Alessandro Bosso
email: alessandro.bosso@unibo.it

Ilario A Azzollini
email: ilario.azzollini@unibo.it

Simone Baldi
email: s.baldi@tudelft.nl

Luca Zaccarian
email: zaccarian@laas.fr

A Hybrid Distributed Strategy for Robust Global Phase Synchronization of Second-Order Kuramoto Oscillators

This work proposes a distributed control strategy for the robust global leader-follower phase synchronization of Kuramoto oscillators with inertia. For a convenient design, the phase angles are represented as elements of the unit circle.

In particular, we exploit a "half-angle" representation inspired by unit quaternions. The ensuing non-Euclidean state space poses some challenges for robust global stabilization, which can be conveniently overcome with dynamic hybrid feedback. For this reason, we propose a hybrid solution obtained by combining a distributed observer with local hysteresis-based tracking controllers. The overall closed-loop system is analyzed through reduction theorems and Lyapunov-based arguments.

I. INTRODUCTION

Numerous synchronization phenomena related to engineering, physics, and biology, can be naturally described through networks of coupled nonlinear oscillators. Among the mathematical models that capture synchronization phenomena, the Kuramoto model [START_REF] Kuramoto | Chemical Oscillations, Waves, and Turbulence[END_REF] is one of the most widespread. This model has received a renewed attention over the last years, mainly because of its applicability to neuronal synchronization [START_REF] Menara | A framework to control functional connectivity in the human brain[END_REF] and synchronization in power networks [START_REF] Guo | Overviews on the applications of the Kuramoto model in modern power system analysis[END_REF], [START_REF] Dörfler | Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators[END_REF].

In networks of coupled oscillators, synchronization phenomena can either naturally arise, can be enforced by designing appropriate control laws, or can be promoted by a pacemaker in a leader-follower architecture [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF]. In all these cases, several studies have been conducted on the various parameters that influence synchronization, such as the network's interaction topology, the diversity of the agents, and the coupling strengths [START_REF] Zhang | Synchronization of Kuramoto oscillators in small-world networks[END_REF], [START_REF] Zhang | Exponential synchronization of the highdimensional Kuramoto model with identical oscillators under digraphs[END_REF]. In particular, necessary [START_REF] Chopra | On exponential synchronization of Kuramoto oscillators[END_REF] and sufficient [START_REF] Chopra | On exponential synchronization of Kuramoto oscillators[END_REF], [START_REF] Dörfler | On the critical coupling for Kuramoto oscillators[END_REF], [START_REF] Zhu | Synchronization of Kuramoto oscillators: A regional stability framework[END_REF] conditions for synchronization were found for Kuramoto networks.

This paper considers a second-order generalization of the classical first-order Kuramoto model, where each oscillator has its own inertia [START_REF] Dörfler | Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators[END_REF], [START_REF] Choi | Complete synchronization of Kuramoto oscillators with finite inertia[END_REF]. Other generalizations have been recently proposed, such as the third-order model [START_REF] Wu | Synchronization conditions for a third-order Kuramoto network[END_REF] and the Lohe model [START_REF] Markdahl | Robust synchronization of heterogeneous robot swarms on the sphere[END_REF], which generalizes the Kuramoto model from the circle to the sphere.

The synchronization of the Kuramoto model is an inherently nonlinear phenomenon. Since the oscillators' angular configurations are equivalent modulo 2π, their natural representation is given by elements of the unit circle. This leads to the N -torus as the state space of the network [START_REF] Scardovi | Synchronization and balancing on the N-torus[END_REF], [START_REF] Bosso | Global frequency synchronization over networks of uncertain second-order Kuramoto oscillators via distributed adaptive tracking[END_REF]. Relevant challenges arise from the control viewpoint, mainly because the unit circle is not contractible, i.e., it is not diffeomorphic to any Euclidean space. This property is known to be a major obstacle to global stabilization. For instance, several works on rigid body control and estimation provide almost global results with continuous feedback [START_REF] Mahony | Nonlinear complementary filters on the special orthogonal group[END_REF], [START_REF] Marco | Output regulation on the special Euclidean group SE(3)[END_REF]. Global results can only be obtained via discontinuous feedback at the cost of sacrificing robustness. In this respect, hybrid systems theory has been proven successful for robust global stabilization on numerous non-contractible spaces, including unit quaternions [START_REF] Mayhew | Quaternion-based hybrid control for robust global attitude tracking[END_REF] and the N -sphere [START_REF] Casau | Robust global exponential stabilization on the n-dimensional sphere with applications to trajectory tracking for quadrotors[END_REF].

Motivated by the above discussion, we present a hybrid distributed strategy for the robust global leader-follower phase synchronization of second-order Kuramoto oscillators. For simplicity, we assume that each node knows its local model parameters. To conveniently solve the problem, we introduce a new representation inspired by unit quaternions and based on "half-angle" expressions.

The proposed approach is composed of two main parts. First, the reference is estimated in each node of the network through a distributed observer, based on embedding the unit circle in R 2 . Then, employing the estimate thus obtained, we introduce a hysteresis-based hybrid controller inspired by [START_REF] Mayhew | Quaternion-based hybrid control for robust global attitude tracking[END_REF] to guarantee global phase tracking for each oscillator. The resulting closed-loop dynamics can be described through a cascade of simple subsystems, each of them analyzed through Lyapunov theory. The overall stability analysis, performed through reduction theorems, not only shows globally robustly stable synchronization but also highlights that no time-scale separation and no tuning constraints are needed.

The paper is organized as follows. In Section II, we present the second-order Kuramoto model. Next, the control objectives are discussed in Section III, while in Sections IV and V we treat the distributed observer and the hysteresisbased controller. The interconnection of these components is then presented in Section VI. Finally, in Section VII, we validate the proposed strategy with a numerical example, while Section VIII provides some concluding remarks. Due to space constraints, some proofs are omitted.

Notation: We denote R ≥0 := [0, ∞), while N * is the set of natural numbers zero excluded. Given column vectors v and w, we often use (v, w) to denote [v w ] . The notation ⊗ indicates the Kronecker product of matrices. For any n ∈ N * , I n is the identity matrix of dimension n and 1 n ∈ R n is the vector of all ones. Finally, we employ the hybrid systems formalism as in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF] and the graph theory formalism of [START_REF] Bosso | Global frequency synchronization over networks of uncertain second-order Kuramoto oscillators via distributed adaptive tracking[END_REF].

II. MODEL DESCRIPTION

A. Second-Order Kuramoto Network

The second-order Kuramoto network is a system of N nonlinear oscillators, coupled through an undirected and connected graph G := {V, E}, where for each node i ∈ V, N i is the set of its neighbors according to E:

θi = ω i , i ∈ V m i ωi =-d i ω i + ω ni - j∈Ni k ij sin(θ i -θ j -ϕ ij ) + u i . (1)
In [START_REF] Kuramoto | Chemical Oscillations, Waves, and Turbulence[END_REF], for each i ∈ V, θ i ∈ R and ω i ∈ R are the phase and the frequency, respectively, u i is the control input, m i > 0 is the oscillator's inertia, d i > 0 is a damping constant, whereas ω ni is the oscillator's natural frequency. Finally, k ij = k ji > 0 and ϕ ij = ϕ ji ∈ [0, 2π) are, respectively, the coupling weight and the phase shift between oscillators i and j (note that ϕ ij is often restricted to the set [0, π/2)).

Define θ := θ 1 . . . θ N and ω := ω 1 . . . ω N . With some abuse of notation, denote by (θ(•), ω(•)) : R ≥0 → R 2N a solution of system (1), for some input signal u(•) and with initial conditions (θ(0), ω(0)). We say that (θ(•), ω(•)) achieves phase synchronization if

lim t→+∞ θ i (t)-θ j (t) ∈ { θ : θ = 2kπ, k ∈ Z}, ∀i, j ∈ V. (2)
Similarly, the solution (θ(•), ω(•)) is said to achieve frequency synchronization if

lim t→+∞ ω i (t) -ω j (t) = 0, ∀i, j ∈ V. (3)
In this respect, the objective of this work consists in designing a distributed control strategy such that phase synchronization and convergence to a specified reference are reached globally, i.e., for any arbitrary initialization of the network (1). This goal is precisely defined in Section III.

B. Quaternion-Inspired Representation

In the following, we introduce a meaningful reformulation for system [START_REF] Kuramoto | Chemical Oscillations, Waves, and Turbulence[END_REF]. Motivated by the equivalence modulo 2π of angular values, our manipulations are based on the adoption of the unit circle S 1 := α β ∈ R 2 : α 2 + β 2 = 1 as a more convenient representation of the phases θ i . Notably, S 1 admits a Lie group structure, isomorphic to the group of planar rotations SO [START_REF] Menara | A framework to control functional connectivity in the human brain[END_REF]. In this respect, the function R :

S 1 → SO(2) maps any element α β ∈ S 1 into the corresponding rotation matrix: R α β := α -β β α , (4)
which is used to compute the group multiplication as R(γ)γ = R(γ)γ, for any γ, γ ∈ S 1 , with identity element e := 1 0 .

