Bruno Yun
email: bruno.yun@abdn.ac.uk

Nir Oren

Representing Pure Nash Equilibria in Argumentation

Keywords: Argumentation, Game theory, Nash equilibrium, Pure strategy

In this paper we describe an argumentation-based representation of normal form games, and demonstrate how argumentation can be used to compute pure strategy Nash equilibria. Our approach builds on Modgil's Extended Argumentation Frameworks. We demonstrate its correctness, prove several theoretical properties it satisfies, and outline how it can be used to explain why certain strategies are Nash equilibria to a non-expert human user.

Introduction

Game theory studies how multiple rational decision-makers should act given interactions between their strategies, and preferences over the resultant outcomes. Game theory has been applied to myriad fields [START_REF] Matsumoto | Game Theory and Its Applications[END_REF]. Within game theory, decision-makers (referred to as players), their strategies, preferences and outcomes are represented within a game, and the solutions to a game identify some form of rational outcome. One such solution concept is that of a dominant strategy, where a player has a strategy or a set of strategies that will always result in the best outcome for them, regardless of what other players do. However, such dominant strategies often do not exist. In this work, we consider instead the notion of a Nash equilibrium, which identifies optimal strategies given that other players also pursue their own optimal strategies. Such Nash equilibria therefore represent a form of best response, and provide a well understood solution concept in game theory. However, finding Nash equilibria is computationally difficult, and it is sometimes difficult for a non-expert to understand why a given strategy is (or is not) a Nash equilibrium. We believe that by providing an argumentation-based representation of games, dialogues can be used to explain a Nash equilibrium to such non-experts. While work such as [START_REF] Fan | On the Interplay between Games, Argumentation and Dialogues[END_REF] has considered game theory in the context of ABA, to our knowledge, this work is the first to link abstract argumentation and Nash equilibria. We consider only so-called pure strategies for normal form games and intend to relax this restriction in future work.

The remainder of the paper is structured as follows. In Section 2, we provide a brief overview of argumentation and game-theory concepts necessary to understand our article. In Section 3, we describe how a normal form game can be encoded using argumen-tation. Section 4 examines some formal properties of our approach. Lastly, we discuss related and future work in Section 5 before concluding.

Background

We begin by providing the necessary background in game theory and argumentation required for the rest of the paper.

Game Theory

In this paper, we use the usual normal form for games [START_REF] Osborne | Introduction to Game Theory: International Edition[END_REF].

Definition 1. (Normal Game) A (normal) game is G = (Ag, Ac, Av, Ou, Ef, ≤) where Ag = {0, 1, . . . , n} is a finite set of players; Ac is a finite set of strategies; Av = [Ac 0 , . . . , Ac n ] with Ac i ⊆ Ac denoting the strategies available to i; Ou = {o 0 , . . . , o m } is a set of possible outcomes; Ef : Ac n → Ou n captures the consequences of the joint strategies for each player; and ≤= [≤ 0 , . . . , ≤ n ] with ≤ i ⊆ Ou×Ou denoting the preference relation for player i.

The notation o

k ≤ i o l means that player i prefers outcome o l to o k . As commonly done, we write o i < i o j iff o i ≤ i o j and o j ≤ i o i 2 .
A pure strategy profile S is a tuple containing one strategy from each player in the game. The set of all such pure strategy profiles is S G = Π i∈Ag Ac i , and represents one joint strategy of all players. A partial strategy profile is a tuple containing a single strategy for a subset of the players. Given any pure strategy profile S = [s 0 , . . . , s n ], we write S -i to denote the partial strategy profile [s 0 , . . . , s i-1 , ∅, s i+1 , . . . , s n ], where the strategy for player i is not specified. We then write S -i ⊕ s i to denote strategy profile S. With a slight abuse of notation, for any S, S ∈ S G we write that S ≤ i S iff Ef (S) i ≤ i Ef (S ) i3 .

Example 1. Let us consider the stag hunt game G = ({0, 1}, Ac, Av, Ou, Ef, ≤), where Ac = {stag, hare}, Av = [Ac, Ac], Ou = {4, 3, 2, 1}, ≤ is the standard less than relation over numbers. Table 1a graphically illustrates this game in normal form, and specifies Ef . For example, the tuple (1, 3) in the column "hare" and row "stag" means that Ef ([stag, hare]) = (1, 3). Given the pure strategy profile S = [stag, hare],

S -0 = [∅, hare] and S -0 ⊕ hare = [hare, hare]. Here [stag, hare] ≤ 0 [hare, hare] because (1, 3) 0 ≤ 0 (2, 2) 0 but [hare, hare] ≤ 1 [stag, hare].
In asking why a player should pursue a some strategy, we must take into account the strategies of others. A Nash equilibrium is the best response a player can make given optimal play by all other players. Definition 2. Let G = (Ag, Ac, Av, Ou, Ef, ≤), we say that S ∈ S G is a Nash equilibrium if for every i ∈ Ag and for any strategy s ∈ Ac i , it holds that S -i ⊕ s ≤ i S. A simple algorithm to identify all Nash equilibrium in the presence of pure strategies involves iterating through every player and identifying the best strategy profile (in terms of Ef for that player) given all other players' possible joint strategies. Any strategy profile which all players consider best is then a Nash equilibrium.

Given a game in normal form, the above algorithm involves -for a two player game -scanning down each column and marking the best strategy for the row player, and then doing the same for each row marking the best strategy for the column player. Each cell marked for both players is a Nash equilibrium. In the remainder of this paper, we show an argumentation-based alternative.

Argumentation

We encode normal form games in terms of arguments and attacks by building on Modgil's Extended Argumentation Frameworks (EAF) [START_REF] Modgil | Reasoning about preferences in argumentation frameworks[END_REF]. We say that: We will use the notation Ext s (AS) (resp. Ext p (AS)) to denote the set of all stable (resp. preferred) extensions.

Definition 3. An Extended Argumentation Framework is a triple A, C, D where A is a set of arguments, C ⊆ A × A, D ⊆ A × C and if (z, (x, y)), (z , (y, x)) ∈ D then (z, z ), (z , z) ∈ C.
• E is conflict-free iff for every x, y ∈ E, if (y, x) ∈ C then (x, y) ∈ C,

Argumentation-based approach for games

We consider an argumentation framework with multi-level arguments. At the base level, we consider all possible strategy profiles as arguments. Since only a single strategy profile can ever occur (as players execute one set of strategies in the interaction), every argument at this level must attack every other argument. We refer to such arguments as game-based arguments, and note that they are equivalent to pure strategy profiles.

Definition 6 (Game-based argument). Let G = (Ag, Ac, Av, Ou, Ef, ≤) be a game, a game-based argument (w.r.t. G) is a pure strategy profile S ∈ S G .
The set of all game-based arguments for a game G is denoted by A g (G).

Next, we introduce preference arguments. Intuitively, these can be interpreted as statements of the form: "Given that the other players are performing a given set of strategies, the remaining player's preferred strategy should be playing x". Definition 7 (Preference argument). Let G = (Ag, Ac, Av, Ou, Ef, ≤) be a game, S ∈ S G be a pure strategy profile and i ∈ Ag. A preference argument (w.r.t. G) is a tuple (S -i , s), where s ∈ Ac i .

The set of preference arguments for a game G is denoted by A p (G). A cluster of preference arguments is a maximal set of preference arguments sharing the same partial strategy profile.

Finally, we introduce valuation arguments, which can be interpreted as statements of the form: "Given that the other players are performing a given set of strategies, it is the case that the outcome of strategy s is better than the outcome of strategy s for the remaining player".

Definition 8 (Valuation argument). Let G = (Ag, Ac, Av, Ou, Ef, ≤) be a game, i ∈ Ag, (S -i , s), (S -i , s ) ∈ A p (G) be two preference arguments and S -i ⊕ s < i S -i ⊕ s. A valuation argument (w.r.t. G) is the pair (S -i , s < s).

The set of valuation arguments for a game G is denoted by A v (G).

Example 3 (Cont'd). The sets of game-based, preference and valuation arguments w.r.t. G are shown in Table 2. The argument a 1 represents the case where player 0 chooses to hunt a stag and player 1 chooses to hunt a hare. The argument a 9 represents the argument: "Given that player 0 chooses to hunt a hare, player 2's preferred strategy should be to hunt a stag". The argument a 16 represents the argument: "Given that player 1 chooses to hunt a hare, the outcome of hunting a hare is better than the outcome of hunting a stag for player 0".

Game-based arguments

Preference arguments Valuation arguments We now turn our attention to attacks. We note that preference and valuation arguments provide reasons why one argument should not attack another, and therefore introduce not only attacks between arguments, but also attacks on attacks.

a 1 = [stag, hare] a 5 = ([stag, ∅], stag) a 13 = ([stag, ∅], stag > hare) a 2 = [stag, stag] a 6 = ([stag, ∅], hare) a 14 = ([∅, stag], stag > hare) a 3 = [hare, stag] a 7 = ([∅, stag],
Definition 9 (Attack). For a game G = (Ag, Ac, Av, Ou, Ef, ≤), a 1 , a 2 ∈ A g (G), a 3 = (S 1 , s 2 ), a 4 = (S 3 , s 4 ) ∈ A p (G) and a 5 = (S 5 , s 6 > s 7 ) ∈ A v (G).
We say that:

• a 1 attacks a 2 , denoted (a 1 , a 2 ) ∈ C r (G), iff a 1 = a 2 . • a 3 attacks a 4 , denoted (a 3 , a 4 ) ∈ C p (G), iff S 1 = S 3 and s 2 = s 4 . • a 3 attacks (a 1 , a 2 ) ∈ C r (G), denoted by (a 3 , (a 1 , a 2 )) ∈ C u (G), iff there exists s ∈ Ac such that S 1 ⊕ s = a 1 and S 1 ⊕ s 2 = a 2 . • a 5 attacks (a 3 , a 4 ) ∈ C p (G), denoted by (a 5 , (a 3 , a 4 )) ∈ C v (G), iff S 5 = S 3 , s 6 = s 4 and s 7 = s 2 .
The first attack captured within Definition 9 is between every two distinct gamebased arguments. As each player has to choose exactly one strategy, different strategy profiles are clearly incompatible. The second bullet point represents attacks between preference arguments. In the stag hunt example for instance, a 5 attacks a 6 (and viceversa) because in the event of player 0 hunting a stag, player 1 can either hunt a stag or a hare. The third type of attack captures attacks from preference arguments to attacks between game-based arguments. Within the stag hunt, a 5 attacks (a 1 , a 2 ) because a 5 states that it is preferable for player 1 to hunt a stag when player 0 is also hunting a stag. Note that in general, the preference argument (S 1 , s 2 ) attacks all attacks against the game-based argument S 1 ⊕ s 2 coming from any other game-based arguments of the form S 1 ⊕ s , for any s ∈ Ac such that s = s 2 . The last type of attack captures attacks from valuation arguments to attacks between preference arguments. Returning to the stag hunt, a 13 attacks (a 6 , a 5 ) as a 13 states that the strategy "hunt a stag" is better than the strategy "hunt a hare" for player 1 when player 0 is hunting a stag.

The arguments and attacks induce a very specific type of extended argumentation framework, where object-level (game-based) arguments have their attacks attacked by meta-arguments (preference arguments) at level one, and where attacks between these meta-arguments are attacked by meta-arguments at level two (valuation arguments).

Definition 10 (Argumentation framework). Let G be a game. The argumentation framework corresponding to G is the tuple For our framework to be an EAF, it must satisfy some constraints, as described in [START_REF] Modgil | Labellings and games for extended argumentation frameworks[END_REF], and we can easily show that this is the case.

AS G = (A, C, D) where A = A g (G) ∪ A p (G) ∪ A v (G), C = C r (G) ∪ C p (G) and D = C u (G) ∪ C v (G).
Proposition 1. Let G be a game and AS G = (A, C, D) be the corresponding argumentation framework, it holds that if (z, (x, y)), (z , (y, x)) ∈ D then (z, z ), (z , z) ∈ C.

Proof. There are only two types of attacks on attacks: (1) attacks coming from valuation arguments to attacks between preference arguments and (2) attacks coming from preference arguments to attacks between game-based arguments. In the rest of this proof, we prove that Proposition 1 is satisfied for the two types of attacks on attacks.

• Considering (1), for a fixed partial strategy profile S i , and fixed strategies s j , s k ∈ Ac, there is exactly one (or no) valuation argument of the form (S i , s j > j k ) or (S i , s k > s j ). As a result, the condition in Proposition 1 is trivially satisfied for attacks coming from valuation arguments.

• We now study the case (2) and show that Proposition 1 is also satisfied for attacks coming from preference arguments on attacks between game-based arguments. Assume that (a 3 , (x, y)), (a 4 , (y, x)) ∈ D, where a 3 = (S 1 , s 2 ), a 4 = (S 1 , s 4 ), x = S 1 ⊕ s 4 and y = S 1 ⊕ s 2 . By Definition 9, s 2 = s 4 thus (a 3 , a 4 ), (a 4 , a 3 ) ∈ C p (G) ⊆ C.

