Mattia Guerini
email: mattia.guerini@unibs.it

Lionel Nesta
email: lionel.nesta@univ-cotedazur.fr

Xavier Ragot
email: xavier.ragot@sciencespo.fr

Stefano Schiavo
email: stefano.schiavo@unitn.it

Clemens Fuest

The Zombification of the Economy? Assessing the Effectiveness of French Government Support during Covid-19 Lockdown

Keywords: H120, H320, J380, G330, L200 Covid-19, zombie firms, job-retention, schemes, microsimulation, policy evaluation

This paper evaluates the risk of zombification of the French economy during the sanitary crisis, as a result of the unconditional financial support provided to firms by public authorities. We develop a simple theoretical framework based on a partial-equilibrium model to simulate the liquidity and solvency stress faced by a large panel of French firms and assess the impact of government support measures. Simulation results suggest that those policies helped healthy but illiquid firms to withstand the shock caused by the pandemic. Moreover, the analysis finds no evidence of a "zombification effect", as government support has not disproportionately benefited less productive companies.

Introduction

At the onset of the COVID-19 pandemic, many governments across the world imposed strict lockdown rules in order to limit the virus diffusion. As a result, demand plummeted and economic activity experienced an unprecedented slowdown. To limit the negative effect on income and employment, and reduce the risks of viable but illiquid companies being forced out of the market, authorities introduced a wide range of policies to support workers and firms (OECD 2020(OECD , 2021b)).

This paper aims to evaluate the risk of "zombification" of the economy. With this term we mean the possibility for inefficient firms to remain active thanks to the unconditional and extensive COVID-related government financial support, which might limit the efficiency of market selection. To assess such risk, the paper combines micro-simulation and regression analyses to investigate the impact of the job-retention scheme enacted by the French government on the liquidity and solvency of companies.

French authorities have been very responsive, setting up two important measures within days of the first lockdown. First, the French Government extended the job retention scheme called Partial Activity Scheme ("Dispositif d'Activité Partielle", AP), initially set up in 2008. Originally, public authorities would indemnify a fixed amount of almost 8 euros per hour not worked, while firms compensated temporally laid-off (TLO) workers up to 70% of the gross wages. With the COVID crisis, the French government has significantly changed the scale of the AP scheme, since public authorities would then compensate TLO workers 70% of the original gross wage, while the remaining wage loss was covered at the discretion of the employer. This allowed companies to lay-off substantial shares of the labour force at virtually no cost. The UNEDIC, the organization responsible for the implementation of AP, estimates the overall cost for public authorities to be e35bn in 2020 and 2021 (UNEDIC 2022). A second important scheme was the State-Guaranteed-Loans ("Prêt Garanti par l'Etat", PGE). In this case, the government pledged e300bn in order to cover the loss in case a company defaulted on a bank loan. In other words, the risk of default is borne by the state rather than the banks. This scheme aims at facilitating ac-cess to additional financial resources for illiquid firms, although the final decision remains within the hands of financial institutions. 1 Benitto et al. (2022) report that as of January 2022, e145bn had been lent out to more than 700,000 firms, and the anticipated loss for the Government is e1.4bn.

Official figures show that firm liquidations dropped by 50% in 2020, and remain stubbornly low during the first half of 2022.2 This led several authors to warn against the possible side effects of public support schemes that, because in most of the cases are unconditional, not discriminating between efficient and inefficient firms, may hamper the cleansing effect of market selection, preventing unproductive firms from being forced out of the market [START_REF] Banerjee | Corporate zombies: Anatomy and life cycle[END_REF][START_REF] Laeven | Zombification in Europe in times of pandemic[END_REF][START_REF] Helmersson | Corporate zombification: post-pandemic risks in the euro area[END_REF][START_REF] Araújo | Policy options for supporting and restructuring firms hit by the covid-19 crisis[END_REF]) and leading to a zombification of the economy. OECD (2021a,b) find that the number of "zombie" firms -defined as companies with an interest coverage ratio lower than one for three consecutive years -has spiked in 2020. Bankruptcies in 2020 were approximately 30% lower than the pre-pandemic average, suggesting that the safety net provided by governments might have prevented large scale failures.

Concerns about the ability of market selection to direct resources toward the most productive companies and the effects of misallocation of credit on productivity and growth is certainly not new [START_REF] Caballero | Zombie lending and depressed restructuring in Japan[END_REF], and has grown louder in the last decade, when very loose monetary policy made it easier for (quasi-)zombie firms to access credit and roll-over debt [START_REF] Mcgowan | The walking dead? Zombie firms and productivity performance in OECD countries[END_REF][START_REF] Acharya | Zombie credit and (dis-)inflation: Evidence from Europe[END_REF][START_REF] Sedláček | Lost generations of firms and aggregate labor market dynamics[END_REF], Schivardi et al. 2020). In fact, productivity slowdown and weak business dynamism observed in several OECD countries are often considered two symptoms of economic malaise associated with a reduction in market selection efficiency [START_REF] Storz | Do we want these two to tango? on zombie firms and stressed banks in europe[END_REF].

Zombification could constrain the post-pandemic recovery by limiting productivity growth and preventing an efficient allocation of resources [START_REF] Sedláček | Lost generations of firms and aggregate labor market dynamics[END_REF].

There is little reason to believe that the sudden fall in firm exit we have seen since the pandemic is permanent. And its slowness to come back to normal levels limits our ability to evaluate the extent of the possible zombification of the economy based on empirical data. The widespread firms reliance on public support schemes conceals the true state of firm financial health. Moreover, because liquidations, bankruptcies and exit procedures are lengthy, and lockdown measures have de facto frozen the work of administrative tribunals, the actual effect of the pandemic on firm demography may take years to materialize. Our contention, then, is that if one wishes to evaluate the risk of zombification associated with Covid-related support measures, one needs to rely on micro-simulations.

The rest of the paper is organized as follows. Next Section describes the simulation model, presents the data and discusses the main results. Section 3 builds on the simulations to study the impact of the job-retention scheme across the productivity distribution and evaluate the risk of zombification. Section 4 concludes.

Simulations

We develop a partial equilibrium model with a finite number of companies, denoted by i = 1, . . . , N , each belonging to a single sector j = 1, . . . , J. The model is set in monthly discrete time t = 1, . . . , T and is calibrated with French firm-level data. The simulation covers the January 2020 -April 2021 period, covering one full year after the first French lockdown in late March 2020.

Simulation model with COVID-19

COVID-19 shock. The COVID-19 pandemics and the related series of lockdown, imposed to limit the spread of viral contagion, generated severe and unprecedented sector-specific demand shocks to many businesses [START_REF] Baqaee | Supply and demand in disaggregated keynesian economies with an application to the covid-19 crisis[END_REF]. In fact, the interruption of economic activity during lockdown was conditional upon the importance that each industry had in the value chain for the production of necessary final products and services (see [START_REF] Ferraresi | On the economic and health impact of the covid-19 shock on italian regions: A value chain approach[END_REF]. We model sector-specific shocks as demand shifters:

Q d i,j,t = Q d i,j,t 0 (1 + ξ j,t ) (1)
where Q d i,j,t is the demand faced at period t by firm i, belonging to sector j. The term

Q d i,j,t 0
is the demand of the same firm, in the period right before the emergence of COVID-19 (i.e. January 2020). The term ξ j,t -either positive or negative depending on the nature of the shock -represents the sector demand shifter with respect to the pre-COVID-19 demand level. The larger the severity of the lockdown in sector j, the lower the value of ξ j,t .

Firms' behaviour. Firms are modeled as price takers, given the short time horizon of the simulation. Although fixed in our model, observed prices may depart from competitive prices in all markets since they may not reflect the equalization of marginal cost and revenue. 3 Given prices, at each period, firms face a level of demand Q d it , which is assumed to be known. Thus, firms solve the following one-period cost minimization problem:

4                            arg min L it ,M it C it = p L L it + p M M it + p K K it subject to Q it ≤ F(K it , L it , M it ) = ω it K β K it L β L it M β M it Q it ≥ Q d it (2)
where the choice variables are the demand for labour (in hours worked, L it ) and intermediate materials (M it ). Given the time horizon of the model, we assume capital stocks to be invariant over the simulation period, such that firms invest in maintenance without investment in additional productive capacity: K it = K i . The objective function is a linear cost function (C it ) accounting for the presence of two variable inputs and one fixed input.

Prices P K , P L and P M refer to the user costs of capital, hourly wage and price of materials, respectively. The first constraint is a Cobb-Douglas production function

F(K it , L it , M it )
with three inputs, where ω i measures the firm-specific total factor productivity (TFP, see Appendix B for its estimation). The second constraint is the demand level, determined by Equation 1.

Variable factor demand. Because of our interest in the short-term firm-level effects generated by COVID-19, we make the reasonable assumption that all prices are fixed.

This allows us to derive the firms' optimal demand of the variables factors:

L ⋆ it = K -β K it Q it ω it P M P L β L β M 1 β L +β M (3) M ⋆ it = K -β K it Q it ω it P L P M β M β L 1 β L +β M
However, we take into account that in most real world situation, companies' adjustments of the flexible inputs are not instantaneous. In fact, companies orders for intermediate inputs are lumpy and workers cannot be immediately dismissed due to the rigidity of many contracts. Furthermore, during the pandemic COVID-19 period it is likely that market imperfections and information asymmetries have been amplified. Therefore, we adopt a partial adjustment model for both variable inputs that take the following form.5

This implies that at every period t firms won't achieve their optimal choices L ⋆ it and M ⋆ it .

Rather they slowly adjust toward these values as follows:

Lit = L it-1 + γ L (L ⋆ it -L it-1 ) (4) Mit = M it-1 + γ M (M ⋆ it -M it-1 )
where the parameters γ L , γ M ∈ [0, 1] describe the speed of adjustment for the flexible factors and the hat (e.g. Lit ) indicates that the variable has been only imperfectly adjusted.

At one extreme, a value γ j = 1, j ∈ {L, M } implies firms can immediately adjust their inputs at their optimal levels. At the other extreme, if γ j = 0, companies cannot adjust and the two inputs are kept fixed (see Appendix A).

Liquidity dynamics. Liquidity holdings (Λ) of firms evolve according to the following law of motion:

Λ it = Λ it-1 + ℓ it (5)
where cash flow ℓ it is defined as:

ℓ it = P Q it -P L L it -P M M it -F C i -T i (6)
Equation ( 6) says that cash flow is the difference between revenues from sales and production costs, which include the wage bill (P L L), the cost of intermediate products (P M M ), the fixed costs F C -defined as rents, interest and principal repayment -and the corporate tax bill T . Both the fixed costs F C and the tax bill T are kept constant over the simulation horizon. Writing Gross Operating Income as

Π it = P Q it -P L L it -P M M it ,
Eq. ( 6) can be rewritten as ℓ it = Π it -F C i -T i , implying that liquidity equals gross operating income minus fixed costs and corporate taxes.6 Finally, it is worth noticing that we abstract from modeling dividends distribution.

Financial stress. A company is said to face liquidity stress whenever Λ it < 0. This implies that the sum of liquidity holdings at the end of the previous accounting period Λ it-1 and the current gross operating profits Π it are smaller than the sum of the firm's fixed costs plus the tax bill:

Λ it < 0 (7) Λ it-1 + Π it < F C i + T i .
The same definition has been recently adopted by [START_REF] Demmou | Corporate sector vulnerabilities during the Covid-19 outbreak: assessment and policy responses[END_REF], [START_REF] Gourinchas | Covid-19 and small-and medium-sized enterprises: A 2021 "time bomb"?[END_REF].

Illiquid firms may continue operating by raising new debt, either via bonds or new loans. However, this cannot continue indefinitely. A company is said to be insolvent when liquidity holding Λ it becomes negative, and in absolute value larger than its equity E it .

This insolvency condition reads:

Λ it + E it < 0 (8) Λ it-1 + Π it + E it < F C i + T i
where the second form of Equation 8suggests that a firm shuts down whenever all liquid and illiquid resources -i.e. the left hand side -are insufficient to cover financial and tax charges -i.e. the right hand side. In this situation, also short-term borrowing would be unfeasible because neither investors in corporate bonds nor a bank would rationally provide funds to such a firm.7

At the end of each accounting period t, both financial conditions are updated, thereby determining the share of illiquid and/or insolvent firms. Note that both liquidity and solvency are the results of a host of factors including productivity, output elasticities, operating profit, and the pre-COVID financial health of the firms in terms of debts and fixed costs.

Job Retention Scheme. We model the AP scheme put forward by the ordinance 2020-346 of March 27, 2020. 8 This policy device allowed companies to obtain a subsidy for the wage costs of the employees not directly involved in production during the COVID-19 series of lockdown. With respect to our model, this implies that such policy instrument allows firms to always reach their optimal level of hours worked L ⋆ it without sluggishness.