Employing the formalism just introduced, the representation adopted in [START_REF] Bosso | Global frequency synchronization over networks of uncertain second-order Kuramoto oscillators via distributed adaptive tracking[END_REF] (also used in [START_REF]Hybrid dynamical systems[END_REF]Ex. 34]) is given by:

ξ i := cos(θ i ) sin(θ i ) ∈ S 1 . (6)
Through straightforward computations, with ( 6), the phase dynamics on both SO(2) and S 1 is obtained as

d dt R(ξ i ) = ω i JR(ξ i ), ξi = ω i Jξ i , i ∈ V, (7)
where

J := 0 -1 1 0
∈ SO(2). Different from [START_REF] Bosso | Global frequency synchronization over networks of uncertain second-order Kuramoto oscillators via distributed adaptive tracking[END_REF] and [START_REF]Hybrid dynamical systems[END_REF], we use here a second representation, which is more convenient for control design. Instead of (6), we represent θ i through

ζ i := η i i := cos(θ i /2) sin(θ i /2) ∈ S 1 . (8)
More precisely, in place of ξ i in ( 6) we employ the element

ζ i ∈ S 1 such that ξ i = φ(ζ i ), where φ : S 1 → S 1 is defined for any ζ i = [η i i ] as follows: φ(ζ i ) := R(ζ i )ζ i = η 2 i -2 i 2η i i . (9)
The map φ can be interpreted as the double-angle formula for elements on the unit circle. Note that φ is two-to-one, because φ(

ζ i ) = φ(-ζ i ).
This property is analogous to the relation between rotation matrices in SO(3) and unit quaternions, whose domain, S 3 , is a double cover of SO(3).

From

ξ i = φ(ζ i ) := R(ζ i )ζ i , we obtain ξi = ω i Jξ i = ∂φ ∂ζi ζi = 2R(ζ i ) ζi , (10)
which leads to the phase dynamics in the new representation, presented both for SO(2) and S 1 :

d dt R(ζ i ) = 1 2 ω i JR(ζ i ), ζi = 1 2 ω i Jζ i , i ∈ V. (11)
Let T N := N i=1 S 1 denote the N -torus, then we can rewrite the network dynamics on T N × R N as follows:

ζi = 1 2 ω i Jζ i m i ωi = -d i ω i +ω ni - j∈Ni k ij φ(ζ i ) Jφ(ζ j )cos(ϕ ij ) + j∈Ni k ij φ(ζ i ) φ(ζ j ) sin(ϕ ij ) + u i , i ∈ V. (12)
In these new coordinates, we observe that condition (2) for phase synchronization coincides with

lim t→+∞ R(ζ i (t)) ζ j (t) ∈ {-e, e}, ∀i, j ∈ V. (13)

III. PROBLEM STATEMENT A. Leader Exosystem

For simplicity, suppose that the graph G, introduced for the physical couplings of network [START_REF] Wu | Synchronization conditions for a third-order Kuramoto network[END_REF], also corresponds to the communication graph among the local controllers acting in each node. Then, we specify the tracking objectives through a leader system (sometimes indicated in the literature as pacemaker [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF]) that delivers to the network some reference signals generated by the following autonomous exosystem: where ζ is the phase reference, w ∈ R n (with n ∈ N * ) is used to generate the frequency reference c w , while c ∈ R n×1 and S ∈ R n×n are constant matrices. It is assumed that S be Poisson stable (namely having only semisimple eigenvalues on the imaginary axis). Therefore, we can assume without loss of generality that the set W ⊂ R n of admissible initial conditions w (0) is compact and forward invariant for ẇ = Sw . Finally, we require the following connectivity property between the leader ( 14) and the network [START_REF] Wu | Synchronization conditions for a third-order Kuramoto network[END_REF].

ζ = 1 2 c w Jζ ẇ = Sw (ζ , w ) ∈ S 1 × W, (14)
Assumption 1. The leader system (14) interacts, by communicating signal (ζ , w ), with at least one node of G.

Let A = [a ij ] ∈ R N ×N and L = [l ij ] ∈ R N ×N
be the adjacency and Laplacian matrices of graph G, and denote by T = [τ ij ] ∈ R N ×N the target matrix, a diagonal matrix such that τ ii = 1 if ( 14) communicates with node i ∈ V and τ ii = 0 otherwise. A scheme of the interaction and communication pattern is shown in Fig. 1.

The control problem of this work, stated below, corresponds to ensuring suitable stability and attractivity properties of the following synchronization set

A s := (ζ , w ), (ζ i , ω i ), i ∈ V : R(ζ i ) ζ ∈ {-e, e}, ω i = c w , ( 15
)
where e is defined in [START_REF] Wang | Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker[END_REF]. Note that A s is compact because [START_REF] Scardovi | Synchronization and balancing on the N-torus[END_REF] evolves in the compact set S 1 × W.

Problem 1. Consider the second-order Kuramoto network (12) and the exosystem [START_REF] Scardovi | Synchronization and balancing on the N-torus[END_REF].

Let m i , d i , ω ni , ϕ ij , k ij , c,
and S be locally known at each node i. Design a distributed control strategy, based on the information exchange specified by G and Assumption 1, such that the network achieves global phase synchronization to the reference ζ . More specifically, design a distributed dynamic control strategy globally asymptotically stabilizing a compact set whose projection in the plant-exosystem direction coincides with the compact set A s in [START_REF] Bosso | Global frequency synchronization over networks of uncertain second-order Kuramoto oscillators via distributed adaptive tracking[END_REF].

The solution that we adopt for Problem 1 comprises hybrid dynamic laws whose data satisfy the so-called hybrid basic conditions of [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF]Assumption 6.5]. As a consequence, our global asymptotic stability result is equivalent to the existence of a uniform class KL bound (see [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF]Thm. 7.12]). Following the robustness results in [20, §7.3], this also implies robust KL-asymptotic stability in the presence of fairly general perturbations of the dynamics. This property was not attained by using the discontinuous approach of [START_REF] Gui | Global finite-time attitude consensus of leader-following spacecraft systems based on distributed observers[END_REF].

B. Control Architecture

Through the parametrization (8), ( 9), [START_REF] Wu | Synchronization conditions for a third-order Kuramoto network[END_REF], Problem 1 addresses the synchronization goal in (2), (3), in a convenient scenario wherein the set to be stabilized is compact. Topological limitations associated to the non-Euclidean nature of the phase dynamics make the solution to Problem 1 challenging. In fact, the N -torus is non-contractible, i.e., it is not diffeomorphic to any Euclidean space, and convergence to [START_REF] Bosso | Global frequency synchronization over networks of uncertain second-order Kuramoto oscillators via distributed adaptive tracking[END_REF], as well as goal [START_REF] Menara | A framework to control functional connectivity in the human brain[END_REF], require convergence to a disconnected set of points. Two main issues arise in this context. First of all, if the control laws u i are designed to stabilize only one of the given configurations, the trajectories in the coordinates ξ i (or θ i ) will display the so-called unwinding phenomenon [START_REF] Mayhew | Quaternion-based hybrid control for robust global attitude tracking[END_REF], causing unnecessary motion when the system is initialized close to synchronization. Additionally, if a static discontinuous feedback is adopted, the global attractivity property of the closed-loop system is not robust, e.g., to measurement noise. This fact, in practice, translates into chattering and high disturbance sensitivity [START_REF] Mayhew | Quaternion-based hybrid control for robust global attitude tracking[END_REF].

Given these considerations, we employ the following hybrid dynamic strategy to robustly globally asymptotically stabilize a compact set containing A s in (15):

• first, we design a distributed observer for the exosystem [START_REF] Scardovi | Synchronization and balancing on the N-torus[END_REF], such that certain local estimates ( ζi , ŵi ) of (ζ , w ) ∈ S 1 ×W are defined as elements of R 2+n ; • for each agent i, we implement a hybrid mechanism for globally tracking ( ζi , ŵi ). Exploiting the resulting cascade structure, the overall solution is analyzed via reduction theorems.

IV. DISTRIBUTED OBSERVER DESIGN

We propose the following observer:

ζi = 1 2 c ŵi J ζi -k ζ e ζ i ẇi = S ŵi -k w e wi i ∈ V, (16)
where ζi ∈ R 2 , ŵi ∈ R n are the local estimates of (ζ , w ) at node i, k ζ and k w ∈ R are gains to be designed, while 17) can be written as [START_REF] Azzollini | Adaptive leader-follower synchronization over heterogeneous and uncertain networks of linear systems without distributed observer[END_REF]:

e ζ i := j∈Ni a ij ( ζi -ζj ) + τ ii ( ζi -ζ ) e wi := j∈Ni a ij ( ŵi -ŵj ) + τ ii ( ŵi -w ) i ∈ V, (17)
e ζ = (B ⊗ I 2 ) ζ, e w = (B ⊗ I n ) w, (18)
where B := L + T is the leader-follower matrix, which is positive-definite since T = 0 from Assumption 1 (see [START_REF] Zhang | Adaptive cooperative tracking control of higher-order nonlinear systems with unknown dynamics[END_REF]). The dynamics of the frequency estimation error w := ŵ -1 N ⊗ w is obtained from ( 14), ( 16), [START_REF] Mayhew | Quaternion-based hybrid control for robust global attitude tracking[END_REF] as

ẇ = (I N ⊗ S) -k w (B ⊗ I n ) w, (19)
leading to the following stability result.

Proposition 1. For any scalar gain k w > 0, the zeroequilibrium of system (19) is globally exponentially stable.