Since -given Proposition 1 -our argumentation system is an EAF, we can use EAF semantics to evaluate it.

Example 5 (Example 4 Contd). In our running example, a 5 defeats a 6 w.r.t. A as (a 5 , a 6 ) ∈ C and there is no argument z ∈ A such that (z, (a 5 , a 6 )) ∈ D. However, a 6 does not defeat a 5 w.r.t. A because (a 13 , (a 6 , a 5 )) ∈ D. All extensions contain arguments {a 16 , a 15 , a 14 , a 13 , a 12 , a 10 , a 7 , a 5 }, while one preferred extension contains {a 2 } and the other contains {a 4 }.

System Properties

Having described our system, we now consider its properties. The most important result we seek to show is the correspondence between argumentation semantics and Nash equilibria, and we begin by laying the groundwork for this. We then consider how many arguments will be generated for an arbitrary normal form game. We begin by considering which preference arguments will appear in a preferred extension. This result is used in later proofs.

Lemma 1. Let G = (Ag, Ac, Av, Ou, Ef, ≤) be a game, and AS G be the corresponding AS. For each preferred extension E of AS G , for each cluster C of preference arguments, there exists a unique argument c ∈ C such that c ∈ E.

Proof. Assume a partial strategy profile S = [s 0 , . . . , s i-1 , ∅, s i+1 , s n ] and the corresponding cluster of preference arguments C. Because our preferences are acyclic, we know that there exists a strategy s * such that for every s ∈ Ac i , S ⊕ s ≤ i S ⊕ s * . From the definition of the valuation argument, there are no valuation arguments attacking the attacks from the preference argument (S, s * ) to other preference arguments. As a result, we conclude that (S, s * ) is in a preferred extension E and that all the other preference in C are not E. Moreover, you need to choose one of such arguments from the cluster C for each preferred extension to satisfy the maximality condition of the semantics.

Next, we show that if there is a preferred extension with game-based arguments, then each such extension has exactly one game-based argument.

Lemma 2. If any preferred extension of AS G contains a game-based argument, then it contains exactly one game-based argument.

Proof. Let E be a preferred extension containing game-based arguments. We prove by contradiction that it is not possible for E to have more than one game-based argument. Assume that E contains two game-based arguments a 1 and a 2 . By definition of the attack relation, there is a symmetric attack between a 1 and a 2 . Hence there must exist two preference arguments p 3 and p 4 with (p 3 , (a 1 , a 2 )), (p 4 (a 2 , a 1 )) ∈ D and (p 3 , p 4 ), (p 4 , p 3 ) ∈ C. It is not possible for both (p 4 , p 3 ) and (p 3 , p 4 ) to be attacked by valuation arguments as this would require an inconsistency or cycle in ≤. By this observation, E contains only p 3 or p 4 . Hence, {a 1 , a 2 } is not conflict-free, contradiction.

We now show that a game-based argument which is not a Nash equilibrium will not appear in any preferred extension of the associated argumentation system. Lemma 3. Let G = (Ag, Ac, Av, Ou, Ef, ≤) be a game, and AS G be the corresponding AS. If S ∈ S G such that S is not a Nash equilibrium then for every preferred extension E, S / ∈ E.

Proof. Assume there is a non-Nash equilibrium game-based argument S = [s 0 , . . . , s n ] in a preferred extension E. Then, from Lemma 2, E does not contain any other gamebased arguments. Since S is not a Nash equilibrium, there exists i ∈ Ag and s ∈ Ac i such that S -i ⊕ s i < i S -i ⊕ s. In the rest of this proof, we consider the strategy s * such that for every s ∈ Ac i , S -i ⊕ s ≤ i S -i ⊕ s * . By definition, the attack from S to S -i ⊕ s * is attacked by the preference argument (S -i , s * ). Moreover, the preference argument (S -i , s * ) attacks all the other preference arguments (S -i , s ), where s ∈ Ac i and s = s. By definition of the valuation arguments, none of the attacks from (S -i , s * ) to those other preference arguments is defeated. As a result, we conclude that there is preferred extension that contains (S -i , s * ). Corollary 1. Let G = (Ag, Ac, Av, Ou, Ef, ≤) be a game, and AS G be the corresponding AS. If E is a preferred extension that contains a game-based argument S, then S is a Nash equilibrium.

Let s + = {s ∈ Ac i | S -i ⊕ s ≤ i S -i ⊕ s * and S -i ⊕ s * ≤ i S -i ⊕ s},
In the next proposition, we show that if there is only one preferred extension that contains a game-based argument, then there is an equivalence between preferred and stable extensions. Proof. We show that if a preferred extension possesses a game-based argument, then it is also a stable extension. Assume E contains a single game-based argument. By Lemma 2, E contains exactly one game-based argument. Therefore, all game-based arguments not in the extension are defeated by the game-based argument within the extension with respect to E, meaning that the game-based argument is a member (at the game-based level) of the stable extension.

It may seem intuitive that the preferred and stable extension should coincide where multiple preferred extensions exist. However, this is not the case, as demonstrated by the following counter-example. Example 6. Consider the matching pennies game G = (Ag, Ac, Av, Ou, Ef, ≤) where Ag = {0, 1}, Ac = {heads, tails}, Av = [Ac, Ac], Ou = {1, -1}, ≤ is defined as the "less-than relation" for each player, and Ef is defined in Table 1b.

The set of arguments is A = {b Furthermore, even when multiple preferred extensions exist, these may not coincide with the stable extensions. 3. Arguments for the matching pennies game Player 0 Player 1 4. Three strategy variant of the matching pennies game.

heads tails edge heads -1 1 1 -1 1 -1 tails 1 -1 -1 1 -1 1 edge 1 -1 -1 1 -1 1 Table
Example 7. Let us consider the following variant of the matching pennies game with three strategies for each player. We have G = (Ag, Ac, Av, Ou, Ef, ≤) where Ag = {0, 1}, Ac = {heads, tails, edge}, Av = [Ac, Ac], Ou = {1, -1}, ≤ is defined as the "less-than" relation for numbers for each player, and Ef is defined in Table 4. This variant of the game has eight distinct preferred extensions, but none contain any gamebased arguments.

We now turn to our main result, namely the equivalence of the Nash equilibrium with the game-based arguments found in the preferred extensions.

Proposition 3 (Equivalence). Let G = (Ag, Ac, Av, Ou, Ef, ≤) be a game, and AS G be the argument framework for the game. A strategy profile S = [s 0 , . . . , s n ] ∈ S G is a Nash equilibrium iff there exists E ∈ Ext p (AS G ) such that S ∈ E.

Proof. We split this proof in two parts:

(⇒) We need to show that if S is a Nash equilibrium, then it is within a preferred extension of AS G . Let us consider the set of arguments

E = {S} ∪ A v (G) ∪ {(S -i , s i ) | i ∈ Ag}.
We now show that E is a preferred extension of AS G . It is clear that E is conflict-free as for every x, y ∈ E, (x, y) / ∈ C. Every argument in A v (G) is acceptable w.r.t. E as valuation arguments are not attacked. Every argument a = (S -i , s i ) is also acceptable w.r.t. E because for every s ∈ Ac i and s = s i , the attacks from a = (S -i , s ) to a, is either not a defeat w.r.t. E (if there is a valuation argument that attacks (a , a)) or it is a defeat but a is defeated by a w.r.t. E. The argument S is also acceptable w.r.t. E because for every S ∈ S G and S = S, the attack from S to S is not a defeat w.r.t. E as the arguments (S -i , s i ) are attacking those attacks. We conclude that the set E is admissible.

Following Lemma 2 and 1, we conclude that E is maximal for set inclusion as it contains all the valuation arguments, one preference argument per cluster and exactly one game-based argument. (⇐) We need to show that if S is within a preferred extension, then S is a Nash equilibrium. This follows directly from the result from Corollary 1.

Returning to the stable extensions, the following result shows that there is a one-toone correspondence between the sets of Nash equilibria and the set of classes of stable extensions 4 , where each Nash equilibrium S corresponds to the class of stable extensions containing argument S. Finally, we consider how many arguments an argumentation system containing representing a normal form game will contain. Proof. The proof is split into three parts.

1. Suppose n players and m strategies per player. Each game-based argument corresponds to a pure strategy profile, i.e., there are m n game-based arguments. 2. Consider the number of the preference arguments. There are m n-1 • n partial strategy profiles. Roughly speaking, a preference argument is obtained from a partial strategy profile by replacing the empty set with a strategy. Hence, there are up to m n-1 • n • m = m n • n preference arguments. 3. We estimate the number of valuation arguments. Each valuation argument is obtained from one partial strategy profile and one pair of different strategies.

There are m n-1 • n partial strategy profiles and up to m • (m -1) pairs of different strategies. Furthermore, if a strategy x is preferred to strategy y, then y is not preferred to x. Thus, there are up to m•(m-1) 2 possible combinations to consider. Hence, the total number of valuation arguments is limited by

m n-1 •m•(m-1)•n 2 which is in O(m n+1 • n). Thus, the total number of arguments is in O(m n ) + O(m n • n) + O(m n+1 • n) which is in O(m n+1 • n).
We note that computing Nash equilibria is known to be computationally difficult, and the result regarding the number of arguments is therefore unsurprising.

Discussion, Related and Future Work

In this paper, we described how normal form games can be given an argumentation-based interpretation so as to allow -via argumentation semantics -for pure Nash equilibria to be computed. Intuitively, a Nash equilibrium identifies the best strategy a player can pursue given others' strategies. However, explaining -to a non-expert -why some set of strategies forms a Nash equilibrium is often difficult, and our argument-based interpretation is the first step towards an explanatory dialogue for such explanation. Other work has shown the utility of providing such dialogue-based explanations [START_REF] Caminada | Scrutable plan enactment via argumentation and natural language generation[END_REF][START_REF] Kristijonas | Explanation for Case-Based Reasoning via Abstract Argumentation[END_REF][START_REF] Oren | Argument-Based Plan Explanation[END_REF]. In the current context, such an explanation could build on Modgil's proof dialogues for extended argumentation frameworks [START_REF] Modgil | Labellings and games for extended argumentation frameworks[END_REF], and could result in a dialogue as follows for the Stag hunt game shown in Figure 1.

User "Why should both players hunt a stag?" (why a 2 ?) System "It is the best response because a 2 defeats all the other game-based arguments, namely a 1 , a 3 and a 4 ".

Assume now that the user agrees that a 2 defeats a 3 and a 4 ; hence they ask further about why a 2 defeats a 1 .

User: "Why should player 1 play stag if player 0 plays stag?" (why does a 2 defeats a 1 ?) System: "Because playing stag gives a better outcome to player 1 if player 0 plays stag" (a 5 defeats the attack (a 1 , a 2 )) User: "Why does player 1 not prefer the outcome when hare is played"? (why not a 6 )? System: "Because of the valuation defined for player 1" (a 13 ) User: "I understand."

In the short term, we intend to formalise the dialogue and empirically evaluate its explanatory capability with human subjects. Other extensions which we intend to investigate include providing an argumentation semantics for mixed Nash equilibria (perhaps through the use of some form of ranking semantics [START_REF] Amgoud | Ranking Arguments With Compensation-Based Semantics[END_REF][START_REF] Bonzon | A comparative study of ranking-based semantics for abstract argumentation[END_REF][START_REF] Matt | A Game-Theoretic Measure of Argument Strength for Abstract Argumentation[END_REF]), and investigating other solution concepts (e.g., Pareto optimality) for more complex types of games. Finally, there are clear links between game theory and group-based practical reasoning. Building on work such as [START_REF] Atkinson | Argument schemes reasoning about the actions of others[END_REF][START_REF] Shams | Argumentation-based reasoning about plans, maintenance goals, and norms[END_REF], we intend to investigate how an argument-based formulation to practical reasoning underpinned by game theory can be created.

Several other authors have investigated some links between game theory and argumentation. For example, in his seminal paper, Dung [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] noted that the stable extension corresponds to the stable solution of an cooperative n-person game, but did not seem to deal with non-cooperative games as we do here. Game theory was also used to describe argument strength by Matt and Toni [START_REF] Matt | A Game-Theoretic Measure of Argument Strength for Abstract Argumentation[END_REF], and Rahwan and Larson [START_REF] Rahwan | Argumentation and Game Theory[END_REF] investigated the links between argumentation and game theory from a mechanism design point of view. Perhaps most closely related to the current work is Fan and Toni's work [START_REF] Fan | On the Interplay between Games, Argumentation and Dialogues[END_REF] exploring the links between dialogue and assumption-based argumentation (ABA). Here, the authors showed how admissible sets of arguments obtained from their ABA constructs are equivalent to Nash equilibria. In contrast to the current work, they only considered two player games and utilised structured argumentation, allowing them to describe a proof dialogue with associated strategies.

Conclusions

In this paper, we provided an argumentation-based interpretation of pure strategies in normal form games, demonstrating how argumentation semantics can be aligned with the Nash equilibrium as a solution concept, and examining some of the argumentation system's properties.