Formally, we implement this policy setting γ L = 1.

Time-line of events. At each time period, the following sequence of events takes place:

1. Firms observe the new level of demand Q d i,j,t as determined by Equation 1;

2. firms determine the optimal flexible inputs (L ⋆ it , M ⋆ it ) (see Equation 3);

3. firms take into account the partial adjustment and use the inputs ( Lit , Mit ) (see Equation 4); 4. firms produce the amount Q it using the Cobb-Douglas technology of Equation 2;

5. firms cash flows are updated according to Equation 5.

Simulation model without COVID-19

To properly evaluate the effectiveness of the policy measure, it is necessary to build a counterfactual (i.e. No-COVID) scenario. Two main differences with respect to the model described in Section 2.1 must be emphasised. The first difference is empirical in nature and concerns the estimation of the demand shocks, had COVID-19 not struck. The second difference instead is theoretical, and concerns firm behaviour in a "business as usual" economic environment.

Demand shocks in the No-COVID scenario. In order to build the No-COVID scenario, we rely on standard time-series techniques. In particular, for each sector j, we employ the monthly time-series of value added in production covering the period from January 2012

to December 2019 (T = 96) with the aim of estimating sector-specific AR(1) processes with monthly dummies: 9

g j,t = ρ s g j,t-1 + δ j,m + ε j,t . ( 9
)
where g j,t measures the growth rate of industrial production in sector j at time t; ρ j represents the first-order autoregressive parameter for a specific sector; δ j,m captures the sector-specific monthly components of demand. Equation 9 can be consistently estimated by OLS.

We then employ the estimates of ρj and the twelve δj,m dummies to carry out iterated one-step ahead predictions of the the growth rate for all subsequent months up to April 2021: ĝj,t+1 = ρj g j,t + δj,m .

(10)

Finally, to infer the demand shocks in the No-COVID scenario, we index the industrial 9 See Hamilton (2020). Note that before estimating the process we transformed all the industrial production series in growth rates to ensure stationarity. All p-values of the sector-specific Augmented Dickey-Fuller (ADF) tests, computed on the residuals of Equation 9 are smaller than 10% (99% of them are below the 5% threshold) indicating that the autocorrelation present in the time series of industrial production growth has been taken into account either by the AR(1) component or by the monthly dummies. production at January 2020 -i.e. null demand shifter ξ j,0 = 0 in January 2020 -and we iteratively forecast the index value of industrial production for all subsequent months using:

(1 + ξ j,t ) = (1 + ξ j,0 ) t k=0 (1 + ĝj,k ) (1 + ξ j,t ) = t k=0 (1 + ĝj,k ) (11)
Firm behaviour in the No-COVID scenario. The second difference concerns the choice of (L it , M it ), as described by the minimization problem described by Equation 2. Absent any COVID shock, we can relax the assumption about the fact that firms gradually adjust by following the observed demand shocks. This assumption is reasonable when uncertainty is high and firms cannot resort to optimal behaviour. Therefore, in the No-COVID scenario, we assume that firms optimally choose L and M according to Equation 3. This is also consistent with [START_REF] Gourinchas | Covid-19 and small-and medium-sized enterprises: A 2021 "time bomb"?[END_REF]. 10

Overall, we end up with three different scenarios: (i) a No-COVID scenario, in which the demand shocks are generated coherently with the variability in the monthly growth rates of industrial production experienced over the 2012-2019 period; (ii) a COVID scenario, which takes into account the actual fall in value added experienced in 2020, as the demand shocks are drawn from the 2020 empirical observation; (iii) a COVID+Gov scenario where the demand shocks are equivalent to the COVID scenario, but in which we also model the intervention of the government by means of the AP scheme. These scenario form the bulk of our simulation exercise.

Firm-level data

The simulation exercise uses balance sheet data for 2019, as reported in the FARE database maintained by the French Statistical Office (INSEE), which represents a snapshot of the situation of French companies before the beginning of the pandemic and is based on tax 10 Gourinchas et al. (2021) use this optimal behaviour also in the presence of the pandemic shock. This choice in turn requires them to assume also supply-side (productivity) shocks in order to generate higher firm exit in the COVID-19 scenario. filings by companies. FARE 2019 includes more than 4 million companies (4,356,764). We exclude from the analysis companies with incomplete information; firms in Agriculture (AZ), Finance and Insurance (KZ) and Public Administration, Education, Human Health and Social Action (OQ) sectors; legal persons and organizations subject to administrative law, as well as self-employees and craftsmen. The reason for leaving these companies aside is that, their decisions on the amount of factors of production employed do not necessarily comply with the logic of our simulation model, which is based on cost minimization and on a production function framework.

The set of firms that are part of our exercise includes 752,603 companies (or 17.2% of FARE's legal units); more than 10.5 million jobs (76.2% of FARE jobs), and e914bn of added value (i.e. 74.1% of FARE). The simulation takes as a reference legal units. Thus we do not model possible resource flows among companies belonging to the same business group (or between parent and subsidiaries), which could affect their financial stress.

Output elasticities (β K , β L , β M ) are estimated using a Cobb-Douglas production function based on balance sheet data (taken from FARE) for the years 2012-2019 (see Appendix B).

The size of the demand shock (ξ j,t ) at the 4-digit industry level (referenced as level A732 in the French nomenclature) is taken from INSEE. 11 In particular, we use industry-specific value-added indexes recorded over the COVID period to capture the size of fluctuations to demand during the pandemic. Exploiting the same information for 2012-2019 period we also build the counterfactual No-COVID scenario presented in Subsection 2.2.

Simulation results

Figure 1 presents the broad trends emerging from the simulations for the three scenarios.

While the upward trend is at odds with the fall in the number of liquidations, we should bear in mind that we are not simulating a failure rate, but rather the unobserved financial health of firms, and that data from firms financial statements are affected by public support measures.

[Figure 1 about here.] Unsurprisingly, the pandemic has a sudden, brutal and sizable impact on the liquidity of French companies. The drastic drop in revenues determined by the lockdown, the presence of frictions in the markets for factors of production and of fixed costs that do not adjust to the level of production (or adjust very slowly) drain the liquidity of non-financial firms. The fraction of companies experiencing solvency (liquidity) issues climbs up to 3.4% (16%) by the end of the simulation. This contrasts with an insolvency (illiquidity) rate of around 1.9% (5%) in April 2021 under the baseline No-COVID scenario.

Figure 1 provides two additional insights. The first one concerns the impact of the partial activity scheme on solvency, which is large and positive. The measure reduces the number of insolvent companies, trimming it by 0.8 percentage points (from 3.4 to 2.6%) in April 2021. 12 The second one is that a number of firms face solvency issues irrespective of the pandemic (1.9%), implying that they are unprofitable even when the economy is growing. These companies are generally smaller, less productive, more indebted and have a lower level of liquidity than the others. This evolution is qualitatively similar to the results presented in OECD (2021a), despite the fact that the analysis is based on a sample of firms with different characteristics.

3 Public support and firm solvency: a policy evaluation exercise

Firm sorting

The first set of questions concerns the financial stress of firms across the different scenarios.

Did public financial support relieve firms from financial stress during the sanitary crisis?

We proceed in two steps. First, we look at the solvency stress of firms under the two scenarios. Table 1 discriminates among firms suffering from solvency stress in No-COVID and in the COVID scenarios without state support. Consistently with Figure 1, it shows that the number of insolvent firms raises from 1.9% to 3.4%. It also shows that the vast majority (726,438 firms, representing 96.5% of our sample) of firms have sufficient resources to escape from insolvency stress in either scenario. Among the 25,697 financially-stressed companies of the COVID scenario, more than half (13,772) show similar problems also in normal times, whereas our simulations reveal that almost 12,000 firms would become insolvent because of the crisis, absent any government support. A small number of firms (468) enjoy an improvement of their financial condition following the pandemic: this stems from the positive demand shock induced by the lockdown on few sectors (including utilities, and a range of telecommunication and transport services).

[Table 1 about here.]

Relative to firms being financially stressed in both scenarios (bottom right quadrant), firms not exposed to the global lockdown (upper left quadrant) are generally larger (P Q 0 ), more productive (ω 0 ), less indebted (Lev 0 ), and face relatively low fixed costs (6.5% of their revenues). Firms suffering from the lockdown (upper right quadrant) appear globally healthy financially, both in terms of fixed costs and leverage. At the same time, they show a relatively low level of productivity (ω 0 = 2.85). 13Our simulations allow retrieving the estimated cumulative AP resources perceived by firms, which altogether amounts to e21.5bn. Most resources (e19,75bn, 97.5% of AP spending) are perceived by firms that would not be threatened by either scenarios, even with the absence of the job retention scheme. In the same vein, firms undergoing solvency stress in both scenarios represent 1.1% of overall AP resources (e220ml). Firms entering into solvency stress due to the COVID crisis gather 1.3% of overall AP resources (e270ml).

Overall, these preliminary figures cast doubts on the necessity of the job retention scheme.

In the end, the bulk of AP spending applied to these 760,000 companies which did not need it in the first place. Our stand on this issue is far more moderate. First and foremost, AP is also a demand policy. Had AP not been implemented, the effect of the sanitary crisis on final demand would presumably have been much fiercer than observed. Second, although most firms can financially cope with the global lockdown, AP maintained their financial health such that they do not cope with solvency stress as defined here. It may still be that their financial health deteriorated.

[Table 2 about here.] The second step raises the question of the effect of government support on solvency stress. If anything, a well-functioning government support should favor financially-healthy firms in normal time (i.e. under the No-COVID scenario) while being unsupportive with insolvent one. Table 2 displays the effect of government aid on solvency for the 25,697 firms made potentially insolvent by the crisis, corresponding to the right column of Table 1. This allows us to address two issues.

One issue relates to a type-II error in government support: did the government support firms that would have experienced solvency stress in any case? Table 2 shows that 307 among the 13, 772 solvency-stressed firms (only 2%) have actually benefited from government support, becoming solvent during the crisis. Hence, government support has, by and large, not supported unhealthy firms. The other question relates to a type-I error: did the government fail to support firms that would not have experienced solvency stress under a No-COVID scenario? This applies to 11,925 firms. We observe that for almost half of them (5,371 companies, amounting 45% of financially health firms under the No-COVID scenario), government support has been key to support them throughout the sanitary crisis.

Table 2 also displays the expected frequency of each type of firms if the effect of government support on firms solvency were random. The gap between the observed (N ) and the expected ( N ) number of firms is an indication of the capacity of the support to sort correctly amongst firms. This suggests that government support, despite not being targeted, did not support financially unhealthy firms and did make a difference for firms that deserved to be supported.14

Zombification

We now move to the core of our empirical exercise, namely the evaluation of the policy measures enacted by the French government to mitigate the impact of the pandemic on firms.

We focus on the 11,925 companies which display no solvency issue under the notional No-COVID scenario, while facing a solvency problem in the baseline partial-adjustment model with no furlough scheme. Among these companies, we further discriminate among those that are made solvent by Activité Partielle (5,371) and the others (6,556). Note that by doing so, we avoid the obvious correlation between the amount of public help received by a firm and its financial fragility that stems from the fact that less productive firms have higher costs and thus are more likely to face liquidity and solvency issues when hit by a negative shock, as reported in Table 1. 15 When we properly account for this selection bias, we immediately see that firms with no solvency issue enjoyed a larger amount of government support as a fraction of their output, beside being on average larger, more productive, and having a stronger financial structure (with all means being significantly different across groups).

To address more directly the question of the effect of AP on French firms, we run a series of probit regression models in which the dependent variable is an indicator that takes value 1 if the company has faced solvency issues at any time in our simulation and the main explanatory variable is the amount of government support via AP scaled by sales.

Columns (1-3) of Table 3 show that the job retention scheme reduces the likelihood of facing solvency issues, and this remains valid when we add additional controls such as productivity, fixed costs and leverage, all of which have the expected sign. Size and productivity reduce the probability of becoming insolvent, while higher leverage and larger fixed costs increase it.

The issue of zombification can be addressed by means of an interaction term between the amount of public support received by the company and productivity. A positive coefficient would signal that AP has benefited mostly low-productivity firms, thus rising the risk of a zombification of the economy. Column (4) of Table 3 provides little support for this hypothesis, because the interaction term displays a positive but not significant coefficient.

[Table 3

about here.]