For notational convenience, define

Ω := diag(c w1 , . . . , c wN ) = diag((I N ⊗ c ) w). ( 20
)
It follows that the dynamics of the phase estimation error 14), ( 16), [START_REF] Mayhew | Quaternion-based hybrid control for robust global attitude tracking[END_REF] as:

ζ := ζ -1 N ⊗ ζ is computed from (
ζ = 1 2 c w (I N ⊗ J) -k ζ (B ⊗ I 2 ) + 1 2 ( Ω ⊗ J) ζ + 1 2 ( Ω ⊗ J)(1 N ⊗ ζ ), (21)
with inputs ζ , w , and w (see [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF]). The next proposition provides an input-to-state stability result for [START_REF]Hybrid dynamical systems[END_REF], used in Section VI to study the cascade between ( 19) and ( 21).

| ζ(t)| ≤ b exp(-λt)| ζ(0)| + σ w(•) ∞ , ∀t ∈ R ≥0 . (22)

V. FULL-INFORMATION SYNCHRONIZATION

In this section, we design a tracking controller in the simplified setup where the observer estimation errors are zero. Define the phase and frequency tracking errors

ζi = ηi ¯ i := R(ζ i ) ζi ∈ R 2 ωi := c ŵi -ω i ∈ R i ∈ V. ( 23
)
In these coordinates, the control objective corresponds to imposing (¯ i , ωi ) → 0, for all i ∈ V. From ( 11), ( 12), ( 14), ( 16), we can compute the error dynamics as

ζi = 1 2 ωi J ζi -k ζ R(ζ i ) e ζ i m i ωi = ψ i -u i -m i k w c e wi i ∈ V, (24)
where (e ζi , e ωi ) are defined in [START_REF] Marco | Output regulation on the special Euclidean group SE(3)[END_REF] and:

ψ i := d i ω i -ω ni + j∈Ni k ij φ(ζ i ) Jφ(ζ j )cos(ϕ ij ) - j∈Ni k ij φ(ζ i ) φ(ζ j )sin(ϕ ij ) + m i c S ŵi . ( 25
)
When ζi = ζ and ŵi = w (i.e., ( ζ, w) = 0), the quantities in (23

) become ζi = R(ζ i ) ζ ∈ S 1 and ωi = c w -ω i ∈ R.
In view of this reduction argument, we begin the design by assuming that the exosystem signals (ζ , w ) are globally known for feedback. This full-information scenario corresponds to the requirement T = I N , which will be removed in Section VI. Assuming that ω i can be arbitrarily assigned by the controller, the phase dynamics in [START_REF] Zhang | Adaptive cooperative tracking control of higher-order nonlinear systems with unknown dynamics[END_REF] reduces to

ζi = 1 2 (c w -ω vi )J ζi , i ∈ V, (26)
where ω vi is a virtual input. Define Q := {-1, 1} and choose any gain k p > 0 and hysteresis margin δ ∈ (0, 1). For each i ∈ V, a hysteresis-based dynamic controller is given by qi = 0, ( ζi

, q i ) ∈ C κ q + i = -q i , ( ζi , q i ) ∈ D κ ω vi = c w + k p q i ¯ i , i ∈ V (27)
where q i ∈ Q is the controller state and

C κ := {( ζi , q i ) ∈ S 1 × Q : ηi q i ≥ -δ} D κ := {( ζi , q i ) ∈ S 1 × Q : ηi q i ≤ -δ}. (28)
The closed-loop error dynamics then corresponds to

ζi = -1 2 k p q i ¯ i J ζi qi = 0 ζi q i ∈ C κ , ζ+ i = ζi q + i = -q i ζi q i ∈ D κ , (29)
which provides an autonomous hybrid dynamics having state ( ζi , q i ) ∈ S 1 × Q and such that the point q i ηi = -1 is not included in the flow set C κ (because δ < 1).

Lemma 1. The attractor

A κ := {(ζ, q) ∈ S 1 × Q : ζ = qe}
is uniformly globally asymptotically stable (UGAS) for (29).

Controller ( 27), (28) can be employed in the design of the feedback laws u i for [START_REF] Zhang | Adaptive cooperative tracking control of higher-order nonlinear systems with unknown dynamics[END_REF], following the energy-based approach proposed in [START_REF] Mayhew | Quaternion-based hybrid control for robust global attitude tracking[END_REF]. Whenever ŵi = w , due to our reduction argument, the dynamics of ωi in [START_REF] Zhang | Adaptive cooperative tracking control of higher-order nonlinear systems with unknown dynamics[END_REF] reduces to

m i ωi = ψ i -u i , i ∈ V, (30)
where ψ i in (25) is known to node i (see Problem 1). The local controllers are given by qi = 0 ( ζi , q i , ωi )

∈ C := C κ × R q + i = -q i ( ζi , q i , ωi ) ∈ D := D κ × R u i = ψ i + k p q i ¯ i + k d ωi , i ∈ V (31)
with C κ and D κ as in (28) and where k d is an arbitrary positive scalar. The closed-loop error dynamics for each node i ∈ V is then described by the following hybrid system

       ζi = 1 2 ωi J ζi qi = 0 m i ωi = -k p q i ¯ i -k d ωi   ζi q i ωi   ∈ C,        ζ+ i = ζi q + i = -q i ω+ i = ωi   ζi q i ωi   ∈ D. (32)
The next result generalizes the argument of Lemma 1.

Proposition 3. For the full-information error dynamics (32), corresponding to (24), (31) with ( ζ, w) = 0, attractor

A x := {(ζ, q, ω) ∈ S 1 × Q × R : ζ = qe, ω = 0} is UGAS.
Proof: We sketch the proof inspired by [START_REF] Mayhew | Quaternion-based hybrid control for robust global attitude tracking[END_REF]. Define x i := ( ζi , q i , ωi ), then consider V (x i ) := 2k p (1 -q i ηi ) + m i ω2 i /2, which is positive definite and radially unbounded w.r.t. A x . Then, straightforward computations show that

∇V (x i ), ẋi = -k d ω2 i ≤ 0, ∀x i ∈ C\A x V (x + i ) -V (x i ) = 4k p q i ηi ≤ -4k p δ < 0, ∀x i ∈ D. (33)
It is then sufficient to notice that the largest invariant set contained in S = {x i ∈ C : ωi = 0} satisfies ¯ i = 0 and ηi q i = -1, i.e., V (x i ) = 0, and apply [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF]Thm 8.8].

VI. OVERALL CONTROL SCHEME

Finally, we present the complete hybrid observer-based controller for each node i, obtained by combining the distributed observer [START_REF] Mahony | Nonlinear complementary filters on the special orthogonal group[END_REF] and the local hysteresis-based controller (31). In this context, we no longer assume ( ζ, w) = 0, thus the dynamics of the tracking errors ( ζi , ωi ) in ( 24) is not simplified as in the full-information scenario.

Define the overall state at node i as

χ i := ( ζi , ŵi , ζi , q i , ωi ) ∈ R n+4 × Q × R. (34)
The controllers solving Problem 1 are given, for i ∈ V, by

       ζi = 1 2 c ŵi J ζi -k ζ e ζ i ẇi = S ŵi -k w e wi qi = 0 χ i ∈ C χ ,        ζ+ i = ζi ŵ+ i = ŵi q + i = -q i χ i ∈ D χ , with: C χ := {χ i ∈ R n+4 × Q × R : ηi q i ≥ -δ}, D χ := {χ i ∈ R n+4 × Q × R : ηi q i ≤ -δ},
and:

u i = ψ i + k p q i ¯ i + k d ωi , (35)
with e ζ i , e wi as in [START_REF] Marco | Output regulation on the special Euclidean group SE(3)[END_REF], ηi , ¯ i ωi defined in [START_REF] Azzollini | Adaptive leader-follower synchronization over heterogeneous and uncertain networks of linear systems without distributed observer[END_REF], and tuning parameters given by the stabilizer gains k p > 0, k d > 0, the observer gains k ζ > 0, k w > 0, and the hysteresis margin δ ∈ (0, 1). The next statement provides formal guarantees for the effectiveness of (35) in solving Problem 1.

Theorem 1. The attractor A, corresponding to global phase synchronization to the reference ζ and given by:

A:={(ζ , w , χ 1 , . . . , χ N ) ∈ S 1 × W × (R n+4 × Q × R) N : ζi = ζ , ŵi = w , ζi = q i e, ωi = 0, ∀i ∈ V},
is globally robustly KL asymptotically stable for the closedloop system, having state (ζ , w , χ 1 , . . . , χ N ) and obtained from the interconnection of the second-order Kuramoto network [START_REF] Wu | Synchronization conditions for a third-order Kuramoto network[END_REF], the exosytem [START_REF] Scardovi | Synchronization and balancing on the N-torus[END_REF], and the local controllers (35).

Proof: The subsystems involved in the stability analysis are shown in Fig. 2, where we highlight the cascade structure of the closed-loop error dynamics. Asymptotic stability of A is sketched through reduction theorems. By Propositions 1 and 2 and standard ISS results, the observer estimation error dynamics is UGAS. In other words, the closed (but not compact) attractor

Â:={(ζ , w , χ 1 , . . . , χ N ) ∈ S 1 × W × (R n+4 × Q × R) N : ζi = ζ , ŵi = w , ∀i ∈ V},
is UGAS. On the set Â, we recover the full-information dynamics (32), thus, by Proposition 3, A is UGAS relative to Â. By [START_REF] Maggiore | Reduction theorems for hybrid dynamical systems[END_REF]Cor 4.8], A is uniformly asymptotically stable for the overall closed-loop system, with basin of attraction given by all initial conditions generating bounded solutions.

