ P E , ρ(γ) ∈ Stab(Rρ(x 0 ) -1 β). Proof. Let γ ∈ P E . Thus there exist k ∈ Z such that L(a k Eγ ∩ E) > 0.
In the proof of Lemma 5.2, we saw that

κ -k ρ(γ) -1 T E = T E .
But since T E = {ρ(x 0 ) -1 β}, this means that ρ(γ) -1 stabilizes the line through ρ(x 0 ) -1 β.

The end of the proof of Theorem 2 is given by the following contradictory Claim.

Lemma 5.5. The space V is one-dimensional.

Proof. The group {g ∈ G : ρ(g) ∈ Stab(Rρ(x 0 ) -1 β)}, is an algebraic group containing P E . By Theorem 4, it follows that ρ(x 0 ) -1 β is a common eigenvector for all elements of ρ(G 0 ) (and so is β). Were the representation ρ irreducible over R, this would be sufficient to conclude; however we assumed only Q-irreducibility, and have no particular rationality assumption on β.

Since G 0 is semisimple and connected, the eigenvalue associated to ρ(x 0 ) -1 β is 1 for every g ∈ G 0 . Let V 1 (g) denote the eigenspace associated to the eigenvalue 1 for the operator ρ(g). Let Z = ∩ γ∈P E V 1 (γ). This subspace Z is defined over Q, because P E consists of integral points. The set of g ∈ G 0 acting trivially on Z is an algebraic subgroup containing P E , so again is G 0 . Since ρ(x 0 ) -1 β ∈ Z, Z is of positive dimension. By Q-irreducibility of ρ, Z = V is an irreducible representation where G 0 acts trivially, so is one-dimensional.

Ergodicity and mixing

We now prove Theorem 3. By assumption, a is triangularizable with positive eigenvalues. We separate the proof in two cases. Let β ∈ V -{0} be an eigenvector for ρ(a), its eigenvalue is 1. Notice that the actions of (e, β) and (a, α) commute: since ρ(a)β = β, we have (a, α)(e, β) = (a, α + ρ(a)β) = (a, ρ(e)α + β) = (e, β)(a, α).

Consider the ergodic decomposition of λ with respect to the action of (a, α): there exists a measure m on the set of ergodic, (a, α)-invariant measures on G/Γ, such that λ = ν dm(ν).

If we apply the projection map

π : G/Γ → G 0 /Γ 0 to this equality, we obtain L = µ = (π) * ν dm(ν),
where (π) * ν are a-invariant. Since µ is a-ergodic, we have that (π) * ν = µ, m-almost surely. Therefore, m is supported on M µ . But since (e, β) commutes with (a, α), (e, β) * m is the measure associated to the ergodic decomposition of (e, β) * λ = λ. This implies that m is (e, β)-invariant, and by Corollary 1.1, m is the Dirac measure at λ. This concludes the proof of the ergodicity.

Case 2: ρ(a) is not unipotent. Thus ρ(a) has some of its eigenvalues different from 1. Since G 0 is semisimple, det(ρ(a)) = 1 so there exist at least one eigenvalue κ < 1. Let β ∈ V be an eigenvector of ρ(a) associated to κ.

Notice that , for k ≥ 0,

(a, α) k (e, β) = (a k , ρ(a) k β + k-1 i=0 ρ(a) i α) = (e, κ k β)(a, α) k ,
and (e, κ k β) → (e, 0) when k → +∞. This implies that the distance (with respect to a distance d G/Γ on G/Γ induced by a right-G-invariant riemannian distance on G) between (a, α) k (x, v)Γ and (a, α) k (e, β)(x, v)Γ tends to zero as k tends to +∞. In other words, the strong stable distribution for (a, α), defined by We first prove the claims about ergodicity and weak-mixing.

Let f ∈ L 2 (G/Γ, λ) -{0} be an eigenvector for the Koopman operator of (a, α), that is f ((a, α)(x, v)Γ) = ωf ((x, v)Γ), for λ-almost every (x, v)Γ, for some ω ∈ C of modulus one. Recall that ergodicity states that any such eigenvector associated to ω = 1 is constant almost everywhere, and weak-mixing that any such eigenvector is constant almost everywhere and moreover ω = 1.

To prove ergodicity or weak-mixing, we may (and will) restrict to the case where f is bounded. By the Hopf argument, and more precisely the version proved by Coudène [START_REF] Coudène | The Hopf argument[END_REF], f is W ss -invariant, meaning that there exists a full measure set Ω ⊂ G/Γ such that for all (x, v)Γ ∈ Ω, (y, w)Γ) ∈ Ω such that (y, w)Γ ∈ W ss ((x, v)Γ), f ((x, v)Γ) = f ((y, w)Γ).