The overall results could, however, conceal important heterogeneity across different sectors of the economy. Indeed, a quick look at the sample of firms included in the analysis reveals that 45% belongs to wholesale and retail trade, and another 41% to hotels and restaurants, two of the sectors that have been more severely affected by the pandemic. To investigate the presence of industry-specific effects, we run the regression model on specific subgroups of firms and obtain interesting insights -see Columns (5-7) of Table 3.

In wholesale and retail trade productivity has no significant effect on the probability to face solvency stress and even if the interaction term is positive and significant, the vast majority of firms in the sample (up to the 95 th percentile of the productivity distribution) lies in the region where AP has no meaningful effect on the dependent variable (see the bottom-left panel of Figure 2). The situation is different in the accommodation and food service sector, where AP reduces the probability to undergo solvency stress and 90% of the firms in the sample seem to benefit from it. Moreover, the interaction term is not significant, suggesting there is little evidence to support a process of zombification among hotels and restaurants. A similar, albeit stronger, effect is found for companies operating in all other sectors, for which the impact of the job retention scheme is positive while the differential effect across the productivity distribution is small and not significant (see the bottom-right panel of Figure 2).

[Figure 2 about here.] This paper has addressed the issue of the possible negative effects of job retention schemes and other support measures enacted by governments in many advanced countries at the onset of the COVID-19 pandemic, when strict lockdown rules and the sharp contraction of aggregate demand threatened an avalanche of business failures. Because most of the policy measures were meant to offer companies a lifeline and did not discriminate among them, the risk of supporting non-viable firms is high. Beside being a waste of public money, such an outcome could undermine the recovery phase since zombie firms would absorb productive resources (such as capital, labour and credit), reduce entry by creating congestion, and act as a drag on productive investment and market reallocation.

We have developed a microfunded simulation framework that replicates the dynamics of liquidity for a sample of 750,000 French firms across different scenarios and sheds light on the effect of support measures implemented by the government. We find that the policies have been successful in significantly reducing the number of firms facing financial distress throughout the first part of the pandemic. Government support has mainly benefited financially healthy firms, whereas companies already under stress did not manage to overcome their problems thanks to additional public funds. Furthermore, we find no evidence of a zombification effect, since the impact of government support on relieving firms from insolvency is constant over the productivity distribution. solvency liquidity 2 0 / 0 2 2 0 / 0 3 2 0 / 0 4 2 0 / 0 5 2 0 / 0 6 2 0 / 0 7 2 0 / 0 8 2 0 / 0 9 2 0 / 1 0 2 0 / 1 1 2 0 / 1 2 2 1 / 0 1 2 1 / 0 2 2 1 / 0 3 2 1 / 0 4 2 0 / 0 2 2 0 / 0 3 2 0 / 0 4 2 0 / 0 5 2 0 / 0 6 2 0 / 0 7 2 0 / 0 8 2 0 / 0 9 2 0 / 1 0 2 0 / 1 1 2 0 / 1 2 2 1 / 0 1 2 1 / 0 2 2 1 / 0 3 2 1 / 0 4 scenarios. Focusing on the red bars, companies experiencing liquidity problems as of April 2021 under the COVID scenario vary between a minimum of 2% (Construction) to a max-

f s = β K k + β L l + β M m (B3)
Observe that function f s is allowed to change across two-digit sectors, as implied by the subscript s. Thus, the parameter vector is composed of three parameters for each sector. The sector decomposition is the two digit level (referenced as level A88 in the French nomenclature).

Different estimators may be used to estimate the production function in equation (B2).

The preferred estimator in this paper is the Wooldridge-Levinsohn-Petrin (WLP) estimator, as derived from [START_REF] Wooldridge | On estimating firm-level production functions using proxy variables to control for unobservables[END_REF] and implemented in [START_REF] Petrin | Measuring aggregate productivity growth using plantlevel data[END_REF].

The main reason is that it corrects for the simultaneous determination of inputs and unobserved productivity by proxying the latter with firm-level material inputs. Moreover, it does not assume constant returns to scale, it is robust to the [START_REF] Ackerberg | Identification properties of recent production function estimators[END_REF] criticism of the [START_REF] Levinsohn | Estimating Production Functions Using Inputs to Control for Unobservables[END_REF] estimator and it is programmed as a simple instrumental variable estimator.

We assume that both labour and materials are a variable input with no rigidity in a business as usual (No-COVID) scenario. We instrument current labour and materials with the first and second lags of labour as well as the second lags of capital and materials. In addition, the WLP estimator requires the variables affecting the productivity process to be specified. We assume that productivity is a function of lagged capital and materials.

Year fixed effects are also included to take into account time-variant shocks common to all firms. All these additional regressors are not included in the function f s . Given fs , estimated total factor productivity ωit eventually reads:

ωit = qit -βK k it + βL l it + βM m it (B4)
Alternatively, one could use the factor shares in revenues, α K , α L and α M , as proxies for their output elasticities (see e.g. [START_REF] Gourinchas | Covid-19 and small-and medium-sized enterprises: A 2021 "time bomb"?[END_REF], for such an assumption). The cost would be that of assuming perfect product and factor markets. We choose instead to assume perfect markets away and let the series of output elasticities depart from their respective revenue shares. Table B1 also reports mean values of revenue shares. We observe a significant wedge between the output elasticities and the revenue shares. This suggest the presence of market imperfections in the product and factor markets. In turn, these relate to various factors affecting perfect composition such as the presence of entry barriers on the various markets, industry structures, the skill composition of labour, etc. [START_REF] Caselli | Imports and labour market imperfections: Firm-level evidence from france[END_REF] provide evidence of imperfect product and labour market imperfections in France based, where they show that product and labour market imperfections can be inferred from the combination of the series of β and α.

= 1-α L -α M .
Table C1: The determinants of the probability to experience liquidity and solvency stress during the COVID global shutdown.

(

  N = 738, 363 P Q0 = 374.03 P Q0 = 243.48 AP € = 19.75 AP € = 0.27 AP/P Q0 = 0.11 AP/P Q0 = 0.13 ω0 = : Business As Usual. COVID: Partial adjustment model under COVID-19 without government support. Yes/No answers refer to whether the firm is experiencing solvency stress. No: Λ it + E it < 0; Yes: Λ it + E it < 0. N : Number of firms; P Q 0 : monthly level of sales in January 2020; AP € : Simulated cumulated amount of AP resources received by firms between March 2020 and April 2021, in billions of euros; AP/P Q 0 : Simulated cumulated amount of AP resources received by firms between March 2020 and April 2021, relative to monthly level of sales in January 2020; ω 0 : estimated level of Total factor Productivity in January 2020; F C 0 : Fixed cost defined as monthly corporate taxes plus payment of principal and interest in January 2020; Lev 0 Leverage, initial level of debts to suppliers and other third party, relative to equity.

  P Q0 = 332.47 P Q0 = 170.50 AP € = 0.17 AP € = 0.10 AP/P Q0 = 0.14 AP/P Q0 = 0.12 ω = 6, 804.6 (p = 0.000). No-COVID: Business As Usual. COVID+Gov: Partial adjustment model under COVID-19 with government support. Yes/No answers refer to whether the firm is experiencing solvency stress. No: Λ it + E it < 0; Yes: Λ it + E it < 0. N : Simulated number of firms. N : Expected number of firms being financially stressed if the effect of government support on firm solvency were random. P Q 0 : monthly level of sales in January 2020; AP € : Simulated cumulated amount of AP resources received by firms between March 2020 and April 2021, in billions of euros; AP/P Q 0 : Simulated cumulated amount of AP resources received by firms between March 2020 and April 2021, relative to monthly level of sales in January 2020; ω 0 : estimated level of Total factor Productivity in January 2020; F C 0 : Fixed cost defined as monthly corporate taxes plus payment of principal and interest in January 2020; Lev 0 Leverage, initial level of debts to suppliers and other third party, relative to equity.

Figure 1 :

 1 Figure 1: Dynamics of cumulative financial stress, as estimated from our simulations and according to the three different scenarios. Top panel: share of firms with liquidity issues. Bottom panel: share of firms with solvency issues.

Figure 2 :Figure A1 :

 2A1 Figure2: Estimated marginal effect of AP on the probability of experiencing solvency stress (black line) with its 95% confidence interval (grey shaded area). The dashed blue horizontal line represents the zero, above (below) which the estimated effect would (would not) imply a zombification of the economy. The dotted red vertical lines represent respectively the 5th and 95th percentile of the productivity distribution in the specific sector. Note: y-axis scale varies across quadrants.

Table 1 :

 1 Solvency stress under the No-COVID and COVID scenarios.

Table 2 :

 2 Solvency stress under the No-COVID and COVID+Gov scenarios, concerning firms facing solvency stress under the COVID scenario.

Table 3 :

 3 The determinants of the probability to experience solvency stress during the COVID global shutdown

		(1)	(2)	(3)	(4)	(5)	(6)	(7)
	VARIABLES		All sectors		Trade	Hotels	Others
	AP/P Q0	-0.166***	-0.172***	-0.218***	-0.230***	-0.502**	-0.685**	-0.330**
		(0.0426)	(0.0427)	(0.0437)	(0.0870)	(0.214)	(0.328)	(0.135)
	ω0		-0.287***	-0.495***	-0.480***	0.762	-0.345	-0.252*
			(0.0575)	(0.0594)	(0.114)	(0.565)	(0.559)	(0.145)
	AP/P Q0 × ω0				0.009	0.664***	0.291	0.034
					(0.0611)	(0.249)	(0.346)	(0.0821)
	F C0			0.209***	0.210***	0.099***	0.420***	0.345***
				(0.0172)	(0.0176)	(0.0232)	(0.0370)	(0.0435)
	Lev0			0.196***	0.196***	0.217***	0.267***	0.037
				(0.0106)	(0.0106)	(0.0151)	(0.0193)	(0.0264)
	P Q0	-0.0835*** -0.0649***	-0.162***	-0.162***	-0.0599*** -0.310*** -0.210***
		(0.0126)	(0.0132)	(0.0159)	(0.0162)	(0.0220)	(0.0404)	(0.0411)
	Constant	-0.323***	0.042	-0.011	-0.029	-0.460	-0.141	-0.029
		(0.0512)	(0.0891)	(0.0934)	(0.150)	(0.458)	(0.489)	(0.215)
	∂ Pr ∂AP	-0.0638*** -0.0658*** -0.0811*** -0.0820***	0.0436	-0.163*** -0.100***
		(-0.0163)	(-0.0163)	(-0.0162)	(0.0171)	(0.0348)	(0.0466)	(0.0234)
	Observations	11,925	11,925	11,925	11,925	5,377	4,895	1,653
	LL	-7968	-7955	-7740	-7740	-3378	-3247	-1039
	LR	478.9	503.8	933.6	933.6	273.4	289.1	90.43
	Pseudo R-squared	0.029	0.031	0.057	0.057	0.039	0.043	0.042

Table B1

 B1 presents the average revenue shares for labour L and materials M . The estimation sample contains 752,653 firms. The factor shares conform to the usual characteristics that materials represent most of the costs (59% of total sales), whereas labour costs represent on average one-third of total sales (35%). The estimated factor elasticities βM and βL amount to 0.73 and 0.21, respectively. Overall, firms operate below constant returns to scale, as the sum of factor elasticities λ = 0.96 lies below unity.Table B1 also reports the estimated output elasticities from a Cobb-Douglas production function by two-digit industry, using the Wooldridge (2009) methodology. There is substantial heterogeneity across industries in the parameter estimates. The average capi-

tal elasticity βK ranges between 0.001 in Professional, scientific, technical, support service activities to 0.07 in Financial and insurance activities. The values for βM range between 0.60 (Professional, scientific, technical, support service activities) and 0.86 (Wholesale and retail trade; repair of motor vehicles and motorcycles), βL takes values ranging between a minimum of 0.103 (Wholesale and retail trade; repair of motor vehicles and motorcycles) and a maximum of 0.39 (Professional, scientific, technical, support service activities). Estimated returns to scale λ are close to unity for most of the sectors, with values ranging between 0.94 (Other manufacturing) and 1.03 (Financial and insurance activities).