Next, we show that all closed-loop solutions are bounded, which then implies UGAS. Note that the state (ζ , w ) of ( 14) evolves in the bounded forward invariant set S 1 × W, thus it is bounded. Similarly, q i is bounded by construction. Due to Propositions 1 and 2, ( ζ, w) converge to zero, therefore ( ζi , ŵi ) are bounded for all i ∈ V. We need to show that ζi and ωi are bounded, for all i ∈ V. Recall that ζi :

= R(ζ i ) ζi , thus ζi is bounded as | ζi | ≤ | ζi |. Regarding ωi , let V ω (ω i ) := m i ω2
i /2 and we have, using [START_REF] Zhang | Adaptive cooperative tracking control of higher-order nonlinear systems with unknown dynamics[END_REF] and (35):

Vω = ωi -k d ωi -k p q i ¯ i -m i k w c e wi ≤ -kd 2 ω2 i + 1 2kd k p q i ¯ i + m i k w c e wi 2 ≤ -kd mi V ω (ω i ) + ν( ζi , w), (36)
where ν(•) is bounded by a positive scalar ν (depending on the initial conditions), because of the obtained bounds on ζi and w. Since Vω < 0 for V ω (ω i ) > m i ν/k d and V + ω -V ω = 0, it follows that ωi is contained in a compact set, which proves global boundedness of solutions. The proof is completed by noticing that the closed-loop system satisfies the hybrid basic conditions, thus [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF]Thm 7.21] ensures global robust KL asymptotic stability.

VII. NUMERICAL EXAMPLE

We consider a second-order Kuramoto network with the communication and interaction structure as in Fig. 1, where k 12 = 0.5, k 13 = 3, k 14 = 1, k 34 = 2, ϕ 12 = π/2, ϕ 13 = π/3, ϕ 14 = π/4, ϕ 34 = π/5. We report in Tab. I the parameters and the initial conditions of each oscillator. The Kuramoto network has been implemented according to (1) and, in addition, each θ i has been wrapped between -2π and 2π to ensure boundedness of the simulation variables. For feedback computation, a continuous selection of ζ i was obtained from [START_REF] Chopra | On exponential synchronization of Kuramoto oscillators[END_REF]. Exosystem [START_REF] Scardovi | Synchronization and balancing on the N-torus[END_REF] has been defined through: We provide in Fig. 3 the results of a simulation run, which confirms the effectiveness of the proposed solution. The quantities in subfigure (c) are computed as ϑ := 2atan2 (ζ 2 , ζ 1 ) , ϑ i := mod (θ i + π, 2π)-π, i ∈ V, where ϑ is the angular reference corresponding to ζ , while ϑ i represents θ i in the interval [-π, π).

S = 0 -1 0 1 0 0 0 0 0 , c = 0 1 1 (37)

VIII. CONCLUSIONS

Fig. 1 .

 1 Fig. 1. Interaction and communication scheme.

  are the local estimation errors, in which a ij and τ ii are the entries of the adjacency matrix A and the target matrix T . To represent the overall network dynamics in a compact form, define the overall states ζ := [ ζ 1 . . . ζ N ] ∈ R 2N and ŵ := [ ŵ 1 . . . ŵ N ] ∈ R N n and the overall estimation errors ζ := ζ -1 N ⊗ζ and w := ŵ-1 N ⊗w . Furthermore, define e ζ := [e ζ1 . . . e ζ N ] ∈ R 2N and e w := [e w1 . . . e w N ] ∈ R N n , which from (

Proposition 2 .

 2 For any scalar gain k ζ > 0, system (21) is finite-gain exponentially input-to-state stable (ISS) with respect to the input w, uniformly in the inputs (ζ , w ). Namely, there exist positive scalars b, λ, σ such that, for any solution (ζ (•), w (•)) of the exosystem[START_REF] Scardovi | Synchronization and balancing on the N-torus[END_REF] and any w(•) ∈ L ∞ , the solutions of system (21) satisfy:

Fig. 2 .

 2 Fig. 2. Sketch of the closed-loop error subsystems, with their interconnections and related stability results.

Fig. 3 .

 3 Fig. 3. Simulation results. (a): distributed observer phase estimation (reference in blue); (b): phase tracking errors; (c): phase angles represented in [-π, π) (reference in blue); (d): frequency tracking errors.

  with initial conditions w (0) = [0 0.5 1] and ζ (0) = [1 0] . Instead, the controllers (35) have been initialized randomly, while the tuning parameters have been selected as δ = 0.5, k p = k d = 1, k ζ = 2.5, and k w = 5.

We suggested a hybrid leader-follower phase synchronization control law for second-order Kuramoto oscillators. Under an undirected and connected graph topology, we provide an observer-stabilizer architecture, where the observer is distributed and the stabilizer exploits the local estimates of the reference motion, following a separation principle type of approach. Our main result, relying on hybrid reduction theorems, not only proves global convergence to the synchronization set but also its robust stability. Future work will provide an adaptive solution addressing unknown parameters in the dynamics of each oscillator.

S. Baldi was partly supported by the Double Innovation Plan Grant 4207012004 and the Special Funding for Overseas talents Grant 6207011901. L. Zaccarian was supported in part by ANR via Grant HANDY, number ANR-18-CE40-0010.


Arnaud Chéritat
email: arnaud.cheritat@math.univ-toulouse.fr

conjugate of Q θ is univalent on D). For all θ ∈ R \ Q, 1 20 rad ∆( Q θ ) inf f ∈S θ rad ∆(f ) rad ∆( Q θ ). ANNALES DE L'INSTITUT FOURIER

Keywords: Small divisors, Siegel disks, linearization. Math. classification: 37F50

value of r ∈ (0, +∞] such that there exists a conformal map ψ : D(0, r) → U with ψ(0) = 0 and ψ (0) = 1. It will be denoted by rad U in this article.

Remark. A consequence of Schwarz's lemma is that if U ⊂ U and U is also open connected and simply connected, then rad U rad U .

Let f be a holomorphic map fixing the origin in C. Assume that its differential at the origin is an aperiodic rotation: f (z) = e 2πiθ z + O(z 2 ) with θ ∈ R \ Q. If f is linearizable at the origin, the Siegel disk ∆(f ) is the maximal domain containing 0 on which f is conjugated to a rotation. Then f has a Siegel disk if and only if the radius of convergence of the normalized formal linearizing power series is positive, see Section 1.1. In this case the conformal radius of its Siegel disk is less than or equal to this convergence radius. If f is not linearizable let us set ∆(f ) = ∅. Let us set by convention that rad ∅ = 0.

Definition 2. -Let Q θ : C → C be the quadratic polynomial defined by: Q θ (z) = e 2πiθ z + z 2 .

In [18], Yoccoz used a technique of Ilyashenko and the polynomial-like map theory of Douady and Hubbard [7] to prove the following result.

Theorem A (Yoccoz). -For all ε > 0, there exists a constant c(ε) > 0 such that the following holds. For all θ ∈ R \ Q and f ∈ S θ , then:

rad ∆(f ) c(ε) • rad ∆(Q θ ) 1+ε .
Our first result, whose proof takes its roots in the one of Yoccoz, asserts that one can choose a constant c(ε) which does not depend on ε. This follows from the results obtained in [3] but there, the techniques are much more elaborate than the ones we present here.

Theorem 3 (main theorem)

. -Assume f ∈ S θ with θ ∈ R \ Q. Then rad ∆(f ) 1 10 rad ∆(Q θ ).
We shall prove this theorem in Section 1. The constant 1/10 is not optimal.

The main theorem can also be presented as follows.

Stated this way, the right hand inequality is trivial. The left hand follows from Theorem 3 and the following elementary formula expressing how rad ∆(f ) changes under a linear conjugacy: if g(z) = λf (z/λ) then rad ∆(g) = |λ| rad ∆(f ).

Y (θ) = +∞ n=0 α 0 • • • α n-1 log 1 α n
where α 0 = Frac(θ) = θ -θ and α n+1 = Frac(1/α n ). Definition 6. -A Brjuno number is an irrational real number θ satisfying Brjuno's condition Y (θ) < +∞. We denote by B the set of Brjuno numbers.

The following results were proved by Yoccoz in [18].

Theorem B (Yoccoz). -There exists a constant C ∈ R such that for all θ ∈ B, for all f ∈ S θ , the Siegel disk of f contains the disk D(0, e -Y (θ)-C ). In particular, log rad ∆(f ) -Y (θ) -C.

Corollary (Yoccoz). -If θ is a Brjuno number, rad ∆(Q θ ) > 0 and

log rad ∆(Q θ ) -Y (θ) -C -log 2.
The termlog 2 comes from the fact Q θ is univalent only on the disk D(0, 1/2).

Theorem C (Yoccoz). -There exists a constant C ∈ R such that for all θ ∈ R \ Q there exists f ∈ S θ , such that

log rad ∆(f ) -Y (θ) + C.
This includes the case θ / ∈ B (i.e. Y (θ) = +∞) if we interpret the above inequality as rad ∆(f ) = 0. Theorems B and C can be presented together as follows:

Corollary (Yoccoz).

- ∀θ ∈ R \ Q, -Y (θ) -C inf f ∈S θ log rad ∆(f ) -Y (θ) + C.
SUBMITTED ARTICLE : NEWYOCCOZ.TEX

Combining Theorem A and Theorem C, Yoccoz obtained the following corollaries.

Corollary (Yoccoz). -For all ε > 0, there exists C ε ∈ R (that a priori may tend to +∞ as ε -→ 0) such that for all θ ∈ R \ Q,

log rad ∆(Q θ ) -(1 -ε)Y (θ) + C ε .
In particular, if θ is not a Brjuno number, then rad ∆(Q θ ) = 0.