We believe that this work has significant application potential in the context of argument-based explanation. At the same time, we recognise that there are significant open avenues for research in this area, but believe that the current work is an important step in investigating the linkages between the two domains.

Example 2 (

 2 Cont'd). There are two Nash equilibria in the stag hunt game: [stag, stag] and [hare, hare]. The strategy profile [stag, stag] is a Nash equilibrium because [hare, stag] ≤ 0 [stag, stag] and [stag, hare] ≤ 1 [stag, stag]. Similarly, [hare, hare] is also a Nash equilibrium as [stag, hare] ≤ 0 [hare, hare] and [hare, stag] ≤ 1 [hare, hare].

Definition 4 (

 4 Defeat). Let AS = (A, C, D) be an EAF, x, y ∈ A and Y ⊆ A. We say that y defeats x w.r.t. Y , denoted y → Y x iff (y, x) ∈ C and there is no z ∈ Y s.t. (z, (y, x)) ∈ D. Definition 5 (Argumentation semantics). Let AS = (A, C, D) be an EAF and E ⊆ A.

Figure 1 .

 1 Figure 1. Argumentation graph corresponding to stag hunt game

Figure 1

 1 represents the game-based, preference and valuation arguments of G using blue, yellow and green nodes respectively. The attacks between arguments (C) and on attacks (D) are represented using solid black arrows and dashed red arrows respectively.

Proposition 2 .

 2 Let G be a game and AS G = (A, C, D) be the corresponding argumentation framework. If E ∈ Ext p (AS G ) and E ∩ A g (G) = ∅ then E ∈ Ext s (AS G ).

  1 , b 2 , b 3 , . . . , b 16 } and are listed in Table 3. There is only one preferred extension {b 16 , b 15 , b 14 , b 13 , b 12 , b 10 , b 8 , b 6 } but no stable extensions.

Game-based arguments Preference arguments Valuation arguments b 1 =

 1 [heads, heads] b 5 = ([heads, ∅], heads) b 13 = ([heads, ∅], tails > heads) b 2 = [heads, tails] b 6 = ([heads, ∅], tails) b 14 = ([∅, tails], tails > heads) b 3 = [tails, tails] b 7 = ([∅, tails], heads) b 15 = ([tails, ∅], heads > tails) b 4 = [tails, heads] b 8 = ([∅, tails], tails) b 16 = ([∅, heads], heads > tails) b 9 = ([tails, ∅], tails) b 10 = ([tails, ∅], heads) b 11 = ([∅, heads], tails) b 12 = ([∅, heads], heads) Table

Corollary 2 .

 2 Let G = (Ag, Ac, Av, Ou, Ef, ≤) be a game, and AS G be the corresponding EAF. There is a bijection betweenY = {S ∈ S G | S is a Nash equilibrium} and {{E ∈ Ext s (AS G ) | S ∈ E} | S ∈ Y }Proof. Follows directly from Proposition 3 and Proposition 2.

Proposition 4 (

 4 Number of arguments). Let G = (Ag, Ac, Av, Ou, Ef, ≤) be a game s.t. |Ag| = n and m = max i∈Ag |Ac i |, the number of arguments in AS G is in O(m n+1 • n).

Table 1 .

 1 Two games in normal form.

  A is acceptable w.r.t. E iff for every y ∈ A s.t. y → E x, there exists z ∈ E s.t. z → E y and there exists R E = {x 1 → E y 1 , . . . , x n → E y n } s.t. for every i ∈ {1, . . . , n}, x i ∈ E, z → E y ∈ R E and for every x j → E y j ∈ R E , for every y s.t. (y , (x j , y j )) ∈ D, there exists x → E y ∈ R E • E is an admissible extension iff every argument in E is acceptable w.r.t. E

	and there • x ∈ • E is a preferred extension iff E is a maximal (w.r.t. ⊆) admissible extension exists z ∈ E s.t. (z, (y, x)) ∈ D. • E is a stable extension iff for every y / ∈ E, there exists x ∈ E such that x → E y.

Table 2 .

 2 Arguments for the stag hunt game

	stag)	a 15 = ([hare, ∅], hare > stag)

We assume that our preferences are acyclic. I.e., if a < i b < i c then c ≤ i a.

The notation Ef (S ) i means the i-th element of Ef (S ).


Fabrizio Bianchi

PARABOLIC IMPLOSION FOR ENDOMORPHISMS OF C 2

We give an estimate of the discontinuity of the large Julia set for a perturbation of a class of maps tangent to the identity, by means of a two-dimensional Lavaurs Theorem. We adapt to our situation a strategy due to Bedford, Smillie and Ueda in the semiattracting setting. We also prove the discontinuity of the filled Julia set for such perturbations of regular polynomials.

Notation. The symbol O(x) will stand for some element in the ideal generated by x. More generally, given any f , O(f ) will stand for some element in the ideal generated by f . Analogously, O(f 1 , . . . , f k ) will stand for some element in the ideal generated by f 1 , . . . , f k .

The notation O 2 (x, y) will be a shortcut for O(x 2 , xy, y 2 ). Given a point p ∈ C 2 , we shall denote its coordinates by x(p) and y(p).

Introduction and results

Parabolic dynamics and the study of parabolic perturbations have been at the heart of holomorphic dynamics in the last couple of decades. Starting with the work by Lavaurs [START_REF] Lavaurs | Systemes dynamiques holomorphes: explosion de points périodiques paraboliques[END_REF], the theory of parabolic implosion has provided useful tools for a very precise control of these perturbations and for the proof of some of the most striking recent results in the field, e.g., the construction of Julia sets of positive area [START_REF] Buff | Quadratic Julia sets with positive area[END_REF][START_REF] Avila | Lebesgue measure of Feigenbaum Julia sets[END_REF], significative steps toward the setting of the hyperbolicity conjecture for quadratic polynomials [START_REF] Cheraghi | Satellite renormalization of quadratic polynomials[END_REF] and the construction of endomorphisms of P 2 (C) with a wandering domain [START_REF] Astorg | A two-dimensional polynomial mapping with a wandering Fatou component[END_REF]. In particular, these techniques have proved extremely useful in the study of bifurcation loci (see, e.g., [START_REF] Shishikura | The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets[END_REF]).

In several complex variables, the study of parabolic perturbations and the theory of parabolic implosion are just at the start, with recent results only in the semiattracting setting [START_REF] Bedford | Semi-parabolic bifurcations in complex dimension two[END_REF][START_REF] Dujardin | Stability and bifurcations for dissipative polynomial automorphisms of C 2[END_REF]. The goal of this paper is to provide a starting point for an analogous theory in the completely parabolic setting, by a precise study of perturbations of germs of endomorphisms of C 2 tangent to the identity at the origin.

Let us briefly recall the foundational results of the one-dimensional theory. We refer to [START_REF] Douady | L'ensemble de Julia dépend-il continûment du polynôme?[END_REF] for a more extended introduction to the subject, as well as to the original work by Lavaurs [START_REF] Lavaurs | Systemes dynamiques holomorphes: explosion de points périodiques paraboliques[END_REF]. Consider an endomorphism of C tangent to the identity at the origin, given by f (z) = z + z 2 + O(z 2 ). The origin is a parabolic fixed point for f . The dynamics is attracting near the negative real axis: there exists a parabolic basin B for 0, i.e., an open set of points converging to the origin after iteration. The origin is on the boundary of B, and the convergence happens tangentially to the negative real axis. The iteration of f on B is semiconjugated to a translation by 1. More precisely, there exists an incoming Fatou coordinate (unique up to postcomposition with a translation) ϕ ι : B → C such that, for every z ∈ B, we have ϕ ι • f (z) = f (z) + 1. One way to construct such a coordinate is to define it as the limit ϕ ι (z) := lim n (w ι 0 (f n (z)) -n). Here w ι 0 (z) := -1 z -q log(-z), where log is the principal branch of the logarithm and q + 1 is the coefficient of z 3 in the expression of f . We will consider this normalization in the sequel.

The same happens for the inverse iteration near the positive real axis: we have a repelling basin R of points converging to 0 under some inverse iteration, and the convergence happens tangentially to the positive real axis. We can construct in this case an outgoing Fatou parametrization, i.e., a map ψ o from a subset of C to R (unique up to precomposition with a translation) such that f • ψ o (z) = ψ o (z + 1). Here we will consider as ψ o the inverse of the map ϕ o arising as the limit ϕ 0 (z) := lim n (w o 0 (f -n (z)) + n), where w o 0 (z) := -1 z -q log(z). It is worth noticing here that the Date: January 2017.

union of B and R contains a full pointed neighbourhood of the origin. The map ψ o can also be extended to C, with image equal to C. Notice that the incoming Fatou coordinate is a map from the dynamical plane to C, while the outgoing Fatou parametrization is a map from C to the dynamical plane. In particular, given any α ∈ C and denoting by t α the translation by α on C, the composition L α := ψ o • t α • ϕ ι is well defined as a function from B to R. Such a map is usually called a Lavaurs map, or a transfer map.

We consider now the perturbation f ε (z) = z + (z 2 + ε 2 )(1 + O(z) + O(ε 2 )) of the system f , for ε real and positive. As ε = 0, the dynamics abruptly changes: the parabolic point splits in two (repelling) points ±iε, and the orbits of points in B can now pass through the "gate" between these two points. Using the Lavaurs map it is possible to give a very precise description of this phenomenon, by studying the dynamics of high iterates of the perturbed maps f ε , as ε → 0. The following definition plays a central role in this study.

Definition 1.1. Given α ∈ C, an α-sequence is a sequence (ε ν , n ν ) ν∈N ∈ (C × N) N such that n ν → ∞ and n ν -π εν → α as ν → +∞. Notice in particular that the definition of α-sequence implies that ε ν tends to the origin tangentially to the positive real axis. More precisely, there exists a constant c such that, for every ν sufficiently large, we have |Im ε ν | ≤ c |ε ν | 2 . The following result gives the limit description of suitable high iterates of f ε . Theorem 1.2 (Lavaurs [START_REF] Lavaurs | Systemes dynamiques holomorphes: explosion de points périodiques paraboliques[END_REF]). Let f ε (z) = z + (z 2 + ε 2 )(1 + O(z) + O(ε 2 )) and (ε ν , n ν ) be an α-sequence. Then f nν εν → L α , locally uniformly on B, as ν → +∞. One of the most direct consequences of Lavaurs theorem is the fact that the set-valued functions ε → J(f ε ) and ε → K(f ε ) are discontinuous at ε = 0 for the Hausdorff topology. Here J(f ε ) and K(f ε ) denote the Julia set and the filled Julia set of f ε , respectively (recall -see e.g. [START_REF] Douady | L'ensemble de Julia dépend-il continûment du polynôme?[END_REF] -that ε → J(f ε ) is always lower semicontinuous, while ε → K(f ε ) is always upper semicontinuous). More precisely, define the Lavaurs-Julia set J(f 0 , L α ) and the filled Lavaurs-Julia set K(f 0 , L α ) by

J(f 0 , L α ) := { z ∈ C : ∃m ∈ N, L m α (z) ∈ J(f 0 ) } K(f 0 , L α ) := { z ∈ C : ∃m ∈ N, L m α (z) / ∈ K(f 0 ) } c
Notice that the Lavaurs-Julia set J(f 0 , L α ) is in general larger than the Julia set of f 0 . On the other hand, the set K(f 0 , L α ) is in general smaller than K(f 0 ). The following Theorem then gives an estimate of the discontinuity of the maps ε → J(f ε ) at ε = 0.

Theorem 1.3 (Lavaurs [14]). Let f ε (z) = z + (z 2 + ε 2 )(1 + O(z) + O(ε 2 )) and (ε ν , n ν ) be an α-sequence. Then lim inf J(f εν ) ⊇ J(f 0 , L α ) and lim sup K(f εν ) ⊆ K(f 0 , L α )
In particular, at ε = 0, (1) the map ε → J(f ε ) is lower semicontinuous, but not continuous;

(2) the map ε → K(f ε ) is upper semicontinuous, but not continuous.

The goal of this paper is to make a step towards the generalization of Theorems 1.2 and 1.3 to the two-variables setting, by studying the perturbation of a class of maps tangent to the identity (i.e., with differential at a fixed point equal to the identity). More precisely, we consider an endomorphism of C 2 of the form

F 0 x y = x + x 2 • α 0 (x, y) y(1 + ρx + O(x 2 , xy, y 2 ))
,

where α 0 (x, y) = 1 + O(x, y) and ρ > 1. For instance, F 0 may be the local expression of an endomorphism of P 2 (e.g., if the two components of F 0 are polynomials of the same degree in (x, y) with 0 as the only common root of their higher-degree homogeneous parts). We shall primarily be interested in this situation. The map F 0 has a fixed point tangent to the identity at the origin, and two invariant lines { x = 0 } and { y = 0 }. By the work of Hakim [START_REF] Hakim | Transformations tangent to the identity, Stable pieces of manifolds[END_REF] (see Section 2), since ρ > 1 we know that [1 : 0] is a non-degenerate characteristic direction, and that there exists an open set B of initial conditions, with the origin on the boundary, such that every point in B is attracted to the origin tangentially to the direction [1 : 0]. Moreover there exists, on an open subset C 0 of B, a (one dimensional) Fatou coordinate ϕ ι , with values in C, such that

ϕ ι • F 0 (p) = ϕ ι (p) + 1 (see Lemma 2.