Table B1 :

 B1 Estimated output elasticities and revenue shares ** N is the number of firms. Source: FARE 2012-2019. Capital shares α K have been computed assuming constant returns to scales such that α

	N αK αL αM βK βL βM λ	24,999 0.080 0.355 0.565 0.025 0.189 0.775 0.989	5,562 0.056 0.296 0.648 0.018 0.206 0.773 0.997	1,605 0.047 0.259 0.694 0.018 0.167 0.805 0.990	46,022 0.068 0.332 0.600 0.035 0.243 0.706 0.984	5,118 0.148 0.221 0.631 0.054 0.178 0.755 0.987	128,605 0.061 0.346 0.594 0.032 0.208 0.732 0.972	207,893 0.054 0.187 0.760 0.018 0.116 0.847 0.981	28,992 0.071 0.359 0.570 0.043 0.244 0.681 0.968	94,079 0.078 0.347 0.575 0.030 0.165 0.830 1.025	24,609 0.017 0.479 0.503 0.027 0.345 0.655 1.026	24,322 0.155 0.408 0.437 0.071 0.298 0.664 1.034	118,218 0.115 0.494 0.391 0.001 0.372 0.608 0.982	42,579 0.069 0.481 0.450 0.019 0.280 0.698 0.997	752,603 0.074 0.338 0.589 0.024 0.217 0.749 0.989	K
	Code Sector name	C1 Manufacture of food products, beverages and tobacco products	C3 Manufacture of electrical, computer, electronic equipment & machinery	C4 Manufacture of transport equipment	C5 Other manufacturing	DE Electricity, Gas, Steam	FZ Construction	GZ Wholesale and retail trade; repair of motor vehicles and motorcycles	HZ Transportation and storage	IZ Accommodation and food service activities	JZ Information and communication	LZ Financial and insurance activities	MN Professional, scientific, technical, support service activities	RU Other services activities	All All economy

*

  Ordered probit model estimated by maximum likelihood methods. Outcome variable Y = 1 if the firm has neither liquidity nor solvency issues. Outcome variable Y = 2 if the firm has liquidity issues. Outcome variable Y = 3 if the firm has solvency issues. *** p<0.01, ** p<0.05, * p<0.1. AP/P Q 0 : estimated perceived amount of Activité Partielle, relative to sales in January 2020; ω 0 : estimated level of Total factor Productivity in January 2020; F C 0 : Fixed cost defined as monthly corporate taxes plus payment of principal and interest in January 2020; Lev 0 Leverage, initial level of debts to suppliers and other third party, relative to equity; P Q 0 : monthly level of sales in January 2020. All explanatory variables are entered in natural logs. Outcome = 1 if firms has neither liquidity nor solvency issues.

		1)	(2)	(3)	(4)
	AP/P Q0	0.806***	0.294***	0.100***	-0.270***
		(0.005)	(0.006)	(0.006)	(0.007)
	ω0		-1.306*** -1.367*** -0.749***
			(0.007)	(0.008)	(0.010)
	AP/P Q0 × ω0				0.458***
					(0.005)
	F C0			0.281***	0.340***
				(0.003)	(0.003)
	Lev0			0.163***	0.170***
				(0.002)	(0.002)
	P Q0	0.095***	0.076***	-0.117*** -0.130***
		(0.002)	(0.002)	(0.003)	(0.003)
	First cutoff	0.000	-0.417***	0.666***	-0.029
		(0.018)	(0.019)	(0.018)	(0.020)
	Second cutoff	0.818***	0.465***	0.666***	0.910***
		(0.018)	(0.019)	(0.019)	(0.020)
	∂ Pr ∂AP (No issues)	-0.124*** -0.042*** -0.014*** -0.033***
		(0.001)	(0.001)	(0.001)	(0.001)
	∂ Pr ∂AP (Liquidity issues)	0.083***	0.029***	0.095***	0.025***
		(0.001)	(0.001)	(0.001)	(0.001)
	∂ Pr ∂AP (Solvency issues)	0.041***	0.014***	0.004***	0.008***
		(0.000)	(0.000)	(0.000)	(0.000)
	Observations	752,603	752,603	752,603	752,603
	LL	-257,009	-240,253	-232,736	-228,259
	LR	93,191	126,702	141,736	150,691
	Pseudo R-squared	0.153	0.209	0.223	0.248

Repayment carries over the 6 forthcoming years, with a possible extension up to 10 years. The first two years may leave companies free from repayment duties.

See e.g. https://www.banque-france.fr/en/statistics/business-failures-france-2022apr.

That is, we do not assume perfect product and factor markets.

In a period of high uncertainty as it was the 2020, characterized by new ordinances at weekly cadence, it was impossible for firms to make forecasts ahead of time. This is the reason for which we have avoid to model firms' behavior as an inter-temporal optimization problem.

Electronic copy available at: https://ssrn.com/abstract=4169384

This setup is similar to one proposed by[START_REF] Schivardi | A simple method to estimate firms' liquidity needs during the covid-19 crisis with an application to italy[END_REF], with the main difference that our adjustment is micro-founded and approaching the optimal values L ⋆ it and M ⋆ it rather than proportional to the demand shock.

This is a reasonable assumption in the short-term.

A condition for which a firm would be able to operate even when this condition is met, is that fresh capital is provided by shareholders, by a controlling firm, or by an acquiring firm. These, however, are situations that go beyond the scope of this paper.

https://www.legifrance.gouv.fr/dossierlegislatif/JORFDOLE000041913361/.

See https://www.insee.fr/fr/recherche?q=Indice+production+industrielle&debut=0.

This would amount to roughly 30,000 companies being relieved from insolvency stress, relative to the 4 millions companies composing the French economy.

This raises the issue of the selection effect of the global lockdown. Table1reveals that amongst the 25 thousand companies being solvency stressed under the sanitary crisis, those that also undergo a solvency stress in normal times are more productive than those remaining financially healthy. Our interpretation is that high productivity does not perfectly equate with financial health. Although related, the two do not map perfectly, so that market selection occurs on forces other than productivity.

Importantly, the χ 2 -value imply that this effect is statistically significant.

In Appendix C we report the estimates from a regression on the whole sample, showing that it does not account for sample selection bias.

Probit model estimated by maximum likelihood methods. *** p<0.01, ** p<0.05, * p<0.1. AP/P Q 0 : estimated perceived amount of Activité Partielle, relative to sales in January 2020; ω 0 : estimated level of Total factor Productivity in January 2020; F C 0 : Fixed cost defined as monthly corporate taxes plus payment of principal and interest in January 2020; Lev 0 Leverage, initial level of debts to suppliers and other third party, relative to equity; P Q 0 : monthly level of sales in January 2020. All explanatory variables are entered in natural logs.

Appendix A. Supplementary material about the simulation

This appendix provides supplementary material about the simulations. Table A1 displays the values of the parameters used in the various scenarios. Figure A1 provides the share of firms in liquidity and solvency stress per sector.

Model Parameters

Cumulative Financial Stress by Sector

Appendix B. Production function estimations

The methodology used to compute unbiased estimates of the output elasticities with respect to our inputs follows [START_REF] Petrin | Measuring aggregate productivity growth using plantlevel data[END_REF] and is related to the use of inputs to control for unobservables in production function estimations, as set out by [START_REF] Olley | The dynamics of productivity in the telecommunications equipment industry[END_REF], [START_REF] Levinsohn | Estimating Production Functions Using Inputs to Control for Unobservables[END_REF], [START_REF] Ackerberg | Identification properties of recent production function estimators[END_REF] and [START_REF] Wooldridge | On estimating firm-level production functions using proxy variables to control for unobservables[END_REF]. The basic idea behind this approach is that the estimation of a production function may suffer from endogeneity bias because of a correlation between unobserved productivity shocks and inputs. This issue is solved by including lagged values of specific inputs as proxies for productivity. The methodology employed in this paper starts with a first step reading:

where q it is the natural logarithm of output of firm i at yearly time t, and k it , l it , and m it are respectively the natural logarithms of capital, labour in terms of hours worked, and materials used by the firm. In equation (B1), we use a third-order polynomial on all inputs and their interaction terms to obtain estimates of expected output, qit , and an estimate for ϵ it . This first step is included to net out pure error term, i.e. measurement errors, in the measure of output and productivity [START_REF] Ackerberg | Identification properties of recent production function estimators[END_REF][START_REF] De Loecker | Markups and Firm-Level Export Status[END_REF].

Then, we use a general production function of the following type:

where our inputs are transformed into the output according to the production function f s , B is the parameter vector to be estimated in order to calculate the output elasticities, ω it is the firm-level productivity term that is observable by the firm but not by the econometrician and ε it is an error term that is unobservable to both the firm and the econometrician. Leaving subscripts i and t aside for simplicity, function f s is assumed to be of a Cobb-Douglas log-form and reads:

Appendix C. Naive Regression with Sample Selection Issue

A naive ordered-probit regression on the whole sample of firms (Table C1) suggests that AP increases the likelihood of a firm facing financial distress, while lowering the chances of it encountering no difficulties. This could mistakenly been interpreted as indicating that the policy has performed very poorly or, actually, in reverse. What is more, an interaction between the measure of policy support and productivity displays a positive and significant coefficient, which implies that the negative effect of the policy is particularly harmful to productive firms, possibly because by interfering with market selection it is propping up inefficient companies and putting additional competitive pressure on the more productive ones. These results, instead, are the outcome of a large sample selection bias.


Nancy Guelman
email: nguelman@fing.edu.uy

Isabelle Liousse
email: isabelle.liousse@univ-lille.fr

Distortion in groups of Affine Interval Exchange Transformations

Keywords: Affine Interval Exchange Transformations, distortion, conjugacy, subgroups

In this paper, we study distortion in the group A of Affine Interval Exchange Transformations (AIET). We prove that any distorted element f of A has an iterate f k that is conjugate by an element of A to a product of infinite order restricted rotations, with pairwise disjoint supports. As consequences, we prove that no Baumslag-Solitar group, BS(m, n) with |m| = |n|, acts faithfully by elements of A; every finitely generated nilpotent group of A is virtually abelian and there is no distortion element in A Q , the subgroup of A consisting of rational AIETs.

Polterovich ( [START_REF] Polterovich | Growth of maps, distortion in groups and symplectic geometry[END_REF]) and ) proved that Diff 1 µ (Σ 2 ) does not contain distortion, where µ is a full support measure on a compact surface Σ 2 .

Novak (in [START_REF] Novak | Discontinuity growth of interval exchange maps[END_REF]) proved that there is no distortion element in the group of Intervals Exchange Transformations: bijections of the unit interval that are piecewise increasing and isometric.

In this work, we deal with a closely related problem, namely the existence of distortion elements inside the group of AIETs: Affine Intervals Exchange Transformations, denoted by A. Roughly speaking an AIET is a bijection of the unit interval that is increasing and affine on a finite number of intervals. If the endpoints of these intervals and their images are rational points, then the AIET is called a rational AIET. The set of rational AIETs is a subgroup of A and it is denoted by A Q .

Finitely generated groups of AIETs have provided several algebraically interesting groups, as for instance classical Thompson's groups F , T and V [START_REF] Cannon | Introductory notes on Richard Thompson's groups[END_REF] as well as some of their generalized versions [START_REF] Higman | Finitely presented infinite simple groups[END_REF], [START_REF] Bieri | On groups of PL-homeomorphisms of the real line[END_REF], the fundamental groups of orientable surfaces [START_REF] Ghys | Sur l'invariance topologique de la classe de Godbillon-Vey[END_REF], the modular group [START_REF] Navas | Sur les groupes de difféomorphismes du cercle engendrés par des éléments proches des rotations[END_REF], wreath products of the form Z Z ... Z ( [START_REF] Bleak | Solvability in Groups of Piecewise-linear Homeomorphisms of the Unit Interval[END_REF], [START_REF] Brin | Elementary amenable subgroups of R. Thompson's group F[END_REF], [START_REF] Navas | Quelques groupes moyennables de difféomorphismes de l'intervalle[END_REF], [START_REF] Navas | Groupes résolubles de difféomorphismes de l'intervalle, du cercle et de la droite[END_REF]) etc.

Our main result proves that most elements of A are in fact undistorted.

Theorem 1. For every distorted element f of A, there exists a positive integer k such that f k is conjugate by an element of A to a product of infinite order restricted rotations, with pairwise disjoint supports. This description of distorted elements of A enables us to prove the following corollaries. Corollary 2. Every torsion free nilpotent subgroup of A is abelian and every finitely generated nilpotent subgroup of A is virtually abelian.

Consequently, the Heisenberg group and thereby SL(3, Z) do not act faithfully via elements of A.

The last two corollaries were proved by Higman for Thompson's group V (see [START_REF] Higman | Finitely presented infinite simple groups[END_REF] and [START_REF] Röver | Subgroups of finitely presented simple groups[END_REF], chapter 2). Corollary 3. There is no distortion element in A Q .

The preceding corollary extends to all groups of rational AIETs, results of Bleak-Bowman-Gordon-Graham-Hughes-Matucci-Sapir (see [START_REF] Bleak | Centralizers in R.Thompson's group V n , Groups[END_REF] or [START_REF] Burillo | Obstructions for subgroups of Thompson's group V[END_REF]) and Hmili-Liousse (see [START_REF] Hmili | Dynamique des échanges d'intervalles des groupes de Higman-Thompson Vr,m[END_REF]) on the non existence of distortion in the Higman-Thompson groups V n . Its main consequence is that any group G containing distortion elements does not admit faithful actions as rational affine interval exchange transformations. Moreover, if G is almost-simple, such actions have finite image. This paper is organized as follows:

• In Section 2, definitions and basic facts are given.