Corollary (Yoccoz).rad ∆(Q θ ) > 0 if and only if θ is a Brjuno number.

Consequence of the main theorem

The second author found an independent proof of "θ ∈ R \ B =⇒ rad ∆(Q θ ) = 0" in [6], working directly in the family Q θ . He looked at how parabolic points explode into cycles and how these cycles hinder each others. The control on parabolic explosion uses the combinatorics of quadratic polynomials, and the Yoccoz inequality on the limbs of the Mandelbrot set. The relative Schwarz lemma of the first author then enabled us to have a good enough control on conformal radii to prove the following result, conjectured by Yoccoz [18], that enhances his estimate.

Theorem 7. -There exists a constant C ∈ R such that for all θ ∈ R \ Q, log rad ∆(Q θ ) -Y (θ) + C.
This article gives a new proof of Theorem 7, as an immediate corollary of Yoccoz's Theorem C and the main theorem, Theorem 3.

Together with the corollary following Theorem B, this gives:

Corollary 8. -There exists a constant C ∈ R such that for all θ ∈ R \ Q, -Y (θ) -C log rad ∆(Q θ ) -Y (θ) + C.
Thus the function

Υ : B → R defined by Υ(θ) = log rad ∆(Q θ ) + Y (θ)
is uniformly bounded. In [4], we proved the stronger statement that this function has a continuous extension to R. This problem is not addressed in the present article.

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Extension to some other families of polynomials and counterexamples

In Section 2, we show that our techniques extend to other families of polynomials. Let us define a class of well-behaved polynomials that was studied by Lukas Geyer in [9]. Given a polynomial P : C → C, a critical orbit tail is an equivalence class in the set of forward critical orbits [START_REF] Bers | Holomorphic families of injections[END_REF] , with the relation z ≡ z ⇐⇒ ∃m, n ∈ N such that P •n (z) = P •m (z ). We say it is infinite if a point of the class (and therefore every point in the class) has infinite forward orbit. Every infinite critical orbit tail falls in one and only one of the following five categories: let

• ti 0 count the number of infinite critical orbit tails falling in a Siegel disk, • ti 1 count those that fall in a super-attracting basin,

• ti 2 those that fall in an attracting basin, not super-attracting,

• ti 3 those that fall in a parabolic basin,

• ti 4 those that belong to J.

Let ti count the total number of infinite critical orbit tails:

ti = ti 0 +• • •+ti 4 . Let
• n 1 be the number of superattracting cycles,

• n 2 be the number of attracting cycles, not super-attracting,

• n 3 be the number of parabolic cycles,

• n 3 n 3 be the number of cycles of parabolic petals,

• n 4 be the number of irrationally indifferent cycles.

By the Fatou-Shishikura inequality (see [7], [15], [8]) n 2 ti 2 , n 3 ti 3 , n 4 ti 4 . Geyer's condition is that n 4 = ti 4 . Note that we always have n 3 + n 4 ti. Definition 9. -A polynomial has property (G) if the number of infinite critical orbit tails is equal to the number of indifferent cycles, i.e. n 3 + n 4 = ti.

Thus, P has property (G) if and only if ti

4 = n 4 , ti 3 = n 3 = n 3 , ti 2 = n 2 = 0, ti 1 =
0 and ti 0 = 0, i.e. there is no attracting cycle, the basin of attraction of super-attracting cycles and Siegel disks contains no infinite critical orbit tail and the basin of attraction of each parabolic cycle contains at most one infinite critical orbit tail. It follows that each parabolic cycle has exactly one cycle of petals and is virtually repelling (see [5]). Property (G) is less general than Geyer's condition. [START_REF] Bers | Holomorphic families of injections[END_REF] or, equivalently, in the set of critical points SUBMITTED ARTICLE : NEWYOCCOZ.TEX Remark. If P has property (G), its iterates do not necessarily. For instance, take a degree 2 polynomial with a period one Siegel disk. Then n 3 (P ) = 0, n 4 (P ) = 1 and ti(P ) = 1. However n 3 (P •n ) = 0 and n 4 (P

•n ) = 1 whereas ti(P •n ) = n.
Lukas Geyer proved optimality of Brjuno's condition for polynomials satisfying n 4 = ti 4 (i.e. polynomials such that the number of infinite critical orbit tails in the Julia set is equal to the number of irrationally indifferent cycles; they are called saturated polynomials), by using the same method as Yoccoz. It is therefore natural that our new observation adapts in a similar yet slightly less general setting.

Definition 10. -The critical orbits of a polynomial P : C → C are the sets P •k (c) k 0 where c is a critical point of P . A point z in a critical orbit is said to be free [START_REF] Buff | Virtually repelling fixed points[END_REF] if for all critical point c , ∀k ∈ N, ∀ ∈ N * ,

P •k (c ) = P • (z) =⇒ k and P •(k-) (c ) = z .
We denote by Z P the set of non-free points of critical orbits.

This strange definition has the following interest: Z P is the smallest subset of the union of all critical orbits of P , on which for all map g : C → C such that g Z P = P Z P , for all critical points c, c of P and all integers

k 0, k 0, if P •k (c) = P •k (c ) then g •k (c) = g •k (c ).
When a critical point c has a finite forward orbit (i.e. when it is periodic or preperiodic), then c and its iterates are non-free points. When it has an infinite forward orbit, then there can be only finitely many non-free points in its orbit: let P •p (c), p 0, be the last point in the orbit of c where a critical orbit joins the orbit of c. The non-free points in the orbit of c are exactly the P •n (c) for 0 n < p. In particular, if no other critical point shares the same tail then every point in the orbit of c is free. Another consequence is that Z P is finite.

Let I P denote the set of all indifferent periodic points of P .

Definition 11. -A polynomial P with an indifferent fixed point at the origin has "property (G) with bound N and margin ε" if in addition to having property (G), the cardinal of I P ∪ Z P is at most N [START_REF] Buff | Upper bound for the size of quadratic Siegel disks[END_REF] and if ∀z ∈ I P ∪ Z P , either z = 0 or |z| ε. [START_REF] Buff | Virtually repelling fixed points[END_REF] this is not a standard terminology [START_REF] Buff | Upper bound for the size of quadratic Siegel disks[END_REF] in a family of polynomials with bounded degrees, it is equivalent to bound the cardinal of Z P and to bound the sum of local degrees at points in Z P

ANNALES DE L'INSTITUT FOURIER

Theorem 12. -Let N ∈ N, ε > 0 and C be a compact set of degree d polynomials h fixing 0 with indifferent multiplier e 2πiθ(h) , having property (G) with bound N and margin ε.

Let C R\Q = h ∈ C θ(h) ∈ R \ Q . Then ∃C ∈ R such that ∀h ∈ C R\Q , log rad ∆(h) log rad ∆(Q θ(h) ) + C.
The following lemma shows that it is sufficient to check the margin on only Z P . Lemma 13. -Any compact set C of degree d polynomials h fixing 0 with indifferent multiplier e 2πiθ(h) , having property (G) and such that ∀P ∈ C, |I P | N , must have a margin for I P (i.e. there must exist ε > 0 such that ∀P ∈ C and ∀z ∈ I P , either z = 0 or |z| ε).

Proof. Assume that this is not the case: then since C is compact, there would be a map P ∈ C, a sequence P n ∈ C and a sequence u n ∈ I P \{0} such that u n -→ 0 and P n -→ P . The point u n would belong to an indifferent cycle of P n with period bounded by |I P | N . So an iterate of P would have multiple fixed point at 0, i.e. P would have a parabolic fixed point at 0, and P n would have an indifferent fixed point at the origin, and an indifferent cycle close to 0. Then either P would have at least two cycles of petals at 0, or the parabolic fixed point at 0 would be virtually indifferent (see [2] for a definition). In both cases, the basin of attraction of 0 would contain at least two critical points (see [5] for a proof), contradicting the fact that P has property (G).

Remark. We did not try to get the most general result possible. For instance, it is possible that the hypothesis that 0 has period 1 is not required. [START_REF]The Brjuno function continuously estimates the size of quadratic Siegel disks[END_REF] Corollary 14. -Under the assumptions of Theorem 12, ∃C ∈ R such that

∀h ∈ C R\Q , -Y (θ(h)) -C log rad ∆(h) -Y (θ(h)) + C.
The lower bound follows from Yoccoz's Theorem B and the compactness of C which implies that there is a ball D(0, r) on which all maps in C are univalent. [START_REF]The Brjuno function continuously estimates the size of quadratic Siegel disks[END_REF] But in this case, the margin on I P is not automatic anymore: the indifferent cycle containing 0 may collapse on itself. Proof. Conjugate the maps in C by a translation to put the fixed point at the origin. We still have a compact family. By the Fatou-Shishikura inequality, the number of indifferent cycles is at most the number of infinite critical orbit tails. Thus Z P is empty and I P is a singleton. So we may apply Corollary 15.

Corollary 17. -Let d 2 be an integer, and C be the family

e 2πiθ z(1 -z) d-1 } θ∈R .
Then the function Υ is bounded on C B .