2).

A similar description holds for the inverse map. Indeed, after restricting ourselves to a neighbourhood U of the origin where F 0 is invertible, we can define the set R of point that are attracted to the origin tangentially to the direction [1 : 0] by backward iteration. There is then a well defined map

ϕ o : -C 0 ∩ U → C such that ϕ o • F 0 (p) = ϕ o (p) + 1
whenever the left hand side is defined. It is actually possible to construct two-dimensional Fatou coordinates (see [START_REF] Hakim | Transformations tangent to the identity, Stable pieces of manifolds[END_REF]), but we shall not need them in this work. In the following, we shall need some specific coefficients of F 0 . We thus write its expression as

(1.1) F 0 x y = x + x 2 (1 + (q + 1)x + ry + O(x 2 , xy, y 2 )) y(1 + ρx + O(x 2 , xy, y 2 )) ,
where ρ is real and greater than 1 and q, r ∈ C. Consider now a perturbation F ε of F 0 of the form (1.2)

F ε x y = x + (x 2 + ε 2 )α ε (x, y) y(1 + ρx + β ε (x, y)) = x + (x 2 + ε 2 )(1 + (q + 1)x + ry + O(x 2 , xy, y 2 ) + O(ε 2 )) y(1 + ρx + O(x 2 , xy, y 2 ) + O(ε 2 )) .
Our goal is to study the dependence of the large Julia set 1 J 1 (F ε ) on ε near the parameter ε = 0. Notice that the line {y = 0} is invariant for all the maps in the family, and that on this line we have a classical (1-dimensional) parabolic implosion phenomenon. Our main result is the following Theorem, which is a partial generalization of Theorem 1.2 to our setting. As in dimension 1, α-sequences play a crucial role. The set C 0 introduced above will be precisely defined in Proposition 2.1, and the Fatou coordinates ϕ ι and ϕ o in Lemma 2.2.

Theorem 1.4. Let F ε be a holomorphic family of endomorphisms of C 2 as in (1.2). Let B be the attracting basin for the origin for the map F 0 with respect to the characteristic direction [1 : 0]. Let α be a complex number and (n ν , ε ν ) be an α-sequence. Then the family F nν εν is normal on a neighbourhood of B ∩ { y = 0 } in B. In addition every limit L has the following properties:

• it is an open holomorphic map;

• it coincides with the 1-dimensional Lavaurs map L α on the invariant line {y = 0};

• it satisfies

(1.3) ϕ o • L(p) = α + ϕ ι (p)
whenever both sides are defined, where

ϕ ι : C 0 → C and ϕ o : -C 0 → C are suitably normalized (1-dimensional) Fatou coordinates for F 0 .
The reason we do not get the normality of the sequence on all the basin B is that, a priori, the neighbourhood where we prove the normality does not necessarily contain a fundamental domain for the dynamics. If this were the case, the result would easily extend to the full basin.

Remark 1.5. Computer experiments suggest that given any α-sequence (ε ν , n ν ) there is a neighbourhood of C 0 in C 0 such that the sequence F nν εν converges to a (unique) limit map L, without the need of extracting a subsequence. 1 i.e., the complement of the Fatou set, which in general is larger than the Julia set defined as the support of the equilibrium measure for endomorphisms of P 2 , see [START_REF] Dinh | Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings[END_REF].

As a consequence, we shall deduce an estimate of the discontinuity of the large Julia set in this context (notice that the discontinuity itself follows from an application of Theorem 1.2 to the invariant line { y = 0 }). We say that, given U ⊂ C 0 , a map L : U → C 2 is a Lavaurs map if there exists an α-sequence (ε ν , n ν ) such that F nν εν → L on U. We then have the following result (see Section 7 for the definition of the Lavaurs-Julia sets J 1 (F 0 , L) in this setting).

Theorem 1.6. Let F ε be a holomorphic family of endomorphisms of P 2 as in (1.2) and L : U → C 2 be a Lavaurs map such that F nν εν → L on U for some α-sequence (ε ν , n ν ). Then

lim inf J 1 (F εν ) ⊇ J 1 (F 0 , L).
Finally, in the last section, we consider a family of regular polynomials, i.e., polynomial endomorphisms of C 2 admitting an extension to P 2 (C). For these maps, it is meaningful to define the filled Julia set K as the set of points with bounded orbit. In analogy with the one-dimensional theory, we deduce from Theorem 1.4 an estimate for the discontinuity of the filled Julia set at ε = 0 (see Section 8 for the definition of the set K(F 0 , L)) and in particular deduce that ε → K(F ε ) is discontinuous at ε = 0. Notice that, differently from the case of the large Julia set, this is not already a direct consequence of the 1-dimensional theory.

Theorem 1.7. Let F ε be a holomorphic family of regular polynomial maps of C 2 as in (1.2) and L : U → C 2 be a Lavaurs map such that F nν εν → L on U for some α-sequence (ε ν , n ν ). Then K(F 0 , L) ⊇ lim sup K(F εν ).

Moreover, ε → K(F ε ) is discontinuous at ε = 0.
The paper is organized as follows. In Section 2 we recall the results by Hakim describing the local dynamics of the map (1.1) near the origin, and introduce the Fatou coordinates associated to the attracting and repelling basins. In Section 3 we define and study suitable perturbations of the Fatou coordinates, that allow to semiconjugate the iteration of F ε to a translation by 1, up to a controlled error. In Sections 4 and 5 we carefully study the orbits of points under iteration by F ε and prove some preliminary convergence result needed for the proof of Theorem 1.4, which is given in Section 6. In Section 7 and 8 we deduce from Theorem 1.4 the estimates of the discontinuity of the large Julia set and (for regular polynomials) of the filled Julia set at ε = 0.

Preliminaries and Fatou coordinates

Following the work of Hakim [START_REF] Hakim | Transformations tangent to the identity, Stable pieces of manifolds[END_REF] (see also [START_REF] Hakim | Analytic transformations of (C p , 0) tangent to the identity[END_REF][START_REF] Arizzi | On Ecalle-Hakim's theorems in holomorphic dynamics[END_REF]), we start giving a description of the local dynamics near the origin for F 0 by recalling some classical notions in this setting. Let Φ be a germ of transformation tangent to the identity at the origin of C 2 . We can locally write it near the origin as

Φ x y = x + P (x, y) + . . . y + Q(x, y) + . . . ,
where P and Q are homogeneous polynomials of degree 2. In the following, we shall always assume that P (x, y) is not identically zero. A characteristic direction is a direction V = [x : y] ∈ P 1 (C) such that the complex line through the origin in the direction [x : y] is invariant for (P, Q). The direction is degenerate if (P, Q) sends the line to the origin, non degenerate otherwise. The homogeneous part of degree 2 of Φ can be seen as a self map of the tangent space at the origin. The characteristic directions correspond to the invariant lines through the origin for this map. When a direction is non degenerate, the homogeneous map induces a holomorphic map defined on (a neighbourhood of this direction in) P 1 (C) and fixing this direction. The director (see [START_REF] Abate | Fatou flowers and parabolic curves[END_REF]Definition 2.4]) of this direction is defined as ρ -1, where ρ is the multiplier of the map at the fixed direction. Equivalently, one can take coordinates such that the characteristic direction is V = [1 : u 0 ]. Notice that the fact that [1 : u 0 ] is a characteristic direction is equivalent to u 0 being a zero of r(u) := Q(1, u) -uP (1, u). The director of the characteristic direction [1 : u 0 ] is thus equal to r (u 0 ) P (1, u 0 ) .

Given a germ Φ and a non degenerate characteristic direction V for Φ we can assume, without loss of generality, that V = [1 : 0] and that the coefficient of x 2 in P (x, y) is 1 (notice that Hakim has the opposite normalization, i.e., with the term -x 2 ). The following result by Hakim ([12, Proposition 2.6]) gives an explicit description of an invariant subdomain of the attracting basin B. In all this work, we will restrict ourselves to points belonging to such an invariant domain.

Proposition 2.1 (Hakim). Let Φ be a germ of transformation of C 2 tangent to the identity (normalized as above), such that V = [1 : 0] is a nondegenerate characteristic direction with director δ whose real part is greater than some 0 < α ∈ R. Then, if γ, s and R are small enough positive constants, every point of the set

C 0 (γ, R, s) := { (x 0 , y 0 ) ∈ C 2 : |Im x 0 | ≤ -γ Re x 0 , |x 0 | ≤ R, |y 0 | ≤ s |x 0 | } is attracted to the origin in the direction V and x(Φ n (x 0 , y 0 )) ∼ -1
n . Moreover we have

(2.1) |x(Φ n (x 0 , y 0 ))| ≤ 2 n and |y(Φ n (x 0 , y 0 ))| |x(Φ n (x 0 , y 0 ))| -α-1 ≤ |y 0 | |x 0 | -α-1 .
Notice that, for a γ 1 slightly smaller than γ, we have

F 0 ( C 0 (γ, R, s)) ⊆ C 0 (γ, R, s).
Let us now consider F 0 as in (1.1). It is immediate to see that [1 : 0] is a non-degenerate characteristic direction, with director equal to ρ -1. This is the reason we made the assumption that ρ > 1. It will be even clearer later (Lemma 5.1) that this a crucial assumption.

An important feature of our setting is that the (local) inverse of a map tangent to the identity shares a lot of properties with the original map (this does not happen for instance in the semi-parabolic situation). In fact, it is immediate to see that the local inverse of an endomorphism tangent to the identity is still tangent to the identity, with the same characteristic directions and moreover the same Hakim directors (this follows since the homogeneous part of degree 2 of F -1 is the opposite of the homogeneous part of degree 2 of F , and thus induces the same map on P 1 (C) ). In our situation, (0, 0) is still a double fixed point for the local inverse G 0 , which has the following form (see for example the explicit description of the coefficients of the inverse of an endomorphism tangent to the identity given in [START_REF] Abate | Formal Poincaré-Dulac renormalization for holomorphic germs[END_REF]),

G 0 x y = x -x 2 (1 + (q -1)x + ry + O 2 (x, y)) y(1 -ρx + O 2 (x, y))
and the stated properties are readily verified.

In the following, we will fix a neighbourhood U of the origin where F 0 is invertible, and consider an invariant domain C 0 as in Proposition 2.1 for F 0 such that -C 0 satisfies the same property for G 0 and both C 0 and -C 0 are contained in U .

We now briefly recall how to construct a (one dimensional) Fatou coordinate ϕ ι on C 0 semiconjugating F 0 to a translation by 1. We notice here that it is actually possible to construct a two-dimensional Fatou coordinate, on a subset of C 0 , with values in C 2 and semiconjugating the system to the translation by (1, 0). Since we will not use it, we do not detail the construction here, but we refer the interested reader to [START_REF] Hakim | Transformations tangent to the identity, Stable pieces of manifolds[END_REF].

The first step of the construction of ϕ ι is to consider the map

(2.2) w ι 0 (x, y) := - 1 x -q log(-x).
Notice that, in the chart w ι 0 , the map F 0 already looks like a translation by 1. Indeed, by (1.1), we have (2.3)

w ι 0 (F 0 (x, y)) = - 1 x(F 0 (x, y)) -q log(-x(F 0 (x, y))) = - 1 x -q log(-x) + 1 + ry + O 2 (x, y) = w ι 0 (x, y) + 1 + ry + O 2 (x, y).
In order to get an actual Fatou coordinate, we consider the functions (2.4) ϕ ι 0,n := w ι (F n 0 (x, y)) -n. The following Lemma proves that the the sequence ϕ ι 0,n converges to an actual Fatou coordinate ϕ ι as n → ∞.

Lemma 2.2. The functions ϕ ι 0,n converge, locally uniformly on C 0 , to an analytic function

ϕ ι : C 0 → C satisfying ϕ ι (F 0 (p)) = ϕ ι (p) + 1. Proof. Set A 0 (x, y) := w ι 0 (F 0 (x, y)) -w ι 0 (x, y) -1 = ϕ ι 0,1 (x, y) -ϕ ι 0,0 (x, y) and notice that A 0 (F n 0 (x, y)) = ϕ ι 0,n+1 (x, y) -ϕ ι 0,n (x, y).
In order to ensure the convergence of the sequence ϕ ι 0,n we can prove that the series with general term A 0 (F n 0 (x, y))'s converges normally on C 0 . It follows from (2.3) that

A 0 (F n 0 (x, y)) = ry(F n 0 (x, y)) + O 2 (x(F n 0 (x, y)), y(F n 0 (x, y))). By Proposition 2.1, we have |x(F n 0 (x, y))| ≤ 2/n and |y(F n 0 (x, y))| ≤ 1/n α+1 , for some α > 0. This implies that the series ∞ n=0 |A 0 (F n (x, y))| converges normally to ϕ ι (x, y) := ϕ ι 0 (x, y) + ∞ n=0 A 0 (F n 0 (x, y)).
The functional relation is also easily verified, since |A 0 (F n (x, y))| → 0.