In Sections 3 to 8, we establish the propositions that are used in the proof of Theorem 1, given in section 8.

• In Section 3, we prove that any element of A having semi-hyperbolic periodic orbits is undistorted.

• In Section 4, it is shown that, given f in A, the sequence whose general term is the number of break points of the iterate f n of f (for simplicity, this sequence will be called the "number of break points of f n " and it will be denoted by #BP (f n )) is either bounded or growths linearly. As a consequence, for any distortion element the number of break points of f n is bounded.

In the following sections, we study f in A without semi-hyperbolic periodic orbit and with bounded number of break points of f n .

• In section 5 (Theorem 2: "Extended "Alternate Version of Li's Theorem"), we establish that such an f has an iterate that is conjugate by an element of E to a product of restricted PL-homeomorphisms f i with bounded numbers of break points of f n i .

• For such a PL-homeomorphism f , in Section 6, we apply results of Minakawa [START_REF] Minakawa | Classification of exotic circles in P L(S 1 )[END_REF] to prove that f is PL-conjugate to a PL-homeomorphism B having at most two distinct slopes.

• In Section 7, under the additional assumption that f is distorted, we derive that B is a rotation, by showing that its slopes are 1.

• In Section 8, Theorem 1 is proved.

• Section 9 is devoted to prove Corollaries 1, 2 and 3 of Theorem 1.

Acknowledgments. We are deeply indebted to Andres Navas for bringing this problem to our attention, for allowing us to resume work on the unpublished manuscript1 and for several stimulating discussions during this work. We gratefully acknowledge several fruitful discussions with Ignacio Monteverde who also pointed out an error in a preliminary version of this work. We thank the support of the IFUM, the MathAmSud Project GDG 18-MATH-08 and PEDECIBA.

2 Preliminaries.

Affine Interval Exchange Transformations of

I = [0, 1). Definition 2.1.
A bijection f of [0, 1) is an Affine Interval Exchange Transformation (or AIET) of [0, 1) if there exists a finite subdivision 0 = a 0 < a 1 < .... < a p = 1 of [0, 1) such that for all i = 0, ..., p -1, one has f

(x) = λ i x + β i for x ∈ [a i , a i+1 ), where λ i ∈ R * + and β i ∈ R.
A break point is either the initial point 0 or a discontinuity of f or a discontinuity of Df , the derivative of f .

The set of break points of f is denoted BP(f ); it can be decomposed as the union of BP 0 (f ), the set consisting of 0 and all discontinuities of f and BP 1 (f ), the set of 0 and all discontinuities of Df .

We define

∆ f (x) = f + (x) -f -(x), if x ∈ (0, 1) and ∆ f (0) = f + (0) -f -(1), where f + (a) = lim x→a+ f (x) = f (a) and f -(a) = lim x→a- f (x).
The λ i 's are the slopes of f .

The jump of f at x is defined by

σ f (x) = D + f (x) D -f (x) if x ∈ (0, 1)
and

σ f (0) = D + f (0) D -f (1)
where

Df + (a) = lim x→a+ f (x)-f+(a) x-a and Df -(a) = lim x→a- f (x)-f-(a) x-a .
The sets of slopes and jumps of f are denoted respectively by Λ(f ) and σ(f ).

An AIET f of [0, 1) is an IET if Λ(f ) = {1}. Definition 2.2.
We denote by A the group consisting of all AIETs of [0, 1).

We denote by E the group consisting of all IETs of [0, 1).

Remark 1.

A homeomorphism f of the circle S 1 = [0, 1]/(0 = 1) can be seen as the bijection of [0, 1) defined by x → f (x) (mod 1).

Definition 2.3.

When this bijection is an AIET, f is called a PL-homeomorphism of [0, 1) even if it may not be continuous at eventually one point of (0, 1).

In what follows, PL-homeomorphisms of S 1 will be seen as AIETs of [0, 1). For example, the circle rotation by α is viewed as the element R α of A with break points 0 and 1 -α and given by f

(x) = x + α if x ∈ [0, 1 -α) and f (x) = x + α -1 if x ∈ [1 -α, 1). We say that R α is a rotation of [0, 1). Definition 2.4.
An IET f is called a restricted rotation if there exists some interval

I = [a, b) ⊂ [0, 1) such that f is supported by I and f (x) = x + δ if x ∈ [a, b -δ) and f (x) = x + δ -b + a if x ∈ [b -δ, b), where δ ∈ R, 0 < δ < b -a. An AIET f is called a restricted PL-homeomorphism if there exists some interval I = [a, b) ⊂ [0, 1) such that f is supported by I and #(BP 0 (f )∩(a, b)) ≤ 1.
Here are some elementary properties of break points sets. Property 1. Let f and g be two elements of A. Then

• BP (f -1 ) = f (BP (f )), • BP (f • g) ⊂ BP (g) ∪ g -1 (BP (f )), • BP (f n ) ⊂ BP (f )∪f -1 (BP (f ))∪...∪f -(n-1) (BP (f )), for all integer n ≥ 0.
These still hold for BP 0 (f ).

Unfortunately, such formulas do not hold for BP 1 (f ); this is due to the following Property 2. Let f , g in A and x ∈ [0, 1), one has

σ f •g (x) = Df + (g + (x))Dg + (x) Df -(g -(x))Dg -(x) = Df + (g + (x)) Df -(g -(x)) × σ g (x). If x / ∈ BP 0 (g) then σ f •g (x) = σ f (g(x)) × σ g (x). For x ∈ BP 0 (g), σ f •g (x) = σ f (g(x)) × σ g (x), in general. Property 3. If f ∈ A and g is a PL-homeomorphism then for all x in [0, 1), σ f •g (x) = σ f (g(x)) × σ g (x) and thereby BP 1 (f • g) ⊂ g -1 (BP 1 (f )) ∪ BP 1 (g).
Indeed, if g is a PL-homeomorphism and g + (x) = g -(x) then g + (x) = 0 and g -(x) = 1 therefore Df+(g+(x))

Df-(g-(x)) = σ f (0).

Interesting subgroups of A.

Numerous generalizations of Thompson's groups have been defined and studied. We recall, for example, the works of Bieri-Strebel [START_REF] Bieri | On groups of PL-homeomorphisms of the real line[END_REF] and Higman [START_REF] Higman | Finitely presented infinite simple groups[END_REF]. Definition 2.5. Let Λ ⊂ R + * be a multiplicative subgroup and A ⊂ R be an additive subgroup invariant by multiplication by elements of Λ and such that 1 ∈ A.

We define V Λ,A as the subgroup of A consisting of elements with slopes in Λ, break points and their images in A, and E A as the subgroup of E consisting of elements with break points in A, in fact

E A = E ∩ V Λ,A .
The subgroup A Q of rational AIETs is V Q>0,Q .

Definition 2.6. Let n be a positive integer, the group V <n>, Z[1/n] is denoted by V n and called a Higman-Thompson's group.

Note that classical Thompson's group V arises as V 2 . Among many things, Higman (see [START_REF] Higman | Finitely presented infinite simple groups[END_REF], [START_REF] Röver | Subgroups of finitely presented simple groups[END_REF]) proved that V n is finitely presented and satisfies the conclusions of Corollaries 1 and 2.

Remark 2. As established in [START_REF] Dahmani | Free groups of intervals exchange transformations are rare[END_REF], Thompson's groups F , T and V are not subgroups of E. Remark 3. Similarly, one can define AIETs of any interval [a, b) ⊂ R. In anticipation of a more general use of results from [START_REF] Li | A criterion for an Interval Exchange Map to be conjugate to an irrational rotation[END_REF], [START_REF] Novak | Discontinuity growth of interval exchange maps[END_REF], [START_REF] Minakawa | Classification of exotic circles in P L(S 1 )[END_REF] stated for [0, 1), we note that there exists a unique direct affine map sending [a, b) onto [0, 1). Thus any AIET of [a, b) is affinely conjugate to an AIET of [0, 1) with the same slopes set. However, the arithmetic properties of break points might not be preserved.

Distortion.

Definition 2.7.

Let Γ be a finitely generated group and S = {s 1 , ..., s r } be a finite generating set of Γ.

The smallest integer l such that g = s 1 i1 ...s l i l , with j ∈ {-1, 1} is called the length of g relatively to S and it is denoted by l S (g).

We set l S (e) = 0. The function l S : Γ → N is symmetric (l S (g -1 ) = l S (g) and satisfies l S (gh) ≤ l S (g) + l S (h). In particular, for all g in Γ, the sequence l S (g n ) is sub-additive, thus the sequence l S (g n ) n converges. This leads to Definition 2.8.

We say that g is distorted (or of distortion or a distortion element) in Γ =< S > if g has infinite order and lim

n→+∞ l S (g n ) n = 0. Remark 4.
The property of being a distortion element is independent of the choice of generating set S. Definition 2.9.

More generally, if G is not finitely generated, an element g of G is distorted in G if g is a distortion element in some finitely generated subgroup of G.

Properties 2.1.

• Let g ∈ G, the following properties are equivalent

1. g is distorted in G, 2. ∃N ∈ Z * : g N is distorted in G, 3. ∀N ∈ Z * : g N is distorted in G. • If Φ : Γ → G is a morphism and g ∈ Γ is distorted in Γ then its image Φ(g) ∈ G is either of finite order or distorted in G.
3 Semi-hyperbolicity prevents distortion.

Definition 3.1. Let f ∈ A, we say that p ∈ [0, 1] is a semi-hyperbolic periodic point of period l, if either: • p is not a break point of f l , f l (p) = p and Df l = 1 (hyperbolic) or • p is a break point of f l , f l + (p) = p and Df l + (p) = 1 or f l -(p) = p and Df l -(p) = 1 (virtual). Proposition 3.1. If f ∈ A has a semi-hyperbolic periodic point then f is undis- torted in A.
Proof. Let p be a semi-hyperbolic periodic point of f .

Without lost of generality, we can suppose that f + (p) = p and the right derivative of f at p: Df + (p) = λ = 1. For clarity, Df + will be denoted by D + f . By absurd, suppose that f is distorted in a subgroup G of A generated by S = {g 1 , ..., g s }. Then f can be written as f n = g i ln ...g i1 with lim n→+∞ l n n = 0.

We have

D + f n (p) = D + g i ln (p ln ) ... D + g i1 (p 1 )
, where p 1 = p and p j = g ij-1 ... g i1 (p), for j = 2, ..., l n .

Then (Inf D + g i ) ln ≤ D + f n (p) ≤ (Sup D + g i ) ln and l n log(Inf D + g i ) n ≤ log(D + f n )(p) n ≤ l n log(Sup D + g i ) n , where Inf D + g i = inf i,x D + g i (x) and Sup D + g i = sup i,x D + g i (x).
As f is distorted, one has lim

n→+∞ log(D + f n )(p) n = 0.
On the other hand, since p is a fix point of f , one has D + f n (p) = λ n and then

lim n→+∞ log(D + f n )(p) n = log λ = 0, this is a contradiction.
4 Alternative for the growth of the number of break points and characterization of distortion.

We recall that BP (f ) the set of break points of f is the union of the two following sets

BP 1 (f ) = {a ∈ [0, 1) : σ f (a) = 1} ∪ {0} and BP 0 (f ) = {a ∈ [0, 1) : ∆ f (a) = 0} ∪ {0}.
We denote by #BP * (f ) the cardinality of BP * (f ).

4.1 Alternative for the growth of the number of break points.

According to Property 1 (Section 2), one has:

#BP (f n ) ≤ #BP (f ) × n and #BP 0 (f n ) ≤ #BP 0 (f ) × n.
The main purpose of this section is to prove the following

Proposition 4.1. If f ∈ A then either • #BP (f n ) has linear growth or • #BP (f n ) is bounded.
The proof of Proposition 4.1 will be done in two steps: in Step 1, we prove

that if #BP 0 (f n ) is not bounded then #BP 0 (f n ) and therefore #BP (f n ) have linear growth; in Step 2 assuming that #BP 0 (f n ) is bounded, we establish that if #BP 1 (f n ) is not bounded then #BP 1 (f n ) and therefore #BP (f n ) have linear growth. In conclusion, either #BP 0 (f n ) and #BP 1 (f n ) are bounded or #BP (f n ) has linear growth.
We start by recalling and giving some basic properties and related definitions. Note that for all integer n, the set BP (f n ) is included in the union over all a ∈ BP (f ) of the sets

BP (f n ) ∩ O f (a) (by Property 1), where O f (a) = {f n (a), a ∈ Z} is the f -orbit of a. Property 4. If a ∈ BP (f ) is f -periodic then for all integer n, the set BP (f n ) ∩ O f (a)
is finite and has cardinality less or equal than the period of a.