Proof. The critical points are z = 1/d and z = 1 (with multiplicity d -2). The second critical point is mapped in one step on z = 0 which is fixed, thus has finite orbit. Therefore the first one has an infinite orbit because [START_REF] Buff | A parabolic Pommerenke-Levin-Yoccoz inequality[END_REF] ti n 3 + n 4 1. Thus the two critical points have disjoint orbits, so the one with an infinite orbit is free. So Z P = {1, 0}. Also, there is only one indifferent cycle because n 3 + n 4 ti = 1. So I P = {0}. Thus we may apply Corollary 15: the family has property (G) with bound N = 2 and margin ε = 1.

Corollary 18. -For the family

f θ (z) = e 2πiθ (z + z d ), the following holds: ∃C > 0 such that ∀θ ∈ R \ Q, - Y (d -1)θ d -1 -C log rad ∆(f θ ) - Y (d -1)θ d -1 + C.
( [START_REF] Buff | A parabolic Pommerenke-Levin-Yoccoz inequality[END_REF] it also follows from Fatou and Mañé's theorems

ANNALES DE L'INSTITUT FOURIER

Proof. The family f θ is semi-conjugated to the previous family: more precisely let φ(z) = -z d-1 , and g θ (z) = e 2πiθ z(1 -z) d-1 . Then [START_REF] Chéritat | Recherche d'ensembles de Julia de mesure de Lebesgue positive[END_REF] φ

• f θ = g (d-1)θ • φ.
The Siegel disk of f θ is thus the preimage by z → z d-1 of the Siegel disk of g (d-1)θ . The claim follows. [START_REF] Douady | On the dynamics of polynomial-like mappings[END_REF] In Section 3, Lemma 36, we will prove that for any integer m 2, the function

θ ∈ B → Y (θ) - Y (mθ) m is
f θ (z) = e 2πiθ (z + z d ), the function Υ : θ ∈ B → log rad ∆(f θ ) + Y (θ)
is unbounded on any interval.

In fact, the family f θ does not satisfy property (G), which is an hypothesis of Corollary 15. Indeed, there is only one indifferent cycle: the origin, whereas there are n -1 infinite critical orbit tails. This can be seen either by using the semi-conjugacy or the fact that f θ commutes with the map z → e 2πi/(d-1) z.

In Section 3, Lemma 37, we will prove that

(∃C > 0) (∀m ∈ N * ) (∀θ ∈ R) Y (θ) Y (mθ) + C log m. It follows that log rad ∆(f θ ) - Y (θ) d -1 + C .
This suggests the following conjecture.

Conjecture 1. -There exists a constant C = C(d) ∈ R such that for all polynomial P of degree d with an indifferent fixed point at the origin,

log rad ∆(P ) - Y (θ) d -1 + log min |c i | + C
where the c i are the critical points of P and θ is the rotation number at the origin. [START_REF] Chéritat | Recherche d'ensembles de Julia de mesure de Lebesgue positive[END_REF] Note how the rotation number changed. [START_REF] Douady | On the dynamics of polynomial-like mappings[END_REF] If U is a connected simply connected open subset of C containing the origin and U

is the preimage of U by z → z k then rad U = k √ rad U .

SUBMITTED ARTICLE : NEWYOCCOZ.TEX

There are possible refinements according to how many recurrent critical points are associated to the indifferent fixed point.

Optimality of the quadratic polynomial

In this section, we prove Theorem 3. Our proof follows closely Yoccoz's proof of Theorem A. It uses the following notion.

The radius of convergence of the linearizing power series

This section defines the notion, relates it to the conformal radius of the Siegel disk and collects a few known results.

Definition 20. -Assume θ ∈ R\Q and f is a holomorphic map defined in a neighborhood of the origin and such that

f (z) = e 2πiθ z + O(z 2 ).
Then, there is a unique formal series

φ f (Z) = Z + +∞ n=2 φ n Z n in C[[Z]] such that φ f (e 2πiθ Z) = f • φ f (Z),
called the linearizing series. We let R(f ) ∈ [0, +∞] be its radius of convergence.

Remark. For a fixed θ, the coefficient φ n is a polynomial in the coefficients f 2 , . . . , f n of the power series expansion of f at the origin:

f (z) = e 2πiθ z + f 2 z 2 + f 3 z 3 + . . .
Let U be an open subset of C containing the origin and f : U → C be a holomorphic map such that f (z) = e 2πiθ z +O(z 2 ), with θ an irrational real number. The Siegel disk ∆ of f is the maximal domain ⊂ U on which f is linearizable. There is a Siegel disk if and only if R(f ) > 0. Note that R(f ) depends only on the germ [START_REF] Epstein | Infinitesimal Thurston rigidity and the Fatou-Shishikura inequality[END_REF] of f , whereas ∆ depends on the representative [START_REF] Epstein | Infinitesimal Thurston rigidity and the Fatou-Shishikura inequality[END_REF] An analytic germ is an equivalence class of analytic maps defined in a neighborhood of 0, two maps being equivalent if and only if they coincide in some possibly smaller neighborhood of 0. Analytic germs are completely characterized by the power series expansion at the origin.

ANNALES DE L'INSTITUT FOURIER

of the germ. Let rad be the conformal radius of ∆ with respect to 0, i.e. the unique r ∈ (0, +∞] such that there exists a conformal bijection ψ : D(0, r) → ∆ mapping 0 to 0 and such that ψ (0) = 1. Conjugating f by ψ gives a conformal self map of D(0, r) fixing the origin with multiplier e 2πiθ . So it is the rotation:

ψ -1 • f • ψ(z) = e 2πiθ z.
By uniqueness of the linearizing formal power series:

• R(f ) rad, • ∀z ∈ D(0, rad), ψ(z) = φ(z), • ∆ = φ(D(0, rad)) ⊂ φ(D(0, R(f ))).
Let inner denote sup r > 0 D(0, r) ⊂ ∆ . By Schwarz's lemma, rad inner. By Koebe's one quarter theorem, inner [START_REF] Bers | Holomorphic families of injections[END_REF] 4 rad. The radii R(f ) and rad are not necessarily equal. An obvious possibility would be that f has an extension to a bigger domain, and that this extension has a bigger linearization domain. But this is not the only thing that can happen, since φ f is not necessarily injective on its disk of convergence. In fact φ f can be any convergent power series of the form z + O(z 2 ). Indeed, for such a φ, we can set f (z) = φ e 2πiθ φ -1 (z) near 0. . . For instance, φ(z) = e z -1 = z + • • • has infinite radius of convergence, and is not injective on C. The map φ can also have critical points.

Let us recall the following fact.

Lemma 21. -If for some r R(f ), φ(D(0, r)) is contained in the domain of definition U of f then r rad and φ(D(0, r)) ⊂ ∆. Thus

rad = sup r R(f ) φ(D(0, r)) ⊂ U .
Proof. First note that the relation f • φ(z) = φ(e 2πiθ z) holds by analytic continuation for all z ∈ D(0, r). To prove the lemma, it is thus enough to prove that φ is injective on D(0, r), for then φ(D(0, r)) is a simply connected subset of U , it is invariant by f , and uniformizing it to a disk conjugates f to an automorphism that fixes the origin so is a rotation. If injectivity did not hold, then there would be a, b ∈ D(0, r) such that a = b and φ(a) = φ(b). Without loss of generality, let us assume that |a| |b| (so b = 0). By iterating f we would get φ(ρ n a) = φ(ρ n b) where ρ = e 2πiθ . Therefore the relation φ(z) = φ((a/b)z) would hold on a set with a point of accumulation in D(0, r). It therefore would hold everywhere on D(0, r) (both hands of the equality are defined when z ∈ D(0, r) because |a| |b|). This would contradict the injectivity of φ near 0. Proposition 22. -Let U be the domain of definition of f . If R(f ) > rad then ∆ is bounded and its boundary in C contains at least one point of ∂U . As a corollary:

• if ∆ U (this includes polynomials and polynomial-like maps [START_REF] Geyer | Linearization of structurally stable polynomials[END_REF] ),

• or f is entire (this includes polynomials too), then R(f ) = rad .

Proof. Since ψ(z) = φ(z) for all z ∈ D(0, rad), the set ∆ is bounded, because it is equal to ψ(D(0, rad)) that is contained in the compact set φ(D(0, rad)). Now if the boundary of ∆ did not contain a point of ∂U then ∆ would be compactly contained in U . Then there would exist some r > rad such that φ(D(0, r)) ⊂ U . By Lemma 21, r rad: contradiction.

Remark. If R(f ) > rad, then ∂∆ is the image of a euclidean circle by a holomorphic map φ. So it is smooth but at finitely many points, and it is locally connected. The map φ may however be non-injective on the circle and may also have critical points of order 2 (think of a rotation inside a cardioid).

Let us state the following extension of a lemma of Yoccoz [18].

Lemma 23. -Let U be the domain of definition of f . If there exist z 0 ∈ ∂U , ε > 0, a holomorphic map g : B(z 0 , ε) → C and a path in U ∩ B(z 0 , ε) ending on z 0 such that g = f on the path, we say that f has a path extension (note that the point z 0 must be accessible from U ). If f has no path extension then

R(f ) = rad .
Proof. Assume that R(f ) > rad. We have seen that, then, ∃z 0 ∈ ∂∆ ∩ ∂U and that ∂∆ = φ(∂D(0, | rad |)). Thus ∃a ∈ C such that |a| = rad and z 0 = φ(a). If a is not a critical point of φ then φ is invertible near a and it is possible to find a path extension of f at z 0 using φ: g(z) = φ(e 2πiθ φ -1 (z)), the path being for instance the internal ray of ∆: φ(ta) for t < 1 and close enough to 1. If a is a critical point of φ, then the function g is multivalued with only z 0 as a singularity, so z 0 cannot be an isolated point of ∂U for otherwise g would have to coincide with f in a neighborhood of z 0 , so g [START_REF] Geyer | Linearization of structurally stable polynomials[END_REF] The definition of polynomial-like maps is recalled in Definition 29. Note that polynomials have polynomial-like restrictions but are not polynomial-like by themselves, because C is not a compact subset of C.