By Theorem 2, the angle map x → ρ(x) -1 β is asynchronous. In particular, for every strict linear rational (closed) subtorus T ⊂ K, the set of x such that ρ(x) -1 β does not belong to T is of full measure. As the set of such subtorus is countable, this implies that for µ-almost every x, the translation on K given by ρ(x) -1 β is ergodic. Via a linear change of variable, this means that for µ-almost every xΓ 0 ∈ G 0 /Γ 0 , the translation by (e, β) on the fiber above xΓ 0 is ergodic.

As f is (e, β)-invariant, for µ-almost every xΓ 0 , f is almost everywhere constant on the fiber above xΓ 0 , and merely depends on xΓ 0 . Write F ∈ L 2 (G 0 /Γ 0 , µ) for its almost everywhere value, that is f = F • π, λ-almost everywhere, where π : G/Γ → G 0 /Γ 0 is the fiber bundle. We have then F (axΓ 0 ) = ωF (xΓ 0 ).

If ω = 1, then F is a-invariant, and by ergodicity of a, F is constant µ-a.e., and so f is constant λ-a.e. . This proves the ergodicity.

Assume now that the action of a is weakly mixing on (G 0 /Γ 0 , µ). Let f ∈ L 2 (G/Γ, λ) be like previously an eigenvector for the Koopman operator of (a, α), F ∈ L 2 (G 0 /Γ 0 , µ) its almost sure value depending on the fiber. By what we saw before, F is an eigenvector for the Koopman operator of a, so ω = 1 by weak-mixing of a. By ergodicity, F is constant, as was to be proved.

We now assume that the action of a on (G 0 /Γ 0 , µ) is strongly mixing, and wish to prove that (a, α) is also strongly mixing. Recall that strong mixing of a is equivalent to the fact that for all F ∈ L 2 (G 0 /Γ 0 , µ), F •a k converges weakly to a constant as k → +∞.

Let f ∈ L 2 (G/Γ, λ). Let g ∈ L 2 (G/Γ, λ) be any weak limit of f • (a, α) k as k → +∞ along a subsequence. By another result of Coudène [START_REF] Coudene | On invariant distributions and mixing[END_REF], generalizing a result of Babillot, g is W ss -invariant. By ergodicity of (e, β) on almost every fiber, g = G•π almost surely, where G ∈ L 2 (G 0 /Γ 0 , µ). Define

F (xΓ 0 ) = R d /ρ(x)Z d f ((x, v)Γ)dHaar R d /ρ(x)Z d (v),
the mean value of f on each fiber. Let H ∈ L 2 (G 0 /Γ 0 , µ) be a testfunction. Then

3. 5 .Proposition 3 . 3 .

 533 Sets of natural density one. Assume f : I → K is asynchronous. Then there exists a set E ⊂ Z of full natural density such that for all µ ∈ M L , lim n→±∞, n∈E

4. 1 .

 1 Closure of HΓ 0 . Lemma 4.1. The set HΓ 0 is closed Proof.

4. 3 .

 3 Invariance of xHΓ 0 . Lemma 4.3. For L-almost every x ∈ E, a ∈ xHx -1 . Proof. By a Theorem of Chevalley [4, Thm 5.1], there exists a finite dimensional representation α of G such that H is the stabilizer of a line D, that is H = {g ∈ G 0 : α(g)D = D}. By Poincaré recurrence Theorem, for L-almost every x ∈ E, there exists a sequence n k → +∞ and γ

4. 4 .Lemma 4 . 4 .

 444 Conclusion of the proof of Theorem 4. We have H = G 0 .

  for m ∈ Z, has positive L-measure. By construction, for such a m ∈ Z,T Fm ⊂ {v ∈ V : n, v = c + m},the right-hand side set being the translate of a proper Q-subspace, E = F m satisfies the Lemma.

Case 1 :

 1 ρ(a) is unipotent.

W

  ss ((x, v)Γ) = {(y, w)Γ : lim k→+∞ d G/Γ ((a, α) k (x, v)Γ, (a, α) k (y, w)Γ) = 0},is invariant under the action of (e, β).

Université Rennes I, IRMAR UMR 6625, Campus de Beaulieu 35042 Rennes cedex -France E-mail address: francois.maucourant@univ-rennes1.fr

Taking the limits in the left-hand side and right-hand side respectively along the subsequence, using the strong mixing property for a, gives:

which implies that G is µ-almost everywhere the constant G 0 /Γ 0 F dµ = G/Γ f dλ. Therefore the only possible weak limit of f • (a, α) k is the above constant. By weak compactness of the ball of radius f 2 in L 2 (G/Γ, λ), this proves that this sequence must converge weakly to G/Γ f dλ, as required.