Property 5. Let a ∈ BP (f ).
1. If a is not f -periodic then there exists a segment S a of the orbit of a of the form S a = {b = f -p (a), f -p+1 (a), ..., a, ..., f l (a) = c}, where b and c belong to BP (f ) and for all k

∈ N * , f -k (b) / ∈ BP (f ) and f k (c) / ∈ BP (f ).
Such a break point b is called an initial break point of f . Therefore:

2. If a is an initial break of f point then S a = { a, f (a), ... , f Na (a) } and

• O f (a) ∩ BP (f n ) ⊂ {f -(n-1) (a), ..., f Na (a)}, for all integer n ≥ 0, • O f (a) ∩ BP (f -n ) ⊂ {f (a), ..., f Na+n (a)}, for all integer n ≥ 0.
In particular:

(a) f -k (a) / ∈ BP (f m ), for all integers k ≥ m ≥ 0, (b) f k (a) / ∈ BP (f m ), for all integers m ≥ 0, k > N a and (c) f -k (a) / ∈ BP (f -p ) = f p (BP (f p )), for all integers p ≥ 0, k ≥ 0.
Indeed, as #BP (f ) is finite, if (1) does not hold then there exist some d ∈ BP (f ), m 1 and m 2 distinct integers such that d = f m1 (a) = f m2 (a). This contradicts the non periodicity of a. We derive the second item from Property 1.

Step 1: Alternative for #BP 0 (f n ).

This part is devoted to the proof of

Lemma 4.1. Let f ∈ A. • If exists a ∈ BP 0 (f ) a non periodic initial break point with ∆ f Na +1 (a) = 0 then #BP 0 (f ) × n ≥ #BP 0 (f n ) ≥ n -N a .
That is #BP 0 (f n ) has linear growth.

• If for all a ∈ BP 0 (f ) non periodic initial break points, ∆ f Na +1 (a) = 0, then

BP 0 (f n ) ≤ a∈A 2N a + a∈B period(a),
where A = {a ∈ BP 0 (f ) non periodic initial} and B = {a ∈ BP 0 (f ) periodic}.

That is

#BP 0 (f n ) is bounded.
Let a ∈ BP 0 (f ) be a non periodic initial break point and S a = {a, f (a), ..., f Na (a)}; for simplicity of notation, we set N = N a . Sublemma 4.1.

• If ∆ f N +1 (a) = 0 then for all integer l ≥ 1, ∆ f N +l (a) = 0. • If ∆ f N +1 (a) = 0 then for all integer l ≥ 1, ∆ f N +l (a) = 0. Proof. Let l ≥ 1, one has f N +l + (a) = f N +l (a) = f l-1 (f N +1 (a)), f N +l - (a) = f l-1 -(f N +1 - (a)). If ∆ f N +1 (a) = 0 then ∆ f N +l (a) = f l-1 (f N +1 (a)) -f l-1 -(f N +1 (a)) = 0, since f l-1 is continuous at f N +1 (a) by Property 5 (2b). If ∆ f N +1 (a) = 0, as f l-1 is continuous at f N +1 (a), the point f l-1 (f N +1 (a)) can not be equal to f l-1 -(c) for some c = f N +1 (a). It follows that f N +l (a) = f l-1 (f N +1 (a)) = f l-1 -(f N +1 - (a)) = f N +l - (a), since f N +1 (a) = f N +1 - (a).
Sublemma 4.2. Let a be an initial break point and n be a positive integer.

∆ f n (f -k (a)) = ∆ f n-k (a), for 0 ≤ k ≤ n -1. Proof. By Property 5 (2), O f (a) ∩ BP (f n ) ⊂ { f -(n-1) (a), ... a, ... , f N (a) }. Let us compute ∆ f n (f -k (a)), for 0 ≤ k ≤ n -1, we get: ∆ f n (f -k (a)) = f n + (f -k (a)) -f n -(f -k (a)) = f n-k (a) -f n-k - (f k -(f -k (a)).
Moreover, for all k ≥ 0, one has

f k -(f -k (a)) = a, according to Property 5 (2a). Finally, ∆ f n (f -k (a)) = f n-k (a) -f n-k - (a) = ∆ f n-k (a).
We turn now to the proof of Lemma 4.1, estimating #BP 0 (f n ) for a given positive integer n. Combining the two preceding sublemmas, we deduce the following alternative for any initial break point a: Step 2: Alternative for #BP (f n ).

• If ∆ f N +1 (a) = 0 then ∆ f n (f -k (a)) = 0, for all n -k > N . Hence #(BP 0 (f n ) ∩ O f (a)) ≥ n -N . • If ∆ f N +1 (a) = 0 then ∆ f n (f -k (a)) = 0, for all n -k > N . Hence #(BP 0 (f n ) ∩ O f (a)) ≤ N + (n -(n -N )) = 2N .
By Lemma 4.1, if #BP 0 (f n ) is not bounded then #BP (f n ) has linear growth therefore in this part, we will focus on the case where #BP 0 (f n ) is bounded and prove the following Lemma 4.2 which will complete the proof of Proposition 4.1.

Lemma 4.2. Let f ∈ A with #BP 0 (f n ) bounded. • If exists a ∈ BP 1 (f ) a non periodic initial break point such that Π a = 1, then #BP (f ) × n ≥ #BP 1 (f n ) ≥ n -N a .
Hence #BP 1 (f n ) and #BP (f n ) have linear growth.

• If for all a ∈ BP 1 (f ) non periodic initial break point, Π a = 1, then

#BP 1 (f n ) ≤ a∈A 2N a + a∈B period(a), where A = {a ∈ BP 1 (f ) non periodic initial} and B = {a ∈ BP 1 (f ) periodic}. Hence #BP 1 (f n ) and #BP (f n ) are bounded.
Proof. Let a ∈ BP (f ) be a non periodic initial break point and S a = {a, f (a), ..., f N (a)} be the smallest segment containing all the break points of f on O f (a).

Let

n ≥ N + 1, recall that BP (f n ) ∩ O f (a) ⊂ {f -(n-1) (a), ..., a, ..., f N (a)}. Let us compute the jump of f n at the point f -k (a) for k ≥ 0 and n-1-k > N , that is for 0 ≤ k < n -1 -N .
Iterating the composition formula given in Property 2, we get:

σ f n (f -k (a)) = Df + (f n-1 + (f -k (a))) ... Df + (f k + (f -k (a))) ... Df + (f -k (a)) Df -(f n-1 - (f -k (a))) ... Df -(f k -(f -k (a))) ... Df -(f -k (a))
.

According to Property 5(2c),

f -k (a) = f -k -(a) and therefore f l -(f -k (a)) = f l -(f -k -(a)) = f l-k -(a), for any integer l ≥ 0; in addition, if l ≤ k then f l -(f -k (a)) = f l-k (a). Therefore, noting that n -1 -k > N , we get σ f n (f -k (a)) = R 1 × R 2 × R 3 , where R 1 = Df + (f n-1-k + (a)) ... Df + (f N +1 + (a)) Df -(f n-1-k - (a)) ... Df -(f N +1 - (a)) , R 2 = Df + (f N + (a)) ... Df + (a) Df -(f N -(a)) ... Df -(a)
and ). Thus, the first ratio R 1 is also trivial. Finally,

R 3 = Df + (f -1 (a)) ... Df + (f -k (a)) Df -(f -1 (a)) ... Df -(f -k (a)) . As BP 1 (f ) ∩ O f (a) ⊂ S a , the third ratio R 3 is trivial. Since #BP 0 (f n ) is
σ f n (f -k (a)) = R 2 = Df + (f N + (a)) ... Df + (a) Df -(f N -(a)) ... Df -(a) =: Π a
Note that this formula also holds for n -1 -k = N , since the first ratio does not appear in σ f n (f -k (a)).

Therefore, the following alternative holds.

• If Π a = 1 then for all k integer such that 0 ≤ k ≤ n -1 -N , f -k (a) ∈ BP 1 (f n ) and #(BP 1 (f n ) ∩ O f (a)) ≥ n -N . • If Π a = 1 then for all k integer such that 0 ≤ k ≤ n -1 -N , f -k (a) / ∈ BP 1 (f n ) and #(BP 1 (f n ) ∩ O f (a)) ≤ N + (n -(n -N )) = 2N .
4.2. If f is distorted in A then #BP (f n ) is bounded.
Proof. By contradiction, suppose that #BP (f n ) is unbounded and f is distorted in a subgroup G of A generated by S = {g 1 , ..., g s }. This means that f n can be written as f n = g i ln ...g i1 with lim n→+∞ l n n = 0. Therefore, by Property 1, we have:

BP (f n ) ⊂ BP (g i1 ) ∪ g -1 i1 BP (g i2 ).... ∪ (g i l n-1 ...g i1 ) -1 BP (g i ln ) then #BP (f n ) ≤ #BP (g i1 ) + ... + #BP (g i ln ) ≤ l n max{#BP (g i ), i = 1, ...s} and therefore l n n ≥ #BP (f n ) n (max{#BP (g i ), i = 1, ...s}) -1 .
Thus lim n→+∞ l n n > 0, since #BP (f n ) has linear growth according to Proposition 4.1, this is a contradiction.

5 Extended "alternative version of Li's Theorem".

The aim of this section is to prove an extended version of the "Alternate Version of Li's Theorem" given in [START_REF] Novak | Discontinuity growth of interval exchange maps[END_REF].

Theorem 2. Let f in A without periodic points and with bounded #BP (f n ) then there exists an integer q such that f q is conjugate in E to a product of restricted PL-homeomorphisms of disjoint supports that are minimal when restricted to their respective supports.

Definition 5.1. Let f ∈ A. We say that f satisfies the pair property if 1. f does not have periodic points, 2. BP 0 (f ) = {β 1 , ....β s , ω 1 , ...., ω s }, where any pair (β i , ω i ), for i = 1, ...s, verifies f (β i ) = ω i and β i / ∈ BP 0 (f 2 ) and 3. the f -orbits of the β i 's are disjoint.

Convention. Eventually re-indexing the ω i 's, we may suppose that 0 = ω 1 < ω 2 < ... < ω s .

We begin by giving some basic properties.

Property 6.

1. If f has the pair property then any associated pair (β i , ω i ), for i = 1, ...s, verifies (a)

β i ∈ BP 0 (f ) \ BP 0 (f 2 ), (b) ω i ∈ BP 0 (f ) ∩ BP 0 (f -1 ).
2. The pair property is invariant under C 0 -conjugation.

3. If f has the pair property with associated pairs (β i , ω i ),for i = 1, ...s, then for any n ∈ N, f n has the pair property with associated pairs (f -n (ω i ), ω i ), for i = 1, ...s.

Using these properties, we get

f -(β i ) ∈ BP 0 (f ) ∩ BP 0 (f -1 ) ∪ {1} so f -(β i
) is an ω j and we can give Definition 5.2. Let π be the permutation of {1, ...s} defined by either

• π(i) = j in the case that f -(β i ) = ω j or • π(i) = 1 when f -(β i ) = 1.
Hence, one has f (ω i ) = f -(ω π(i) ), for i = π -1 (1), otherwise f (ω i ) = f -(1), for i = π -1 (1). Definition 5.3. (see [START_REF] Li | A criterion for an Interval Exchange Map to be conjugate to an irrational rotation[END_REF]) Let f ∈ A with the pair property.

A pair (β i , ω i ) is said removable if either:

• ω π(i) < ω i or

• ω π(i) > ω i and there exists ω j ∈ (ω i , ω π(i) ).

Lemma 5.1. Let f ∈ A without periodic points and with bounded #BP 0 (f n ) then there exists an iterate of f that satisfies the pair property.

Proof. Let IBP (f ) be the set of all initial break points of f . According to Section 4, one has BP 0 (f ) ⊂

a∈IBP (f ) {a, ...., f Na (a)}, ∆ f Na +1 (a) = 0 and ∆ f n (f -k (a)) = 0, for all k ≥ 0, n -k > N a . (1)
Let N be the maximum of the {N a } over all a ∈ IBP (f ) and F = f N +1 . It holds that BP 0 (F ) ∩ O f (a) ⊂ {f -N (a), ..., a, ..., f Na (a)}.