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would be single-valued; then there exists an accessible z 1 ∈ ∂U near z 0 , at which f has a path extension.

Lemma 24. -Given a non empty connected open subset U of C, the set of holomorphic functions f : U → C that have no path extension is dense for the notion of uniform convergence on compact subsets of U .

Proof. If ∂U = ∅ there is nothing to prove. Otherwise let f 0 : U → C be holomorphic. Let K be a compact subset of U and ε > 0. Let u n ∈ U be a dense sequence. For each Proof. Let r = lim sup n→+∞ rad ∆(f n ). If r = 0 there is nothing to prove. Otherwise let θ n and ψ n be to f n what θ and ψ are to f . By compactness of the set of Schlicht functions, extracting a subsequence one can assume that ψ n converges on every compact subset of D(0, r) to some injective map ζ with ζ(0) = 0 and ζ (0) = 1. One can assume that θ n converges to some θ . Then the relation f n • ψ n (z) = ψ n (e 2πiθn z) passes to the limit:

u n ∈ U choose some v n ∈ ∂U such that |v n -u n | = dist(u n , ∂U ). There exists a sequence ε n such that h(z) := n ε n z -v n
f • ζ(z) = ζ(e 2πiθ z
) for all z ∈ B(0, r). By computing the derivative at the origin, one gets θ = θ mod 2π. Therefore ζ(B(0, r)) is a linearization domain, thus contained in ∆(f ). By Schwarz's lemma, r rad ∆(f ).

The following result can be found in [18], page 20.

Corollary 26 (Yoccoz).

- inf f ∈S θ R(f ) = inf f ∈S θ rad ∆(f ).
SUBMITTED ARTICLE : NEWYOCCOZ.TEX Proof. The following argument is very similar that of [18]. First, from R(f ) rad ∆(f ) it follows at once that inf f ∈S θ R(f ) inf f ∈S θ rad ∆(f ). To prove the reverse inequality, fix a function h : D → C having no path extension, provided for instance by Lemma 24. Given some f ∈ S θ , consider the sequence of maps

f n : D → C, z → 1 n z 2 h(z) + f (1 -1 n )z 1 -1 n
defined on D: they fix the origin and have multiplier e 2πiθ . They have no path extension. They also tend to f uniformly on compact subsets of D. By Lemma 25,

lim sup rad ∆(f n ) rad ∆(f ). By Lemma 23, R(f n ) = rad ∆(f n ).
There exists a sequence

ε n -→ 0 such that f n is injective on D(0, 1 -ε n ). Therefore the restriction u n to D of z → u n ((1 -ε n )z)/(1 -ε n ) belongs to S θ . So R(u n ) inf u∈S θ R(u). Also, R(f n ) = (1 -ε n )R(u n ).
Putting it all together we get:

rad ∆(f ) inf u∈S θ R(u).
Since this holds for all f ∈ S θ , this proves the corollary.

The following rule is elementary: if g(z) = λf (z/λ) holds in a neighborhood of 0 then

φ g (X) = λφ f (X/λ).
In particular, R(g) = |λ|R(f ).

Holomorphic motions

The following notion is defined in [11].

Definition 27. -A holomorphic motion of a subset X of the Riemann sphere S with parameter in a complex manifold Λ and base parameter λ 0 ∈ Λ is an analytic map h : Λ × X → S such that

• ∀z ∈ X, h(λ 0 , z) = z, • ∀z ∈ X, the function λ → h(λ, z) is holomorphic, • ∀λ ∈ Λ, the function z → h(λ, z) is injective.

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Then it can be proved, see [11], that h is continuous and in fact ∀λ ∈ Λ, the map z → h(λ, z) extends to a (non unique) quasiconformal homeomorphism of the Riemann sphere. The λ-lemma of Mañé, Sad and Sullivan in [11] states that a holomorphic motion of X has a unique extension to a holomorphic motion of the closure of X in S.

A family of sets X λ is said to undergo a holomorphic motion if there exists a holomorphic motion such that ∀λ, h(λ, X λ0 ) = X λ . The base parameter can then easily be forced to be any element of Λ.

Let B be a complex manifold. In the present article, an analytic family parameterized by B will refer to a family of maps g b , with b ∈ B, such that the map (b, z) → g b (z) is defined on an open subset of B × C, is analytic, and takes values in C.

An analytic family g b is said to undergo a parabolic bifurcation if there exists b 0 and n 1 such that z → g •n b (z) -z has a root that splits when b varies away from b 0 . Such a root is necessarily multiple and thus is a parabolic point of g b0 . But a parabolic point does not necessarily bifurcate.

The following lemma was formulated by Mañé, Sad and Sullivan in [11] for rational maps, but it is known to work alike for polynomial-like maps.

Lemma 28. -Let B be a simply connected complex manifold. Assume that (g b ) b∈B is an analytic family of polynomial-like maps. Assume that this family does not undergo a parabolic bifurcation. Then the Julia set of g b undergoes a holomorphic motion.

Proof. The Julia set is the closure of the set of repelling periodic points. Because the maps g b are polynomial-like of the same degree, the projection (b, z) → b is proper from the set

P n = (b, z) g •n b (z) = z to B.
The assumption that there is no parabolic bifurcation implies that roots of g •n b (z)-z can be locally followed as a continuous functions of b. Properness implies that they can be followed as global functions over the universal cover of B. Since B is simply connected, it is its own universal cover. These functions are in fact holomorphic as solutions of one dimensional analytic equations. So P n consists in the disjoint union of finitely many graphs of holomorphic functions from B to C. Let C be one of these graphs, let (b, z) ∈ C and µ = (g n b ) (z) be the corresponding multiplier. If µ = 1, then the multiplier must be constant over C itself, for otherwise there would be a parabolic bifurcation. If µ is a k-th root of unity, it is also the case, since C is also one of the graphs composing P kn . If µ is irrationnally indifferent is it also the case, for otherwise a nearby parameter would be a root of unity. From all this, it follows that if |µ| > 1 then the cycle is repelling on all of C. The repelling points can therefore be followed holomorphically on B an stay repelling. Moreover two different repelling periodic point cannot collide as b varies in B, otherwise the iterate of g b of order the GCD of their periods would have a multiple fixed point that splits. So the set of repelling periodic points undergoes a holomorphic motion. By the λ-lemma of [11], its closure also undergoes a holomorphic motion.

Proof of the main theorem

We will first compare the radius of convergence of the linearizing series of a univalent function f on D with the corresponding radius of Q θ (which is univalent on D(0, 1/2)).

Let us assume that θ ∈ R \ Q and R(Q θ ) > 0, since otherwise, there is nothing to prove. Consider a univalent function f : D → C fixing 0 with multiplier e 2πiθ . Following Ilyashenko and Yoccoz, consider the oneparameter families of maps

f a (z) = f (z) + az 2 and g b (w) = 1 b f 1/b (bw) = 1 b f (bw) + w 2 .
The family g b extends analytically at b = 0 by g 0 = Q θ . We have:

(1.1) (∀b ∈ C * ) R( g b ) = 1 |b| R(f 1/b ).
The following notion is defined in [7].

Definition 29. -A polynomial-like map is a proper holomorphic map (and thus a ramified covering) f : U → V of degree at least 2, between two simply connected domains U V of C. When the degree is 2, it is also called quadratic-like.

The following observation is essentially due to Yoccoz [18].

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Proof. Let b 1 = 2/19, w 1 = 4 and ζ 1 = 492/121. These number satisfy: [START_REF] Hardy | An introduction to the theory of numbers[END_REF] b

1 w 1 < 1 (1.2) w 1 (1 -b 1 w 1 ) 2 w 2 1 -ζ 1 (1.3) w 1 < ζ 1 (1.4)
Since f is univalent, we have, for all z ∈ D:

f (z) |z| 1 -|z| 2
(the Koebe function is extremal with this respect). It follows that when

|b| < b 1 , |w| = w 1 and |ζ| < ζ 1 , then |bw| < 1 and ( g b (w) -ζ) -(w 2 -ζ) = 1 b f (bw) w 1 (1 -b 1 w 1 ) 2 w 2 1 -ζ 1 < |w 2 -ζ|.
Thus by Rouché's theorem, every ζ ∈ D(0, ζ 1 ) = V has exactly two preimages by g b in D(0, w 1 ), counted with multiplicity. Therefore g b is proper holomorphic of degree 2. [START_REF] Mañé | On the dynamics of rational maps[END_REF] If U b were not connected, the component of U b containing 0 would be mapped biholomorphically to V , which, by Schwarz's lemma, is not possible since g b (0

) = 1. Last, U b is compactly contained in V because w 1 < ζ 1 .
We saw in Section 1.1 that R(f ) is equal to the conformal radius of the Siegel disk when f is polynomial-like.