In the repelling basin the situation is completely analogous. Setting w o 0 := -1 x -q log(x) on -C 0 and ϕ o 0,n := w o 0 (F -n 0 (x, y)) + n, we have ϕ o 0,n → ϕ o locally uniformly on -C 0 , where

ϕ o : C 0 → C satisfies the functional relation ϕ o • F 0 (p) = ϕ o (p) + 1.
We notice that the Fatou coordinates are not unique. For instance, we can add any constant to them and still have a coordinate satisfying the desired functional relation. In the following (and in Theorem 1.4), we shall use as coordinate the one obtained in Lemma 2.2 above.

The perturbed Fatou coordinates

We consider now the perturbation (1.2) of the system F 0 . The goal of this section is modify the Fatou coordinate ϕ ι built in Section 2 to an approximate coordinate for F ε . More precisely, we are going to construct some coordinates ϕ ι ε (with values in C) that, on suitable subsets of C 0 :

(1) almost conjugate F ε to a translation by 1, in the sense that the error that we have in considering F ε as a translation in this new chart will be bounded and explicitly estimated; and (2) tend to the one-dimensional Fatou coordinates ϕ ι for F 0 as ε → 0. We shall only be concerned with ε small and satisfying

(3.1) Re ε > 0 |Im ε| < c ε 2 .
Notice that this means that ε is contained in the region, in a neighbouhood of the origin, of the points with positive real part and bounded by two circles of the same radius centered on the imaginary axis and tangent one to the other at the origin. Notice in particular that, by definition, every sequence ε ν associated to an α sequence (ε ν , n ν ) (see Definition 1.1) satisfies the above property. First of all, we fix a small neighbourhood U of the origin, such that F ε is invertible in U , for ε sufficiently small. In this section, we shall only be concerned with this local situation. Then, fix sufficiently small γ < γ , R and s such that Proposition 2.1 holds on C 0 (γ, R, s) and C 0 (γ , R, s) for both F 0 and H 0 , where H 0 (x, y) := -F -1 0 (-x, y). By taking γ and γ sufficiently close, we can assume that F 0 ( C 0 (γ , R, s)) and H 0 ( C 0 (γ , R, s)) are contained in C 0 . Denote by C 0 , C 0 ⊂ U (dropping for simplicity the dependence on the parameters) these sets and by C 0 , C 0 their projections on the x-plane. We shall assume that Rρ 1, and so that C 0 ⊂ C 0 U . We consider the classical 1-variable change of coordinates on x (and depending on ε) given by

(3.2) u ε (x) = 1 ε arctan x ε = 1 2iε log iε -x iε + x defined on C \ { ±iεt : t ≥ 1 },
where log is the branch of the logarithm such that u ε (0) = 0. The geometric idea behind this map is the following: for ε small as in (3.1), circular arcs connecting the two points ±iε are sent to parallel (and almost vertical) lines. In particular, the image of the map u ε is contained in the strip

-π 2|ε| < Re ε |ε| w < π 2|ε|
and the image of the disc of radius ε centred at the origin is the strip

-π 4|ε| < Re ε |ε| w < π 4|ε| . Notice that the inverse of this function on -π 2|ε| < Re ε |ε| w < π 2|ε|
is given by w → ε tan (εw). We gather in the next Lemma the main properties of u ε that we shall need in the sequel. Lemma 3.1. Let u ε be given by (3.2). Then the following hold.

(1) For every compact subset C ⊂ C 0 there exist two positive constants M -(C) and M + (C) such that, for every x ∈ C, we have

(3.3) - π 2 |ε| + M -< Re ε |ε| u ε (x) < - π 2 |ε| + M + for every ε sufficiently small. (2) If -π 2|ε| < Re ε |ε| u ε (x) < -π 4|ε| , then |x| ≤ 1 π 2|ε| +Re( ε |ε| uε(x))
. Proof. For the first assertion the main point is to notice that, by the compactness of C, we have

u ε (x) + π 2ε → - 1 x
uniformly on C, as ε → 0. From this we deduce the existence of constants M -, M + such that (3.3) holds for every x ∈ C.

For the second one, we exploit the inverse of u ε on {-π 2|ε| < Re ε |ε| w < π 2|ε| }, which is given by w → ε tan(εw). We have

π 4 < |Re w| < π 2 ⇒ |tan w| ≤ tan |Re w| < 1 π 2 -
|Re w| and the assertion follows putting w = εu ε (x).

We define now, by means of the functions u ε , different regions in the dynamical plane. In order to do this, we have to define some constants (independent on ε) that we shall repeatedly use in the sequel.

First of all, fix some 1 < ρ < ρ. Then, fix some 1 < ρ < 5/4 such that

4π(ρ -1) tan (4π(ρ -1)) > 1 ρ .
This is possible since ρ > 1. In particular, ρ may be very close to 1. Finally, set

(3.4) K := 2π(ρ -1) and τ := tan - π 2 + K 2 .
Without loss of generality, we can take ρ small enough to ensure that K ≤ π/4. Moreover, we shall assume that γ and s are small enough such that

(3.5) ρ < ρ 1-γ √ 1+γ 2 , 4τ s < 1.
Denote by D ε the subset of C given by (3.6) x

∈ D ε ⇔ - π 2 |ε| + K |ε| < Re ε |ε| u ε (x) < π 2 |ε| - K 2 |ε| .
Notice the asymmetry in the definition of D ε . This will be explained in Lemma 6.2.

iε

-iε 0 π 2ε -π 2ε -π 2ε + K ε π 2ε -K 2ε u ε
D |ε| × D 2e 4πρτ |ε| ⊂ D ε ⊂ D τ |ε| × D 2e 4πρτ |ε| .
Notice in particular that the ratios τ and 2e 4πρτ are independent of ε.

Set C ε := ε |ε| C 0 \ D ε and C ε := ε |ε| , 1 • C 0 \ D ε the rotations of C 0 and C 0 of ε |ε| around the y plane. Notice that C ε → C 0 and C ε ∪ D ε → C 0 as ε → 0. Morevover, we have C ε ⊂ C 0 for ε sufficiently small (and satisfying (3.1))
The following Lemma will be very useful in the sequel. Lemma 3.2. For ε sufficiently small, we have

F ε ( C ε ) ⊂ C ε ∪ D ε .
Proof. By the choice of C 0 and C 0 , we have

F 0 ( C 0 ) ⊂ C 0 . Moreover, F ε = F 0 + O(ε 2
) and F ε uniformly converges to F 0 on compact subsets of C 0 . The assertion then follows from the the first inclusion in (3.7).

The first step in the construction of the almost Fatou coordinates consists in considering the functions u ε given by u ε (x, y) := u ε (x). The following lemma gives the fundamental estimate on u ε : in this chart, the map F ε (x, y) approximately acts as a translation by 1 on the first coordinate. Here and in the following, it will be useful to consider the expression

γ ε (x, y) := α ε (x, y) 1 + xα ε (x, y)
,

where α ε (x, y) is as in (1.2). It is immediate to see that γ ε (x, y) = 1 + qx + ry + O 2 (x, y) + O(ε 2 ). Lemma 3.3. Take p = (x, y) ∈ C ε ∪ D ε . Then u ε (F ε (p)) -u ε (p) = 1 + qx + ry + O 2 (x, y) + O(ε 2 ).
In particular, when γ, R, s and ε(γ, R, s) are small enough, for p = (x, y) ∈ C ε ∪ D ε we have

| u ε (F ε (p)) -u ε (p) -1| < ρ -1 and ε |ε| ( u ε (F ε (p)) -u ε (p)) -1 < ρ -1. Proof. Since x(F ε (x, y)) = x + (x 2 + ε 2 )α ε (x, y), it follows that iε -x(F ε (x, y)) iε + x(F ε (x, y)) = (iε -x) (1 + (x + iε)α ε (x, y)) (iε + x) (1 + (x -iε)α ε (x, y)) and so iε + x iε -x iε -x(F ε (x, y)) iε + x(F ε (x, y)) = 1 + iεγ ε (x, y) 1 -iεγ ε (x, y) .
The desired difference is then equal to

u ε (F ε (p)) -u ε (p) = 1 2iε log 1 + iεγ ε (x, y) 1 -iεγ ε (x, y) = 1 iε iεγ ε (x, y) + 1 3 (iεγ ε (x, y)) 3 + O(ε 4 ) = γ ε (x, y) + O(ε 2 ) = 1 + qx + ry + O 2 (x, y) + O(ε 2 )
and the assertion is proved.

The next step is to slightly modify our coordinate u ε to a coordinate w ι ε satisfying the following two properties:

(1) w ι ε → w ι 0 (with w ι 0 as in (2.2)) as ε → 0, and (2) w ι ε (F n ε (p)) -n → ϕ ι when ε → 0 and n → ∞ satisfying some relation to be determined later. We also look for functions w o ε satisfying analogous properties on -C 0 . Recall that the functions w ι 0 (x, y) and w o 0 (x, y) almost semiconjugate the (first coordinate of the) system F 0 to a translation by 1 (by (2.3)).

We set

w ε (x, y) := u ε (x, y) - q 2 log(ε 2 + x 2 ) = 1 2iε log iε -x iε + x - q 2 log(ε 2 + x 2 ).
and consider their incoming and outgoing normalizations w ι ε and w o ε given by

w ι ε (x, y) := 1 2iε log iε -x iε + x - q 2 log(ε 2 + x 2 ) + π 2ε , w o ε (x, y) := 1 2iε log iε -x iε + x - q 2 log(ε 2 + x 2 ) - π 2ε .
It is immediate to check that the first request is satisfied, i.e., that w ι ε (x, y) → w ι 0 on C 0 (and w o ε (x, y) → w o 0 on -C 0 ) as ε → 0. In the next proposition we estimate the distance between the reading of F ε in this new chart w ε and the translation by 1. We want to prove, in particular, that now the error has no linear terms in the x variable. Indeed, notice that also for the system F 0 we had to remove this term (see Lemma 2.2) to ensure the convergence of the series with general term A 0 (F n 0 (p)), by the harmonic behaviour of x(F n 0 (p)). For convenience of notation, we denote this error by A ε (x, y) := w ε (F ε (x, y)) -w ε (x, y) -1 We then have the following estimate.

Proposition 3.4. A ε (x, y) = ry + O 2 (x, y) + O(ε 2 ).
Notice that, differently from [START_REF] Bedford | Semi-parabolic bifurcations in complex dimension two[END_REF], here the error is still linear in y. The reason is that we do not add any correction term in y in the expression of w ε . On the other hand, by our assumptions we do not have any linear dependence in ε.

Proof. The computation is analogous to the one in [START_REF] Bedford | Semi-parabolic bifurcations in complex dimension two[END_REF]. By the definition of w ε and the analogous property of u ε (Lemma 3.3) we have

w ε (F ε (x, y)) -w ε (x, y) = u ε (F ε (x, y)) -u ε (x, y) - q 2 log(ε 2 + x(F ε (x, y)) 2 ) + q 2 log(ε 2 + x 2 ) = 1 + qx + ry + O 2 (x, y) + O(ε 2 ) - q 2 log ε 2 + x(F ε (x, y)) 2 ε 2 + x 2 .
It is thus sufficient to prove that

ε 2 + x(F ε (x, y)) 2 ε 2 + x 2 = 1 + 2x + O 2 (x, y) + O(ε 2 ). But ε 2 + x(F ε (x, y)) 2 = ε 2 + x 2 + (x 2 + ε 2 ) 2 α 2 ε (x, y) + 2x(x 2 + ε 2 )α ε (x, y) = (x 2 + ε 2 )(1 + 2xα ε (x, y) + O(x 2 , ε 2 )) = (x 2 + ε 2 )(1 + 2x + O 2 (x, y) + O(ε 2 ))
and the assertion follows.

Let us finally introduce the incoming almost Fatou coordinate, by means of the w ι ε , as it was done for the map F 0 in (2.4). Set

(3.8) ϕ ι ε,n (p) := w ι ε (F n ε (p)) -n = w ι ε (p) + n-1 j=0 A ε (F j ε (p)).
We shall be particularly interested in the following relation between the parameter ε and the number of iterations.

Definition 3.5. A sequence (ε ν , m ν ) ⊂ (C × N) N such that ε ν → 0 will be said of bounded type if π 2εν -m ν is bounded in ν.
Notice that, given an α-sequence (ε ν , n ν ), the sequence (ε ν , n ν /2) is of bounded type.

The following result in particular proves that the coordinates w ι ε satisfy the second request. This convergence will be crucial in order to prove Theorem 1.4. Here ϕ ι denotes the Fatou coordinate on C 0 given by Lemma 2.2. Proposition 3.6. Let (ε ν , m ν ) ν∈N be a sequence of bounded type. Then ϕ ι εν ,mν → ϕ ι locally uniformly on C 0 .