After noting that ∆ F (f -(N +1)+Na+l (a)) = 0, (for l = 1, ..., N -N a + 1) by

formula (1), BP 0 (F ) ∩ O f (a) ⊂ Na l=1 {f -(N +1)+l (a), f l (a)}.
We claim that any pair (β, ω) = (f -(N +1)+l (a), f l (a)) for l = 1, ..., N a satisfies F (β) = ω and β / ∈ BP 0 (F 2 ). Indeed, obviously F (β) = ω, we now compute ∆ F 2 (β): Obviously, F satisfies Conditions ( 1) and ( 3) of the pair property.

∆ F 2 (β) = ∆ f 2N +2 (f -(N +1)+l
Lemma 5.2. Let F ∈ A with the pair property and (β i , ω i ) be a removable pair of F , then there exists E ∈ E such that #BP 0 (EF E -1 ) ≤ #BP 0 (F ) -2 and EF E -1 has also the pair property. Moreover, BP 0 (E) ⊂ BP 0 (F ) ∩ BP 0 (F -1 ).

Proof.

We begin by considering the case where ω π(i) < ω i . Let E be in E given by BP 0 (E) = {0, ω π(i) , ω i } and with associated permutation (1, 2, 3) → (1, 3, 2).

According to Property 6(1b), BP 0 (E) ⊂ BP 0 (F ) ∩ BP 0 (F -1 ). One has that

BP 0 (EF E -1 ) ⊂ EF -1 (BP 0 (E)) ∪ E(BP 0 (F )) ∪ BP 0 (E -1 ).
As F -1 (BP 0 (E)) ⊂ BP 0 (F ) and BP 0 (E -1 ) = E(BP 0 (E)) ⊂ E(BP 0 (F )), it holds that BP 0 (EF E -1 ) ⊂ E(BP 0 (F )).

We prove that E(β i ) and E(ω i ) do not belong to BP 0 (EF E -1 ) by computing the right and left values at these points:

• EF E -1 (E(β i )) = EF (β i ) = E(ω i ) = ω π(i) and since β i / ∈ BP 0 (E), • (EF E -1 ) -(E(β i )) = E -F -(β i ) = E -(ω π(i) ) = ω π(i) .
This proves that E(β i ) / ∈ BP 0 (EF E -1 ).

• EF E -1 (E(ω i )) = EF (ω i ) and

• (EF E -1 ) -(E(ω i )) = (EF E -1 ) -(ω π(i) )) = (EF ) -(ω π(i) ). In addition, since F (ω i ) = F -(ω π(i) ) does not belong to BP 0 (E), we have (EF ) -(ω π(i) ) = E(F (ω i )). We thus get EF E -1 (E(ω i )) = (EF E -1 ) -(E(ω i )) which proves that E(ω i ) / ∈ BP 0 (EF E -1 ). Finally, #BP 0 (EF E -1 ) ≤ #BP 0 (F ) -2.
We claim that EF E -1 has the pair property with associated pairs of the form (E(β j ), E(ω j )). Indeed, as (F -2 (ω i ), ω i ) is a pair for F 2 , it holds that BP 0 (E) ⊂ BP 0 (F 2 ) ∩ BP 0 (F -2 ) and the same arguments as in the beginning of this proof show that BP 0 (EF 2 E -1 ) ⊂ E(BP 0 (F 2 )).

Obviously EF E -1 (E(β j )) = E(ω j ). Now we argue by contradiction assuming that E(β j ) ∈ BP 0 (EF 2 E -1 ) then β j ∈ E -1 (BP 0 (EF 2 E -1 )) ⊂ BP 0 (F 2 ), a contradiction.

Clearly EF E -1 also satisfies Conditions ( 1) and ( 3) of the pair property. Now, we consider the case where (β i , ω i ) is a removable pair with the following properties: ω π(i) > ω i and there exists ω j ∈ (ω i , ω π(i) ).

We identify 0 to 1 to get a circle and then we cut this circle at the point ω j . We are in the previous case. This ends the proof of Lemma 5.2.

Proof of Theorem 2.

Let f ∈ A without periodic points and with bounded #BP 0 (f n ). By Lemma 5.1, there exists some integer N such that f N has the pair property.

Starting from f N and applying Lemma 5.2 a finite number of times, we get that

G = E r .....E 1 f N E -1 1 ...E -1
r has the pair property and no removable pair. Since the associated pairs (β i , ω i ) of G are not removable, it holds that ω π(i) > ω i , for any i and that the intervals (ω i , ω π(i) ) are pairwise disjoint and disjoint from the last interval (ω s , 1).

Note that s = π -1 (1), indeed if not then ω s < ω π(s) < 1, a contradiction.

We claim that for any 1 ≤ i < s, there exists a unique discontinuity point of G in (ω i , ω π(i) ) and this still holds for (ω s , 1). Indeed as G + (ω i) ) = G -(ω π(i) ), the interval (ω i , ω π(i) ) contains at least one point of BP 0 (G) and a similar argument shows that the same holds for (ω s , 1).

As the associated pairs are unremovable, this discontinuity point is a β j . Moreover, since the number of β i 's is exactly the number of intervals of the form (ω i , ω π(i) ) or (ω s , 1), then β j is unique.

Therefore G([ω i , ω π(i) )) = [ω j , ω π(j) ), since G -(β j ) = ω π(j) and G + (β j ) = ω j , if j = π -1 (1) = s. If j = s, then G([ω i , ω π(i) ) = [ω s , 1). This implies that R = s-1 i=1 [ω i , ω π(i) ) [ω s , 1) is G-invariant. We claim that it is [0, 1[.
Indeed, if not, the complementary of R is a finite union of half open intervals that is G-invariant and G is continuous on each interval (since such intervals do not contain β's and ω's the discontinuity points of G). Thus, these intervals are periodic which contradicts that G (f ) does not have periodic points.

Moreover, there exists an iterate of

G such that G l ([ω i , ω π(i) )) = [ω i , ω π(i) ), G l ([ω s , 1) = [ω s , ω 1
) and the restriction of G l to any [ω i , ω π(i) ),i = 1, ..., s -1 and to [ω s , 1) has just one interior discontinuity point.

Finally, G l is a product of restricted PL-homeomorphisms Γ i with disjoint supports and it is conjugated by E = E r .....E 1 to f lN .

As f and then G l has no periodic points, by Denjoy's Theorem for Class P circle homeomorphisms (see [START_REF] Herman | Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations[END_REF]), each Γ i is minimal when restricted to its support.

This ends the proof of Theorem 2.

Remark 5. Note that the discontinuities of the E i 's belong the f -orbit of BP 0 (f ) and the restricted PL-homeomorphisms Γ i are supported by intervals whose endpoints also are points of the f -orbit of BP 0 (f ). In particular, if f ∈ A Q then the Γ i 's are supported by intervals whose endpoints are rational and E ∈ A Q .

PL conjugation.

Next proposition is due to Minakawa [START_REF] Minakawa | Classification of exotic circles in P L(S 1 )[END_REF].

Proposition 6.1. Let f ∈ A be a PL-homeomorphism having bounded #BP (f n ).
Then there exists a PL-homeomorphism

H ∈ A such that B := H • f • H -1 is an AIET of 2 intervals.
Remark 6. An AIET of 2 intervals B is a PL-homeomorphisms and it is uniquely determined by its slope pair (λ 1 , λ 2 ), so it is denoted by B λ1,λ2 . It verifies that

BP (B) = {0, B -1 (0)}, DB(x) = λ 1 on [0, B -1 (0)) and DB(x) = λ 2 on [B -1 (0), 1
). These maps B λ1,λ2 were studied in [START_REF] Boshernitzan | Dense orbits over rationals[END_REF]. There it was proven that either:

• λ 1 = λ 2 and B λ1,λ2 is a rotation or

• λ 1 = λ 2 and B λ1,λ2 is C 0 -conjugate to a rotation R ρ by a map of the form

x → (ω x -1) (ω -1)
for some positive ω distinct from 1.

Remark 7. We shall give a refinement of Minakawa's proof which will enable us to preserve the arithmetic properties of f , this will be explained in Remark 8. An alternative proof using a "PL pair property" can be found in [START_REF] Liousse | PL Homeomorphisms of the circle which are piecewise C 1 conjugate to irrational rotations[END_REF].

Proof. As #BP (f n ) is bounded, Lemma 4.2 indicates that there exists a sub-

set {a i , i ∈ I} of BP 1 (f ) such that BP 1 (f ) is contained in i∈I S i with S i = {f k (a i ), k = 0, ..., N i } and Π ai = Df + (f Ni + (a i ))....Df + (a i ) Df -(f Ni -(a i ))....Df -(a i ) = 1, ∀i ∈ I.

Note that

Df+(f k + (ai)) Df-(f k -(ai)) = σ f (f k (a i )), since f k + (a i ) = f k -(a i ) or f k + (a i ) = 0 and f k -(a i ) = 1. Then Π ai = c∈Si σ f (c) = 1, for all i ∈ I.
We let H f = H be a PL-homeomorphism of [0, 1) with the following properties:

• The break points of H are the points f (a i ), . . . , f Ni (a i ), for i ∈ I.

• The associated jumps are σ

H (f k (a i )) = σ f N +1 (f k (a i )) for k = 1, . . . , N i , where N = max{N i , i ∈ I}. Note that σ H (a i ) = σ f N +1 (a i ) = N n=0 σ f (f n (a i )) = 1.
At the end of this proof, we will indicate a general lemma (Lemma 6.1) about the existence of PL-homeomorphisms with prescribed break points and slopes. It implies that a necessary and sufficient condition for the existence of such a homeomorphism H is that the product of the H-jumps is trivial, that is

Π(f ) := i∈I,0≤k≤Ni σ f N +1 (f k (a i )) = 1.
• If Π(f ) = 1, then we can define a map H as above and normalize it by setting H(0) = 0.

• If Π(f ) = 1, then we add a break point c / ∈ {a i , ..., f Ni (a i )} and require that σ H (c) = (Π(f )) -1 ; we normalize H by setting H(c) = 0. Now, since f and H are PL-homeomorphisms, Property 3 implies that

• the set BP 1 (H • f • H -1 ) satisfies BP 1 (H • f • H -1 ) ⊂ BP 1 (H -1 ) ∪ H(BP 1 (f )) ∪ H • f -1 (BP 1 (H)) ⊂ {H(a i ), ..., H(f Ni (a i )), i ∈ I} ∪ {H(c), H(f -1 (c))}, • for i ∈ I, 0 ≤ k ≤ N i , the jump of H • f • H -1 at H(f k (a i )) is equal to σ H•f •H -1 H(f k (a i )) = σ H (f k+1 (a i )) × σ f (f k (a i )) σ H (f k (a i )) = σ f N +1 (f k+1 (a i )) × σ f (f k (a i )) σ f N +1 (f k (a i )) = 1, • σ H•f •H -1 (H(c)) = Π(f ) and • σ H•f •H -1 H(f -1 (c)) = Π(f ) -1 .
In conclusion.

• If Π(f ) = 1, the AIET B = H • f • H -1 has no break of slopes, it is a rotation. • If Π(f ) = 1, the AIET B = H • f • H -1
has exactly two breaks of slopes at 0 = H(c) and

B -1 (0) = H(f -1 (c)), that is Λ(B) = {λ 1 , λ 2 }.
Lemma 6.1. Given 0 = c 0 < c 1 < ... < c p < 1 points in [0, 1) and σ 0 , ..., σ p positive real numbers such that p i=0 σ i = 1, there exists a PL-homemorphism H such that :

• BP 0 (H) = {c 0 , c 1 , ..., c p } and

• σ H (c i ) = σ i , for i = 0, ..., p.
Moreover, given d ∈ [0, 1) the map H is unique provided that H(d) = 0.

The proof is left to readers, however we indicate some elements of the construction of H. Setting Λ(H) = {λ 1 , ..., λ p+1 }, one has

λ i = σ 0 ...σ i-1 λ 1 and λ 1 = (|I 1 | + σ 1 |I 2 | + ... + σ 1 ...σ p |I p+1 |) -1 by computation of the total length of H([0, 1)).
In particular, if c i and σ i are rational numbers then λ i ∈ Q. Moreover, we can choose H such that H(c j ) ∈ Q, for some c j and then H ∈ A Q . Remark 8. We have described explicitly the conjugating PL-homeomorphism H, we can deduce that if f ∈ A Q then the break points of H and the jumps of H belong to Q, provided that the point c is chosen in Q. Therefore, if f ∈ A Q then the conclusions of Proposition 6.1 hold with H and B belonging to A Q .

Note that if

7 The case Λ(B) = {λ 1 , λ 2 }.

Let α 1 , ..., α s generating a rank s free abelian multiplicative subgroup Λ of R + * . Therefore, given λ ∈ Λ, there exists a unique (n 1 , ..., n s ) ∈ Z s , such that λ = α n1 1 ... α ns s and we define N j (λ) = n j , for all j ∈ {1, ..., s}.