We can also define the outgoing almost Fatou coordinates on -C 0 as

ϕ o ε,n (p) := w o (F -n ε (p))
+ n (recall that by assumption -C 0 is contained in a neighbourhood U of the origin where F ε is invertible, for ε sufficiently small). The following convergence is then an immediate consequence of Proposition 3.6 applied to the inverse system. Corollary 3.7. Let (ε ν , m ν ) be a sequence of bounded type. Then ϕ o εν ,mν → ϕ o locally uniformly on -C 0 .

To prove Proposition 3.6, we need to estimate the series of the errors in (3.8). In particular, we need to bound the modulus of the two coordinates of the orbit F j ε (p), for p ∈ C 0 and j up to (approximately) π/(2 |ε|). This is the content of the next section. The proof of Proposition 3.6 will then be given in Section 5.

In our study, we will need to carefully compare the behaviour of F ε in C 0 and the one of

F -1 ε on -C 0 . Notice that F -1 ε is given by F -1 ε x y = x -(x 2 + ε 2 )(1 + (q -1)x + ry + +O(ε 2 ) + O 2 (x, y)) y(1 -ρx + O(ε 2 ) + O 2 (x, y))
In order to compare the behaviour of the orbits for F -1 ε with the ones for F ε , it will be useful to consider the change of coordinate (x, y) → (-x, y) and thus study the maps (3.9)

H ε x y = x + (x 2 + ε 2 )(1 + (-q + 1)x + ry + +O(ε 2 ) + O 2 (x, y)) y(1 + ρx + O(ε 2 ) + O 2 (x, y)) = x + (x 2 + ε 2 )α H ε (x, y) y(1 + ρx + β H ε (x, y))
In this way, we can study both F ε and H ε in the same region of space. Notice that the main difference between F ε and H ε is that the coefficient q has changed sign.

The estimates for the points in the orbit

In this section we are going to study the orbit of a point p ∈ C 0 under the iteration of F ε . In particular, since the main application we have in mind is the study of F nν εν when (ε ν , n ν ) is an α-sequence, we shall be primarily interested in the study of orbit up to an order of π/ |ε| iterations.

Recall that the set C 0 is given by Proposition 2.1 and in particular consists of points that converge to the origin under F 0 tangentially to the (negative) real axis of the complex direction [1 : 0]. We shall still assume (by taking R 1 small enough) that C 0 is contained in a small neighbourhood U of the origin where F 0 and F ε are invertible, for ε sufficiently small. By Lemma 3.1, for every compact C ⊂ C 0 there exist two constants M -(C) and M + (C) such that (4.1)

- π 2 |ε| + M -(C) ≤ Re ε |ε| u ε (p) ≤ - π 2 |ε| + M + (C) ∀p ∈ C, ∀ε ≤ ε 0 .
Without loss of generality, we will assume that M -and M + are integers and 1 (since R 1). We shall divide the estimates of the coordinates of F j ε (p) according to its position with respect to the set D ε , i.e., according to the position of x(F j ε (p)) with respect to D ε as in (3.6). The following notation will be consistently used through all our study. Definition 4.1. Given p ∈ C 0 and ε such that p ∈ C ε , we define the entry time n p (ε) and the exit time n p (ε) by

(4.2) n p (ε) := min { j ∈ N : F j ε (p) ∈ D ε } n p (ε) := min { j ∈ N : F j ε / ∈ C ε ∪ D ε }
The next Proposition gives the bounds on n p (ε) that we shall need in the sequel.

Proposition 4.2. Let C ⊂ C 0 be a compact subset and M -, M + be as in (4.1). Then, for every p = (x, y) ∈ C and ε sufficiently small,

K ρ |ε| - M + ρ ≤ n p (ε) ≤ K (2 -ρ ) |ε| - M - 2 -ρ .
In particular,

F j ε (p) ∈ C ε for 0 ≤ j < K ρ |ε| -M + ρ . Proof. Notice that, since F ε ( C ε ) ⊂ C ε ∪ D ε (by Lemma 3.
2), we only have to study the first coordinate of the orbit. Since C ε → C 0 , we have that C ⊂ C ε for ε sufficiently small. From Lemma 3.3 it follows that

2 -ρ < Re ε |ε| u ε (F ε (p)) -Re ε |ε| u ε (p) < ρ .
Thus, we deduce that

(4.3) - π 2 |ε| + M -+ (2 -ρ )j < Re ε |ε| u ε (F j ε (q)) < - π 2 |ε| + M + + ρ j
and the assertion follows from the definition of D ε (see (3.6)).

4.1. Up to n p (ε). Given p in some compact subset C ∈ C 0 , here we study the modulus of the two coordinates of the points in the orbit of p for F ε until they fall in D ε , i.e., for a number of iteration up to n p (ε). We start estimating the first coordinate. Here we shall make use of the definition of K (see (3.4)).

Lemma 4.3. Let C ⊂ C 0 be a compact subset and M -be as in (4.1). Then

x(F j ε (p)) ≤ 2 j + M -
for every p ∈ C, for ε small enough and j ≤ n p (ε).

Proof. The statement follows from Lemma 3.1 (2) and the (first) inequality in (4.3). Indeed, we have (recall that 3/4 < 2 -ρ < 1)

x(F j ε (p)) < 1 π 2|ε| + Re( ε |ε| u ε (F j ε (p))) ≤ 1 π 2|ε| -π 2|ε| + (2 -ρ )j + M -≤ 1 2 -ρ 1 j + M -≤ 2 j + M -.
and the inequality is proved.

We now come to the second coordinate. Estimating this is the main difference between our setting and the semiparabolic one. Notice that, by (1.2), in order to bound the terms y(F j ε (p)) , we will need to get an estimate from below of the first coordinate. This will be done by means of the following lemma. Lemma 4.4. Let C ⊂ C 0 be a compact subset and M -be as in (4.1). Let p, q ∈ C and set q j := ε (tan (ε( u ε (q) + j))) and q j := ε (tan (ε( u ε (q) + |ε| j/ε))). Then, for some positive constants C depending on C and C ε depending on C and ε, and going to zero as Re ε → 0, (4.4) x(F j ε (p)) -q j < C

1 + log(M -+ j) (M -+ j) 2 and (4.5) x(F j ε (p)) -q j < C 1 + log(M -+ j) (M -+ j) 2 + C ε 1 M + + j for every 0 ≤ j ≤ n p (ε).
Notice in particular that the two estimates reduce to the same for ε real.

Proof. The idea is to first estimate the distance between the two sequences u ε (F j ε (p)) and u ε (q) + j (and between u ε (F j ε (p)) and u ε (q) + |ε| j/ε) and then to see how this distance is transformed by the application of the inverse of u ε . Notice that, since j ≤ n p (ε), by definition of

D ε (see (3.6)) we have Re ε |ε| u ε (F j ε (p)) < -π 4|ε|
for the points in the orbit under consideration (since K ≤ π/4).

We first prove that (4.6) u ε (F j ε (p)) -u ε (q) -j ≤ C 1 1 + log(M -+ j) . Notice that this is an improvement with respect to the estimate obtained in Lemma 3.3, but that we shall need both that estimate and the bound from above obtained in Lemma 4.3 in order to get this one.

By the definition of M -, we have that |x(p)| and |x(q)| are bounded above by 2/M -. Recalling that |y| ≤ s |x| for every (x, y) ∈ C ε , Lemma 3.3 gives

u ε (F j ε (p)) -u ε (p) -j ≤ c 1 i<j x(F i ε (p)) + c 2 i<j x(F i ε (p)) 2 + |ε| 2 .
Since by Lemma 4.3 we have x(F j ε (p)) ≤ 2/(j + M -) and the maximal number of iterations n p (ε) is bounded by a constant times 1/ |ε|, this gives

u ε (F j ε (p)) -u ε (p) -j ≤ C 2 1 + log(M -+ j)
for some positive C 2 , and the estimate (4.6) follows since the two sequences ( u ε (p) + j) j and ( u ε (q) + j) j obviously stay at constant distance.

We then consider the sequence q j . Using (4.6), it is immediate to see that

(4.7) u ε (F j ε (p)) -u ε (q) -|ε| j/ε ≤ C 1 1 + log(M -+ j) + |arg(ε)| j,
since the distance between the two sequences u ε (q) + j and u ε (q) + |ε| j/ε. is bounded by the last term.

We now need to estimate how the errors in (4.6) and (4.7) are transformed when passing to the dynamical space, and in particular recover the quadratic denominator in (4.4). By (4.7) we have

Re ε |ε| u ε (F j ε (p)) ≥ - π 2 |ε| + M -+ j -C 1 1 + log(M -+ j) -|arg ε| j > - π 2 |ε| + C 3 (M -+ j)
for ε sufficiently small (as in (3.1)), j ≤ n p (ε) and some C 3 > 0. So, given L > 0, it is enough to bound from above the modulus of the derivative of the inverse of u ε on the strip

-π 2|ε| + L ≤ Re ε |ε| w < -π 4|ε| by (a constant times) 1/ |L| 2 .
This can be done with a straightforward computation. Recall that u ε (z) = 1 ε arctan z ε , so that its inverse is given by ε tan(εw). The derivative of this inverse at a point -π/2ε + w is thus given by ψ ε (w) = ε 2 (cos (εw)) -2 . On the strip in consideration, ψ ε takes its maximum at w = -π 2ε + L, where we have

ψ ε (-π 2ε + L) = ε 2 / sin 2 (
1 ρ -C ε 1 M + + j -C 1 + log(M -+ j) (M -+ j) 2 ≤ x(F j ε (p)) ≤ 2 j + M -
for every p ∈ C, for ε small enough and j ≤ n p (ε).

Proof. The second inequality is the content of Lemma 4.3. Let us then prove the lower bound. By Lemma 4.4, it is enough to get the bound

1 ρ (M + + j) ≤ | q j |
where q j := ε tan (ε(Re( u ε (p)) + |ε| j/ε)) as in Lemma 4.4. Notice that we arranged the points ε |ε| u ε ( q j ) to be on the real axis. Since we have Re ε |ε| u ε (q 0 ) < -π 2|ε| +M + (and thus Re ε

|ε| u ε ( q j ) ≤ -π 2|ε| + M + + j), it follows that | q j | ≥ |ε| tan ε - π 2 |ε| + (M + + j) |ε| ε = |ε| tan (M + |ε| + j |ε|)
.

We thus have to prove that, for ε sufficiently small and j ≤ n p (ε),

|ε| M + + |ε| j tan (M + |ε| + j |ε|) > 1 ρ .
The left hand side is decreasing in j, so we can evaluate it at j = n p (ε), which is less or equal than K (2-ρ )|ε| by Proposition 4.2. We thus need to prove that, for ε sufficiently small,

|ε| M + + K 2-ρ tan M + |ε| + K 2-ρ > 1 ρ .
This follows since |ε| M + + K 2-ρ < 2K for |ε| 1 and, by assumption, K satisfies 2K tan(2K) > 1 ρ . This concludes the proof.

We can now give the estimate for the second coordinate. Proposition 4.6. Let C ⊂ C 0 be a compact subset and M + be as in (4.1). There exists a positive constant c 1 , depending on C, such that for p ∈ C and J ≤ n p (ε),

y(F J ε (p)) ≤ c 1 |y(p)| M + +J-1 l=M + 1 - ρ l for some 1 < ρ < ρ ρ 1-γ √ 1+γ 2 .
Notice that 1 < ρ ρ 1-γ √ 1+γ 2 by the assumption (3.5).

Proof. We shall make use of both estimates obtained in Proposition 4.5. Since the part of orbit which we are considering is in C ε (at least) up to J -1, we have y(F j ε (p)) ≤ s x(F j ε (p)) and

x(F j ε (p)) > |ε|, for j ≤ J -1. So, by the expression of y(F ε (p)) in (1.2), we get

y(F J ε (p)) ≤ |y(p)| J-1 j=0 1 + ρx(F j ε (p)) + O(x 2 (F j ε (p)) ≤ |y(p)| J-1 j=0 1 + ρx(F j ε (p)) + c 1 x 2 (F j ε (p))
for some positive c 1 . For ε sufficiently small, we have

C ε ⊂ C 0 = C 0 (γ , R, s) (see Proposition 2.1
). This implies that Im x(F j ε (p)) < γ Re x(F j ε (p)) for every j < n p (ε). Thus

1 + ρx(F j ε (p)) ≤ 1 -ρ Re x(F j ε (p)) + ρ Im x(F j ε (p)) ≤ 1 -ρ(1 -γ ) Re x(F j ε (p)) ≤ 1 -ρ 1 -γ 1 + γ 2 x(F j ε (p))
and thus, by the estimates on x(F j ε (p)) in Proposition 4.5 we deduce that (for ε sufficiently small)

y(F J ε (p)) ≤ |y(p)| J-1 j=0 1 -ρ 1 -γ 1 + γ 2 1 ρ -C ε 1 M + + j + c 1 1 + log(M -+ j) (M -+ j) 2 ≤ c 1 |y(p)| J-1 j=0 1 -ρ 1 M + + j where ρ is some constant such that 1 < ρ < ρ ρ 1-γ √ 1+γ 2
, and the assertion follows.