Proposition 7.1. Let B = B λ1,λ2 ∈ A with (λ 1 , λ 2 ) ∈ Λ 2 \ {(1, 1)}.
There exist j ∈ {1, ..., s} and x ∈ [0, 1) such that

Nj (D+B n (x)) n → ν = 0.
Proof. Noting that B satisfies that DB(x) = λ 1 on [0, a) and DB(x) = λ 2 on [a, 1), with a = B -1 (0), one has

D + B n (x) = λ N1(x,n) 1 λ N2(x,n) 2
, where

N 1 (x, n) = #{x, f (x), ..., f n-1 (x)} ∩ [0, a) = n-1 k=0 I [0,a) (f k (x))
and

N 2 (x, n) = #{x, f (x), ..., f n-1 (x)} ∩ [a, 1) = n-1 k=0 I [a,1) (f k (x)).
The map B has a unique invariant probability measure µ, since it is C 0conjugate to an irrational rotation R ρ , by [START_REF] Boshernitzan | Dense orbits over rationals[END_REF] (see Remark 6). More precisely, consider h such that h

• B • h -1 = R ρ , one has µ(A) = λ(h(A), for all measur- able set A. In particular, µ([0, a]) = µ([0, B -1 (0))) = λ([h(0), h • B -1 (0))) = λ([h(0), R -1 ρ (h(0)))) = (1 -ρ) and µ([a, 1
)) = ρ. The Birkhoff Ergodic Theorem implies that for µ-almost every point x ∈ [0, 1), one has

lim n→+∞ N 1 (x, n) n = µ([0, a]) and lim n→+∞ N 2 (x, n) n = µ([a, 1]).
Now, let us write λ 1 and λ 2 in the basis α 1 , ..., α s of Λ that is λ 1 = α β1 1 ...α βs s and λ 2 = α δ1 1 ...α δs s . We now compute the coordinates N j (D + B n (x)) of D + B n (x) in this basis.

As

D + B n (x) = λ N1(x,n) 1 λ N2(x,n) 2 = α β1.N1(x,n)+δ1.N2(x,n) 1 ... α βs.N1(x,n)+δs.N2(x,n) s , one has N j (D + B n (x)) = β j .N 1 (x, n) + δ j .N 2 (x, n).
It follows that

N j (D + B n (x)) n = β j . N 1 (x, n) n + δ j . N 2 (x, n) n → β j .(1 -ρ) + δ j ρ = ρ(δ j -β j ) + β j .
Finally, suppose that (λ 1 , λ 2 ) = (1, 1) then necessary λ 1 = λ 2 and there exists j such that δ j = β j . Therefore, ν = ρ(δ j -β j ) + β j = 0, as ρ / ∈ Q.

8 Proof of Theorem 1.

Let f be distorted in A, as f has no a semi-hyperbolic periodic point, its periodic points are not isolated. Using in addition that BP 0 (f ) is finite, we get that the set P er(f ) of f -periodic points is the union of a finite collection of half open intervals with endpoints in the orbits of BP 0 (f ). Thereby, there exists some positive integer p such that P er(f ) = P er(f p ) = F ix(f p ). It is easy to check that there exists S ∈ E whose discontinuities are endpoints of connected components of P er(f ) and such that F ix(Sf p S -1 ) is an interval P = [0, a) and the restriction of Sf p S -1 to M = [a, 1) has no periodic points.

Applying Theorem 2 to the restriction of Sf p S -1 to M , there exist q ∈ N *

and

E ∈ E such that ES • f pq • (ES) -1 = p i=1 f i , where f i are restricted PL- homeomorphisms with pairwise disjoint supports I i = [a i , b i ), f i | Ii is minimal and #BP (f n i ) is bounded (since f is distorted). Let i ∈ {1 ... p}, applying Proposition 6.1 to f i | Ii , we get that f i | Ii is conjugate by a PL-homeomorphism H i of I i to B i with Λ(B i ) = {λ i,1 , λ i,2 } and BP (B i ) = {a i , B -1 i (a i )}. Since f i | Ii is minimal, B i also is minimal and according to Remark 6, B i is C 0 -conjugate to an infinite order rotation R ρ of I i . Let H ∈ A defined by H(x) = H i (x), if x ∈ I i and H(x) = x, if x / ∈ ∪I i and let B = (HES) • f pq • (HES) -1 .
It is easy to check that B| Ii = B i and B is distorted in a subgroup G =< g 1 , ..., g q > of A, since f is distorted in A.

Let Λ G be the free abelian multiplicative subgroup Λ of R + * generated by { Dg k (x), x ∈ [0, 1), k ∈ {1 ... q} }. It has finite rank s, we consider a basis α 1 , ..., α s of it.

Let i ∈ {1 ... p}, note that N j (D + B n (y)) = N j (D + B n i (y)), ∀y ∈ I i . We suppose that (λ i,1 , λ i,2 ) = (1, 1).

On one hand, by Proposition 7.1, there exist j ∈ {1, ..., s} and x ∈ I i such that

Nj (D+B n (x)) n → ν = 0.
On the other hand, since B is distorted in G, its iterates B n can be written

B n = g i ln ... g i1 with lim n→+∞ l n n = 0. Hence D + B n (x) = D + g i ln (x ln ) ... D + g i1 (x 1 )
, where x m = g im-1 ... g i1 (x). Then

|N j (D + B n (x))| ≤ ln m=1 |N j (D + g im (x m ))| ≤ l n S, where S = max{ |N j (D + g k (y))|, y ∈ [0, 1), 1 ≤ k ≤ q }. Finally, l n n ≥ |N j (D + B n (x))| nS → |ν| S > 0
, this is a contradiction.

Consequently, for any i ∈ {1 ... p}, (λ i,1 , λ i,2 ) = (1, 1) and thereby B i is an infinite order rotation of I i . Thus B is a product of infinite order restricted rotations with pairwise disjoint supports.

In conclusion, we have proved that when restricted to M , there exists an iterate of f that is conjugate in A to a product of infinite order restricted rotations with pairwise disjoint supports. We conclude by noting that f |M c = Id |M c . 9 Proof of Corollaries 1, 2 and 3.

Proof of Corollary 1.

Let a, b in A such that ba m b -1 = a n with m, n integers and |m| = |n|. We will prove that a has finite order. By absurd, since a is distorted, eventually passing to a power of a and conjugating a and b by an element of A we can suppose that a is a product of infinite order restricted rotations R αi of disjoint supports I i , according to Theorem 1. We denote a = p i=1 (R αi , I i ).

The main tool of the proof is the following Let X = ∪I i . In what follows we identify a with its restriction to X. By unique ergodicity of irrational rotations, one has that the ergodic a pinvariant probabilities on X are Leb|I i , for all p ∈ Z.

As Supp(ba m b -1 ) = b(Supp(a m )) and Supp(a n ) = Supp(a m ), then b(X) = X and b is identified to its restriction to X.

The image by b of an ergodic a m -invariant measure is an ergodic a n -invariant measure. Hence, for some permutation σ, b (Leb|I i ) = Leb|I σ(i) .

Thus there exists an integer s such that b s (Leb|I i ) = Leb|I i .

The spectrum of an irrational rotation R α viewed as an IET of I is Sp(R α , I, Leb|I) =< e 2iπ α l > where l = |I|.

A consequence of the identity b s a m s b -s = a n s and Lemma 9.1 is that b s |I i sends the generator of Sp(a m s , I i , Leb|I i ) into a generator of Sp(a n s , I i , Leb|I i ). Then e .

Finally, αi li m s = ± αi li n s (mod Z). This is a contradiction since αi li / ∈ Q and |m| = |n|.

Therefore a has finite order, hence any action of BS(m, n) with |m| = |n| by elements of A is not faithful.

Proof of Corollary 2.

Let G be a nilpotent subgroup of A. Suppose by absurd that G is either non abelian torsion free or finitely generated and not virtually abelian.

Since G is nilpotent there exist u, v ∈ G such that c = [u, v] commutes with u and v. Furthermore, we can choose c of infinite order because G is either non abelian torsion free or finitely generated and not virtually abelian. This implies Claim 9.1. For any integers p and q, it holds that [u p , v q ] = c pq .

In particular c n 2 = [u n , v n ], so c is distorted. Hence, by Theorem 1, eventually passing to a power of c and conjugating by an element of A we can suppose that c is a product of infinite order restricted rotations R αi of disjoint supports I i . We denote c = Therefore u su |I i and v sv |I i are IET which commute with the rotation R αi . Finally, by Lemma 5.1 of [START_REF] Novak | Discontinuity growth of interval exchange maps[END_REF], u su |I i and v sv |I i are rotations so they commute.

According to Claim 9.1, [u su , v sv ] = c susv , so c susv = Id on I i for any i = 1, ..., p. It follows that c susv = Id. This contradicts that c has infinite order.

Corollary 1 .

 1 The Baumslag-Solitar groups BS(n, m) =< a, b | ba n b -1 = a m > with m, n integers and |m| = |n| do not act faithfully via elements of A.

Lemma 4 .

 4 1 follows immediately from this and Property 4.

  bounded, by Sublemmas 4.1 and 4.2, for all m ≥ N + 1 the point f m (a) = f m + (a) = f m -(a) and does not belong to BP 1 (f ) (by Property 5 (2)

Lemma 4 .

 4 2 follows immediately from this and Property 4.

4. 2

 2 Characterization of distortion.

Proposition

  (a)) = 0 by formula (1) with n = 2N +2, k = N +1-l. It follows that either f -(N +1)+l (a) ∈ BP 0 (F ) and f l (a) ∈ BP 0 (F ) or f -(N +1)+l (a) / ∈ BP 0 (F ) and f l (a) / ∈ BP 0 (F ). So BP 0 (F ) is a finite union of pairs of the form (f -(N +1)+l (a), f l (a)).

σ

  i = 1 such a H does not exist since it should satisfy that λ p+1 = σ 0 ...σ p λ 1 and σ p+1 = λp+1 λ1 .

Lemma 9 . 1 .

 91 Let a, b ∈ A with a = p i=1 (R αi , I i ) and ba m b -1 = a n then there exists an integer s such that b s maps I i to itself preserving the Lebesgue measure on I i , denoted by Leb|I i .

(

  R αi , I i ). Applying Lemma 9.1 with m = n, a = c and b = u [resp. b = v], there exists s u [resp. s v ] such that u su (Leb|I i ) = Leb|I i and v sv (Leb|I i ) = Leb|I i .

Introduction.In the recent years, notions of distortion have attracted the interest of many people working on geometric group theory as well as rigidity theory (see[START_REF] Franks | Distortion in groups of circle and surface diffeomorphisms. Dynamique des difféomorphismes conservatifs des surfaces : un point de vue topologique[END_REF] for a survey).On one hand, some results established the existence of distorted elements in transformations groups. For instance, D. Calegari and M. Freedman, in[START_REF] Calegary | Distortion in transformation groups. With an appendix by Yves Du Cornulier[END_REF], showed that all homeomorphisms of spheres are distorted. Moreover, in the case of the unit circle, they proved that every irrational Euclidean rotation is distorted inside the group of C

2-ε -diffeomorphisms for any ε > 0. Requiring smoothness, Avila proved in[START_REF] Avila | Distortion elements in Diff ∞ (R/Z)[END_REF] that irrational rotations are distorted in Diff ∞ (S 1 ). In higher dimensions, Militon (see[START_REF] Militon | Éléments de distorsion de Dif f ∞ 0 (M )[END_REF], Theorem 1) showed that irrational translations of the d-dimensional torus are distorted in Diff ∞ (T d ).On the other hand, a significant consequence of the non existence of distortion is the proof of Zimmer's conjecture in dimension 2: "any action of SL(3, Z) by area preserving diffeomorphisms on a surface has finite image". For instance,

Liousse, I. & Navas, A. Distortion elements in P L + (S 1 ) (2008).

We first check that the conjugating maps H, E and S are in A Q . By the definition of S, the break points of S are endpoints of connected components of P er(f ) so they belong to the f -orbit of BP 0 (f ) which is contained in Q. Therefore S ∈ A Q According to Remark 5, E ∈ A Q and the endpoints of the I i 's are rational. Hence, by Remark 8, H ∈ A Q .

Therefore (HES)f pq (HES) -1 ∈ A Q and then R i ∈ A Q . This is a contradiction.

Proof of Corollary 3.

In this section we prove that there is no distortion element in A Q .

Let f ∈ A Q distorted in A Q , by Theorem 1, there exist a positive integer s and

of infinite order restricted rotations of disjoint supports I i .