4.2.

From n p (ε) to n p (ε). Notice that D ε needs not to be F ε -invariant. In this section we estimate the second coordinate for points in an orbit entering D ε (and in particular explain the constant e 4πρτ in the definition of D ε ). Our goal is prove a lower bound on n p (ε) (and moreover to prove that the orbit cannot come back to C ε ). This will in particular give an estimate for the coordinates of the point in the orbit for j up to the lower bound of n p (ε) (since in D ε both |x| and |y| are bounded by (a constant times) |ε|).

Proposition 4.7. Let C ⊂ C 0 be a compact subset. Then, for every p ∈ C, and n p (ε) < j ≤ n p (ε), we have

y(F j ε (p)) ≤ e 4πρτ y(F np(ε) ε (p)) ≤ e 4πρτ |ε|
Proof. Recall that τ = tan -π 2 + K 2 and that by the assumption (3.5) we have 4sτ < 1. Since the part of orbit under consideration is contained in D ε (and thus x(F j ε (p)) ≤ τ |ε|, by (3.7)), we have

y(F j ε (p)) ≤ y(F np(ε) ε (p)) j-1 i=np(ε) (1 + 2ρτ |ε|) ≤ y(F np(ε) ε (p)) π-K/2 (2-ρ )|ε| i=np(ε) (1 + 2ρτ |ε|) .
The product is bounded by (1+2ρτ |ε|) 2π/|ε| ≤ e 4πρτ as ε → 0. Moreover, we have y(F

np(ε) ε (p)) ≤ y(F np(ε)-1 ε (p)) 1 + ρx(F np(ε)-1 ε
) ≤ 4sτ |ε| < |ε|. This gives the assertion.

We can now give the estimate on n p (ε).

Proposition 4.8. Let C ⊂ C 0 be a compact subset and M -, M + be as in (4.1). Then, for every

p ∈ C, π -K/2 ρ |ε| - M + ρ ≤ n p (ε) ≤ π -K/2 (2 -ρ ) |ε| - M - 2 -ρ .
Moreover, we have y(F j ε (p)) ≤ e 4πρτ |ε| for n p (ε) ≤ j < n p (ε) and

Re ε |ε| u ε F n p (ε) ε ≥ π -K 2 |ε| .
In particular, once entered in D ε , the orbit cannot come back to C ε .

Proof. By Proposition 4.7, the modulus of the second coordinate of the points of the orbit is bounded by e 4ρπτ |ε| for n p (ε) < j ≤ n p (ε). Since for j ≤ n p (ε) it is bounded by s x(F j ε (p)) , the assertion follows from Equation (4.3).

4.3.

After n p (ε). In order to study the behaviour of F ε after D ε , we shall make use of the family H ε introduced in (3.9). The following proposition is an immediate consequence of the analogous results for F ε (first assertion of Lemma 4.4). We denote by n H p (ε) the entry time for H (see Definition 4.1). Lemma 4.9. Let C ⊂ C 0 be a compact subset and M -be as in (4.1). Let p, q be contained in some compact subset C ⊂ C 0 . Then, for ε sufficiently small,

x(F j ε (p)) -x(H j ε (q)) < C 1 + log(M -+ j) (M -+ j) 2
for every 0 ≤ j ≤ min(n p (ε), n H q (ε)), for some positive constant C.

We will get the estimates on the second coordinate in this part of the orbit directly in Section 6, when proving Theorem 1.4, by applying Proposition 4.6 to both F ε and H ε .

A preliminary convergence: proof of Proposition 3.6

In this section we prove Proposition 3.6. Namely, given a sequence (ε ν , m ν ) of bounded type (see Definition 3.5), we prove that ϕ ι εν ,mν → ϕ ι and ϕ o εν ,mν → ϕ o , locally uniformly on C 0 and -C 0 , where ϕ ι and ϕ o are the Fatou coordinates for F 0 given by Lemma 2.2. Recall that by assumption these two sets are contained in a neighbourhood U of the origin where F ε is invertible, for ε sufficiently small, ans thus in particular where ϕ o is well defined. We shall need the following elementary Lemma. Lemma 5.1. Let a ∈ R, be strictly greater than 1. Then, for every j 0 ≥ l 0 ≥ 1 such that 0 < 1 -a l < 1 for every l ≥ l 0 , the series

∞ j=j 0 j l=l 0 1 - a l converges.
Notice that the Lemma is false when a = 1, since the series reduces to an harmonic one. In our applications a will essentially be ρ, which we assume by hyphotesis to be greater than 1.

Proof. As in [START_REF] Weickert | Attracting basins for automorphisms of C 2[END_REF]Lemma 4], let us set P j := j l=l 0 (1 -a l ) and notice that the P j 's admit an explicit expression as

P j = c Γ(j + 1 -a) Γ(j + 1)
for some constant c = c(l 0 ), where Γ is the Euler Gamma function. Since Γ(j + 1 -a) ∼ 1 j a j! as j → ∞, we deduce that P j ∼ c 1 j a , and so j P j converges.

We can now prove Proposition 3.6. The proof follows the main ideas of the one of [6, Theorem 2.6]. The major issue (and the main difference with respect to [START_REF] Bedford | Semi-parabolic bifurcations in complex dimension two[END_REF]) will be to take into account the errors due the O(y)-terms in the estimates. This will be done by means of the following Lemma, which relies on Propositions 4.6 and 4.7. ≥ 3π 5|ε| for ε sufficiently small. So, in particular, the orbit up to time n(ε) is contained in C ε ∪ D ε . On the other hand, we have n

p (ε) + M - 2-ρ ≤ K (2-ρ )|ε| ≤ π/4
(2-5/4)|ε| -M - 2-ρ ≤ π 3|ε| . So, in particular, the assumption of Lemma 5.2 is satisfied when (ε ν , n(ε ν )) is of bounded type.

Proof. We start with the first point. The convergence of the second part of the series is immediate from Proposition 2.1, by the harmonic behaviour of x(F j 0 (p)) and the estimate (2.1). Let us thus consider the first part. Here we split this series in a first part, with the indices up to n p (ε) and in the remaining part starting from n p (ε) + 1. The sum is thus given by

np(ε) j=1 y F j ε (p) + n(ε) j=np(ε)+1 y F j ε (p)
and, by Propositions 4.6 and 4.7, this is bounded by (a constant times)

np(ε) j=1 M + +j-1 l=M + 1 - ρ l +   np(ε)-1+M + j=M + 1 - ρ j   • n(ε) j=np(ε) e 4πρτ
where M + is as in (4.1) and ρ is (as in Proposition 4.6) a constant greater than 1. By the lower estimates on n p (ε) in Proposition 4.2 and the asymptotic behaviour proved in Lemma 5.1, the last expression is bounded by

∞ j=1 j-1+M + l=M + 1 - ρ l + 3π 5 |ε| • e 4πρτ • 1 K ρ |ε| -M + ρ -1 + M + ρ .
The first term is bounded, again by Lemma 5.1, and the second one (which, up to a constant, is in particular a majorant for the sum in the second point in the statement) goes to zero as ε → 0 (since ρ > 1). This proves both statements.

Proof of Proposition 3.6. First of all, recall that by Lemma 2.2 the sequence

ϕ ι 0,mν = w ι 0 + mν -1 j=0 A 0 (F j 0 (p))
converges to a (1-dimensional) Fatou coordinate ϕ ι (for this we just need that m ν → ∞). It is then enough to show that the difference ϕ ι εν ,mν -ϕ ι 0,mν goes to zero as ν → ∞. Here we shall make use of the hypothesis that the sequence (ε ν , m ν ) is of bounded type. The difference is equal to

ϕ ι εν ,mν (p) -ϕ ι 0,mν (p) = w ι εν (p) -w ι 0 (p) + mν -1 j=0 A εν (F j εν (p)) -A 0 (F j 0 (p))
and we see that the first difference goes to zero as ν → ∞. We thus only have to estimate the second part, whose modulus is bounded by

I + II := mν -1 j=0 A 0 (F j εν (p)) -A 0 (F j 0 (p)) + mν -1 j=0 A εν (F j εν (p)) -A 0 (F j εν (p)) .
Let us consider the first sum. First of all, we prove that the majorant

mν -1 j=1 A 0 (F j εν (p)) + A 0 (F j 0 (p))
converges. This follows from the fact that A 0 (p) = O(x 2 , y) by Proposition 3.4, the estimates on x(F j 0 (p)) and x(F j εν (p)) in Propositions 2.1 and 4.5 and from Lemma 5.2 (1). Indeed, with M + as in (4.1), we have (for some positive constant K 0 ),

I ≤ mν -1 j=1 A 0 (F j εν (p)) + A 0 (F j 0 (p)) ≤ K 0 mν -1 j=1 x(F j εν (p)) 2 + x(F j 0 (p)) 2 + K 0 mν -1 j=1 y(F j εν (p)) + y(F j 0 (p)) ≤ K 0 mν -1 j=1 8 (j + M + ) 2 + |ε ν | 2 + K 0 mν -1 j=1 y(F j εν (p)) + y(F j 0 (p)) ≤ B
where in the last passage we used the assumption that the sequence (ε ν , m ν ) is of bounded type to estimate the sum of the |ε ν | 2 's and in order to apply Lemma 5.2 (1) for the second sum. We now prove that I goes to zero, as ν → ∞. Given any small η, we look for a sufficiently large J such that the sum

mν -1 j=J A 0 (F j εν (p)) -A 0 (F j 0 (p))
is less than η for |ε ν | smaller than some ε 0 . The convergence to 0 of I will then follow from the fact that A 0 (F j εν (p)) -A 0 (F j 0 (p)) → 0 as ν → ∞, for every fixed j. As above, this sum is bounded by For J sufficiently large, the first sum is smaller than η/3 (uniformly in ε), since (ε ν , m ν ) is of bounded type. The same is true for the third one, by the harmonic behaviour of x(F j 0 (p)) and the estimate (2.1). We are thus left with the second sum of (5.1). We split it as in Lemma 5.2:

(5.2)

mν -1 j=J y(F j εν (p)) ≤ np(εν ) j=J y(F j εν (p)) + mν -1 j=np(εν )+1
y(F j εν (p)) .

Lemma 5.2 (2) implies that the second sum of the right hand side goes to zero as ε ν → 0. We are thus left with the first sum in the right hand side of (5.2). We estimate it by applying twice Proposition 4.6 and Lemma 5.1:

np(εν ) j=J y(F j εν (p)) ≤ c 1 np(εν ) j=J y(F J εν (p)) j-1+M + l=J+M + 1 - ρ l ≤ c 1 y(F J εν (p)) ∞ j=J j-1+M + l=J+M + 1 - ρ l ≤ C 1 y(F J εν (p)) ≤ C 2 |y(p)| J-1+M + l=M + 1 - ρ l .
We can then take J large enough (and independent from ε) so that the last term is smaller than η 6 . Notice in particular the independence of J from ε (for ε sufficiently small).

So, until now we have proved that I goes to zero as ν → ∞. It is immediate to check that the same holds for II . Indeed,

II ≤ mν -1 j=0 A εν (F j εν (p)) -A 0 (F j εν (p)) ≤ mν -1 j=0 K 1 |ε ν | 2
for some positive constant K 1 . The assertion then follows since (ε ν , m ν ) is of bounded type.

The convergence to the Lavaurs map

In this section we prove Theorem 1.4. We shall exploit the 1-dimensional Theorem 1.2, i.e., the convergence of the restriction of F nν εν on C 0 = C 0 ∩ { y = 0 } to the 1-dimensional Lavaurs map L α . Lemma 6.1. Let p 0 ∈ C 0 ∩ { y = 0 } and (ε ν , n ν ) be an α-sequence. Assume that q 0 := L α (p 0 ) belongs to -C 0 ∩ { y = 0 }. Then for every δ there exists η such that (after possibly shrinking C 0 )

ϕ o -C 0 ∩ F nν εν C 0 ∩ ϕ ι -1 D( ϕ ι (p 0 ), η) ⊂ D( ϕ o (q 0 ), δ)
for every ν sufficiently large.

The need of shrinking C 0 is just due to the fact that Proposition 3.6 and Corollary 3.7 give the convergence on compact subsets of C 0 (and -C 0 ).

ϕ o εν ,m o ν • F nν εν (p) = w εν F -m o εν (F nν εν (p)) - π 2ε ν + m o ν = w εν F m ι εν εν (p) - π 2ε ν -m ι ν + n ν = ϕ ι εν ,m o ν (p) + n ν - π ε ν whenever F nν εν (p) ∈ -C 0 .
The assertion follows from Proposition 3.6 and Corollary 3.7.

Lemma 6.2. Let p 0 ∈ C 0 ∩ { y = 0 } and (ε ν , n ν ) be an α-sequence. Assume that q 0 := L α (p 0 ) belongs to -C 0 ∩ { y = 0 }. Then, for every polydisc ∆ q 0 centered at q 0 and contained in -C 0 there exists a polydisc ∆ p 0 centered at p 0 and contained in C 0 such that F nν εν (∆ p 0 ) ⊂ ∆ q 0 for ν sufficiently large.

Proof. Set ∆ q 0 = D 1 q 0 × D 2 q 0 and analogously ∆ p 0 = D 1 p 0 × D 2 p 0 . By Lemma 6.1 it is enough to prove that, if ∆ p 0 is sufficiently small, for every ν sufficiently large we have max

D 1 p 0 ×∂D 2 p 0 y(F m ι ν εν ) ≤ 1 2 min D 1 q 0 ×∂D 2 q 0 y(F -m o ν εν
) .

We shall use the estimates collected in Section 4. First of all, notice that, by Proposition 4.7, it is enough to prove that max

p∈D 1 p 0 ×∂D 2 p 0 y(F np(εν ) εν ) ≤ c min q∈-D 1 q 0 ×∂D 2 q 0 y(H nν -n p (εν ) εν )
for some constant c, where H ε is as in (3.9). Geometrically, we want to ensure that the vertical expansion in the third part of the orbit (i.e., after n p (ε)) is balanced by a suitable contraction during the first part (i.e., up to n p (ε)). This means proving that, for every p ∈ D 1 p 0 × ∂D 2 p 0 and q ∈ -D 1 q 0 × ∂D 2 q 0 , (6.2)

np(ε) j=0 1 + ρx(F j εν (p)) + β εν (x(F j εν (p)), y(F j εν (p))) ≤ c nν -n p (εν ) j=0
1 + ρx(H j εν (q)) + β H εν (x(H j εν (q)), y(H j εν (q)))

for some positive c . First of all, we claim that there exists a constant K 1 (independent from ν) such that K 1 + n p (ε ν ) ≥ n ν -n p (ε ν ), i.e., the number of points in the orbit for F ε before entering in D εν (and thus in the contracting part) is at least the same (up to the constant) of the number of points in the expanding part. Indeed, recalling the definition (3.4) of K, we have

K ρ |εν | ≥ π |εν | -π-K/2 ρ |εν | . So,
+ ρ + n p (ε ν ) ≥ 1 + |α| + M + ρ + K ρ |ε ν | - M + ρ ≥ n ν - π |ε ν | + π |ε ν | - π -K/2 ρ |ε ν | ≥ n ν -n p (ε ν )
for ν sufficiently large, and the desired inequality is proved. The inequality (6.2) now follows from Lemma 4.9 (and Proposition 4.5), and the assertion follows.

We can now prove Theorem 1.4.

Proof of Theorem 1.4. First of all, we can assume that p 0 belongs to C 0 = { y = 0 } ∩ C 0 . Indeed, there exists some N 0 such that F N 0 0 (p 0 ) ∈ C 0 . So, we can prove the Theorem for the (α -N 0 )-sequence (ε ν , n ν -N 0 ) and the base point F N 0 0 (p 0 ) and the assertion then follows since F N 0 εν → F N 0 0 . For the same reason, we can assume that q 0 := L α (p 0 ) belongs to -C 0 . By Lemma 6.2, there exists a polydisc ∆ p 0 centered at p 0 such that the sequence F nν εν is bounded on ∆ p 0 . In particular, up to a subsequence, this sequence converges to a limit map L, defined in ∆ p 0 with values in -C 0 . Notice that the limit must be open, since the same arguments apply to the inverse system. The relation (1.3) then follows from (6.1) and the assertion follows.

In the following, given a subset U ⊂ C 0 , we denote by T α (U) the set T α (U) := { L : U → C 2 : ∃(ε ν , n ν ) α -sequence such that F nν εν → L on U } We denote by T α the union of all the T α (U)'s, where U ⊂ C 0 , and call the elements of T α Lavaurs maps. Theorem 1.4 can then be restated as follows: every compact subset C 0 ⊂ C 0 has a neighbouhhood U C 0 ⊂ C 0 such that every T α (U C 0 ) is not empty.

The discontinuity of the large Julia set

In this section we shall prove Theorem 1.6. By means of the Lavaurs maps L, we first define a 2-dimensional analogous of the Julia-Lavaurs set J 1 (F 0 , L), and use this set to estimate the discontinuity of the Julia set at ε = 0. Definition 7.1. Let U ⊂ C 0 and L ∈ T α (U). The Julia-Lavaurs set J 1 (F 0 , L) is the set J 1 (F 0 , L) := { z ∈ P 2 |∃m ∈ N : L m (z) ∈ J 1 (F 0 ) }.

The condition L m (z) ∈ J 1 (F 0 ) means that we require L i (z) to be defined, for i = 0, . . . m. In particular, we have z, . . . , L m-1 (z) ∈ U.

From the definition it follows that J 1 (F 0 ) ⊆ J 1 (F 0 , L), for every L ∈ T α . The following result gives the key estimate for the lower-semicontinuity of the large Julia sets at ε = 0. The proof is analogous to the 1-dimensional case, exploiting the fact that the maps L are open. Theorem 7.2. Let L ∈ T α be defined on U ⊂ C 0 and (n ν , ε ν ) be an α-sequence such that F nν εν → L on U. Then lim inf J 1 (F εν ) ⊇ J 1 (F 0 , L).

Proof. The key ingredients are the lower semicontinuity of J 1 (F ε ) and Theorem 1.4. By definition, the set of all z's admitting an m such that L m (z) ∈ J 1 (F 0 ) is dense in J 1 (F 0 , L). Thus, given z 0 and m satisfying the previous condition, we only need to find a sequence of points z ν ∈ J 1 (F εν ) such that z ν → z 0 , for some sequence ε ν → 0. Set p 0 := L m (z 0 ). By the lower semicontinuity of ε → J 1 (F ε ) we can find a sequence of points p ν ∈ J 1 (F εν ) such that p ν → p 0 . By Theorem 1.4 we have F mnν εν → L m uniformly near z 0 , and this (since L is open) gives a sequence z ν converging to z 0 such that F mnν εν (z ν ) = p ν ∈ J 1 (F εν ). This implies that z ν ∈ J 1 (F εν ), and the assertion follows.

Notice the function ε → J 1 (F ε ) is discontinuous at ε = 0 since, by means of just the onedimensional Lavaurs Theorem 1.2, we can create points in C 0 ∩ { y = 0 } (which is contained in the Fatou set) satisfying L α (p) ∈ J 1 (F 0 ). Indeed, the following property holds: (7.1) ∀p ∈ C 0 ∩ { y = 0 } there exists α such that p ∈ J 1 ((F 0 ) |y=0 , L α ).

where L α is the 1-dimensional Lavaurs map on the invariant line { y = 0 } associated to α. Indeed, since ∂B ⊆ J 1 (F 0 ) and B intersects the repelling basin R, we can find q ∈ J 1 (F 0 ) ∩ { y = 0 } in the image of the Fatou parametrization ψ o for (F 0 ) |{ y=0 } . The assertion follows considering α such that L α (p) = q.

In our context, given any p ∈ C 0 and q ∈ -C 0 as above, by means of Theorem 1.4 we can consider a neighbourhood of p where a sequence F nν εν converges to a Lavaurs map L (necessarily coinciding with L α on the line { y = 0 }). Since L is open, we have that L -1 (J 1 (F 0 )) is contained in the liminf of the Julia sets J 1 (f εν ). This gives a two-dimensional estimate of the discontinuity.

The discontinuity of the filled Julia set

For regular polynomial endomorphism of C 2 it is meaningful to consider the filled Julia set, defined in the following way. Definition 8.1. Given a regular polynomial endomorphism F of C 2 , the filled Julia set K(F ) is the set of points whose orbit is bounded.

Equivalently, given any sufficiently large ball B R , such that F -1 (B R ) B R , the filled Julia set is equal to

K(F ) := n≥0 F -n (B R ).
In this section we shall prove that, if the family (1.2) is induced by regular polynomials, then the set-valued function ε → K(F ε ) is discontinuous at ε = 0.

Recall that the function ε → K(F ε ) is always upper semicontinuous (see [START_REF] Douady | L'ensemble de Julia dépend-il continûment du polynôme?[END_REF]). Here the key definition will be the following analogous of the filled Lavaurs-Julia set in dimension 1 ( [START_REF] Lavaurs | Systemes dynamiques holomorphes: explosion de points périodiques paraboliques[END_REF]). Definition 8.2. Given U ⊂ C 0 and L ∈ T α (U), the filled Lavaurs-Julia set K(F 0 , L) is the complement of the points p such that there exists m ≥ 0 such that L m (p) is defined and is not in K(F 0 ). Notice in particular that K(F 0 , L) ⊆ K(F 0 ) and coincides with K(F 0 ) outside U. Moreover, notice that K(F 0 , L) is closed. Theorem 8.3. Let L ∈ T α be defined on some U ⊂ C 0 . and let (ε ν , n ν ) be an α-sequence such that F nν εν → L on U. Then K(F 0 , L) ⊇ lim sup K(F εν ).

Figure 1 .

 1 Figure 1. The region D ε , for a real ε Let us now move to C 2 . Let D ε be the product D ε × D 2e 4πρτ |ε| ⊂ C 2 (the constant e 4πρτ will be explained in Proposition 4.7), where D r ⊂ C denotes the open disc of radius r. By definition, since K ≤ π/4, we have (3.7) D |ε| × D 2e 4πρτ |ε| ⊂ D ε ⊂ D τ |ε| × D 2e 4πρτ |ε| .

Proposition 4 . 5 .

 45 εL). The estimate then follows since x ≤ 2 sin(x) on [0, π/4]. Let C ⊂ C 0 be a compact subset, M -, M + be as in (4.1) and C, C ε as in Lemma 4.4. Then

Lemma 5 . 2 .

 52 Let p ∈ C 0 and n p (ε) be as in (4.2). Let n(ε) be such that n p (ε) ≤ n(ε) ≤ 3π 5|ε| . Then the following hold:(1) the function ε → n(ε) j=1 y F j ε (p) + y F j 0 (p)is bounded, locally uniformly on p, for ε sufficiently small; (2) lim ε→0 n(ε) j=np(ε)+1 y(F j ε (p)) = 0, locally uniformly on p.Notice that, by Proposition 4.8, n p (ε) ≥ π-K/2 ρ |ε| -M +

  M + ) 2 + |ε ν | 2 +

Proof.

  Let m o ν and m ι ν be sequences of bounded type such that m ι ν + m o ν = n ν . By definition of ϕ ι ε,n and ϕ o ε,n we have (6.1)

  by Propositions 4.2 and 4.8 we have, with M + as in (4.1),

	1 + |α| +	M

Acknowledgments. It is a pleasure to thank my advisor Marco Abate for suggesting me this problem and for his many helpful comments. I would also like to thank Eric Bedford for helpful discussions and the referee whose comments helped improving the exposition of this paper.

This research was partially supported by the ANR project LAMBDA, ANR-13-BS01-0002 and by the FIRB2012 grant "Differential Geometry and Geometric Function Theory", RBFR12W1AQ 002.

Proof. Since the set-valued function ε → K ε is upper-semicontinuous, there exists a large closed ball B such that, for ν ≥ ν 0 , we have ∪ ν K(F εν ) ⊂ B. Without loss of generality we can assume that ν 0 = 1. Let us consider the space

and its subset X given by

By [10, Proposition 2.1] and the fact that P is compact, it is enough to prove that X is closed in P . This follows from Theorem 1.4. Indeed, let z be in the complement of K(F 0 , L). Since this set is closed, a small ball B z around z is outside K(F 0 , L), too. By definition, this means that, for some m, we have L m (B z ) ⊂ K(F 0 ) c . Theorem 1.4 implies that, up to shrinking the ball B z , we have

, for ν large enough. So, B z ⊂ K(F εν ) c and this gives the assertion.

Corollary 8.4. Let F ε be a holomorphic family of regular polynomials of C 2 as in (1.2). Then the set-valued function ε → K(F ε ) is discontinuous at ε = 0.

Proof. The argument is the same used to prove the discontinuity of J 1 (F ε ) in Section 7. If the function ε → K(F ε ) were continuous, Theorem 8.3 and the fact that K(F 0 , L) ⊆ K(F 0 ) for every Lavaurs map L would imply that all the K(F 0 , L)'s were equal to K(F 0 ). Since C 0 ⊆ K(F 0 ), it is enough to find p ∈ C 0 and α such that p / ∈ K(F 0 , L). To do this, it is enough to take any point q in { y = 0 } not contained in K(F 0 ) (recall that K(F 0 ) is compact) and then consider a point p ∈ C 0 ∩ { y = 0 } and α such that L α (p) = q. The existence of such points is a consequence of the property (7.1). Then, consider a neighbourhood U of p such that some sequence F nν εν converges to a Lavaurs map L on U. The assertion follows since L is open and coincides with L α on the intersection with the invariant line { y = 0 }.