Proposition 31. -Let θ ∈ R\Q. Let g b : U b → C be an analytic family of maps [START_REF] Pommerenke | With a chapter on quadratic differentials[END_REF] of the form g b (z) = e 2πiθ z + O(z 2 ) (so the rotation number is independent of b). Assume that they all have a Siegel disk ∆(g b ) ⊂ U b , and that its boundary in C is not empty and undergoes a holomorphic [START_REF] Hardy | An introduction to the theory of numbers[END_REF] The constants are not optimal, but this is not really important: for the purpose of this article, we need only the existence of positive solutions to the three equations. We chose b 1 very close to its maximal possible value. For b 1 close to 0, it is very easy to find solutions. [START_REF] Mañé | On the dynamics of rational maps[END_REF] A holomorphic map f : U → V between open sets, such that every point has d preimages counted with multiplicity, is proper (and has degree d by definition). Indeed, let K be a compact subset of V and L = f -1 (K). Let un ∈ L be a sequence: f (un) ∈ K. Passing to a subsequence we can assume that f (un) converges to some v ∈ V . The point v has d preimages in U counted with multiplicity. For all such preimage u (call m its multiplicity), let U ⊂ U be a compact neighborhood of u. Then by Rouché, there exists a neighborhood V of v such that for all v ∈ V \ {v}, U contains at least m preimages of v counted with multiplicity. Thus near the preimages of v, one still finds all preimages of points near enough to v. Therefore, there must be a subsequence of un converging in U . Its limit u satisfies f (u) = v thus belongs to L. Thus L is compact. Thus f is proper. [START_REF] Pommerenke | With a chapter on quadratic differentials[END_REF] that we do not assume polynomial-like SUBMITTED ARTICLE : NEWYOCCOZ.TEX motion (we do not require that the holomorphic motion commutes with the dynamics, nor even that ∆(g b ) U b ). Let rad ∆(g b ) be the conformal radius of ∆(g b ) with respect to 0. Then the function b → -log rad ∆(g b ) is harmonic.

Proof. The following proof of this lemma of Sullivan was communicated to us by Saeed Zakeri. First, note that the conformal radius varies continuously. Indeed, the holomorphic motion implies that the function b → ∂∆(g b ) is continuous for the Hausdorff topology on compact subsets of C. It implies the sequential continuity of b → (∆(g b ), 0) for the notion of Carathéodory convergence. The latter implies the continuity of the conformal radius. Then, consider an extension [START_REF] Ransford | Potential theory in the complex plane[END_REF] of the holomorphic motion to a holomorphic motion of all the plane, but which does not necessarily commute with the dynamics. Let b 0 be any parameter and w n be any sequence in the Siegel disk of parameter b 0 , converging to a point in the boundary of ∆(g b0 ). For b close to b 0 let w n (b) be the point to which w n is transported by the motion. Now look at

u n (b) = ψ -1 g b w n (b)
where

ψ
b, w) → b, ψ g b (w) is bi-analytic (its inverse is (b, w) → b, ψ -1 g b (w) ), so b → u n (b) is ana- lytic. Therefore, b → log |u n (b)
| is harmonic (it does not vanish). Now, the map b → log rad ∆(g b ) is the pointwise limit of this sequence of harmonic functions. This sequence is bounded from above by the continuous function b → log rad ∆(g b ), therefore there is, locally, a uniform upper bound. We can thus apply Harnack's theorem: the limit is locally uniform and harmonic.

Let us go back to the specific family g b we were studying. [START_REF] Ransford | Potential theory in the complex plane[END_REF] Slodkowsky's theorem [16] provides one, but we can also use the Bers-Royden [1] or the Sullivan-Thurston [17] version since this argument is local in terms of the parameter.

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Proposition 32. -The map b → log rad ∆(g b ) is well defined and harmonic in D(0, 2/19).

Proof. Let r 0 = 2/19. By Lemma 30, g b : U b → V is quadratic-like for all b ∈ D(0, r 0 ). The radius of convergence of φ g b coincides with the conformal radius of the Siegel disk ∆(g b ) by Proposition 22. The maps g b all have an indifferent fixed point. This is the only non repelling cycle of the quadratic-like map g b (there can be at most one, see [7]). So there is never a parabolic periodic point. So by Lemma 28, the Julia set of g b undergoes a holomorphic motion as b varies in D(0, r 0 ). The Siegel disk of g b is the connected component containing 0 of the complement of its Julia set. Thus the boundary of the Siegel disk undergoes a holomorphic motion too. So by Proposition 31, the map b → log rad(∆(g b )) is harmonic. By equation (1.1):

|b| = 1/10 =⇒ R(g b ) = 10R(f 1/b ). Thus (1.5) log rad ∆(Q θ ) = log 10 + avg |a|=10 log R(f a ). Proposition 34. -We have log R(f ) avg |a|=10 log R(f a )
Proof. Look at the formal linearizing power series of f a :

φ fa (Z) = Z + +∞ n=2 φ n (a)Z n . By Hadamard's theorem, 1 R(f a ) = lim sup n→+∞ n | φ n (a)|.
SUBMITTED ARTICLE : NEWYOCCOZ.TEX By looking at the formal equation defining φ, one can see that the coefficients φ n (a) are polynomials in a (see the lemma p. 59 in [18]). Therefore the maps a → log | φ n (a)| are subharmonic, so:

1 n log φ n (0) avg |a|=10 1 n log φ n (a) .
By Lemma 30, for |a| = 10, the map f a has a quadratic-like restriction. In that case, the linearizing map φ fa takes its values in D(0, 4/|a|) ⊂ D and it follows from the Cauchy inequalities that

φ n (a) 1 R(f a ) n .
By Proposition 32 b → R(g b ) is, in particular, a continuous non vanishing function on the circle |b| = 1/10. Thus, when |a| = 10, R(f a ) = R(g b )/10 reaches a minimum c > 0 and

1 n log φ n (a) log 1 R(f a ) log 1 c .
This uniform upper bound allows us to apply Fatou's lemma:

-log R(f ) = lim sup n→+∞ 1 n log | φ n (0)| avg |a|=10 lim sup n→+∞ 1 n log | φ n (a)| = -avg |a|=10 log R(f a ).
Equality (1.5) and Proposition 34 yield:

log R(f ) avg |a|=10 log R(f a ) = log rad ∆(Q θ ) -log 10, whence R(f ) 1 10 rad ∆(Q θ ).
Therefore, inf R(f )

1 10 rad ∆(Q θ ), with the infimum taken over S θ . By Corollary 26, inf rad ∆(f ) = inf R(f ). Q.E.D.

Other families of polynomials

In this section, we shall first prove Theorem 12. Assume N ∈ N and C is a compact set of degree d polynomials P fixing 0 with indifferent multiplier e 2πiθ(P ) , having property (G) with bound N and margin ε. Let Z P be the set of non-free points of critical orbits (see Definition 10). Let I P be the Proof. The particular form of G has the following consequences. First, every critical point of P is a critical point of g P,b with the same local degree. Since g P,b is polynomial-like of degree d, it has d -1 critical points counted with multiplicity, which is the same as for P . Therefore, the critical points of P and g P,b are the same and have the same degree. Recall that Z P contains the finite critical orbits. Thus for all point x in a finite critical orbit of P , for all b, g P,b (x) = P (x). In particular, a critical point of P with a finite orbit is a critical point of g P,b with a finite orbit. For all non-free point of a critical orbit of P , g P,b (x) = P (x). As a consequence, any two critical point in the same orbit tail of P are two critical points in the same orbit tail of g P,b . Therefore the number of infinite critical orbit tails of g P,b is at most that of P , and the latter is equal to the number of indifferent cycles of P by the assumption (G). So g P,b can not have more indifferent cycles than P . But the particular form of G implies also that all indifferent cycle of P is also a cycle of g P,b , with moreover the same multiplier. Therefore, g P,b must have exactly the same indifferent periodic points as P . Thus there can be no parabolic bifurcation.

Therefore by Lemma 28 the Julia set of g P,b undergoes a holomorphic motion. So, the same analysis as in Proposition 32 holds, with the triple (P, g P,b , r/2) playing the role of (Q θ , g b , 1/10), and we can thus write: We have not yet used the margin hypothesis: the points in (Z P ∪I P )\{0} stay bounded away from 0 when P varies in C. It is also the case for the points of C P for otherwise there would exist a P with 0 ∈ C P contradicting the fact that |P (0)| = 1. Now 0 ∈ I P thus G P (0) = 2 (-w) 1+deg P (w) with the product taken over w ∈ (Z P ∪I P ∪C P )\{0}. As a corollary, G P (0) is bounded away from 0 when P varies in C.

∀P ∈ C, log R(P ) = avg
Let a = 1/b and f P,a (z) = bg P,b (b -1 z) = a -1 g P,1/a (az). Then, as a -→ 0, f P,a tends (pointwise) to the degree 2 polynomial

f P (z) = e 2πiθ(P ) z + G P (0) 2 z 2
The same analysis as in Proposition 34 also holds, with (f P , f P,a , 2/r) playing the role of (f, f a , 10), and yields log R(f P ) avg Since G P (0) is bounded away from 0, we get Theorem 12.

Q.E.D.

Estimates on Yoccoz's Brjuno function

Lemma 36. -For any integer m 2, the function

θ ∈ B → Y (θ) - Y (mθ) m is unbounded on any interval.
Proof. The proof relies on the following fact. For any rational number p/q with p and q coprime, any integer k 1, and any Brjuno number θ,

(3.1) Y p q + k N + θ = N →+∞ log N q + O(1)

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where N denotes an integer. Let us first show how this enables us to conclude. Assume p/q is a rational number with p and q coprime and assume q and m are coprime. Choose a Brjuno number θ and set

θ N = p q + 1 N + θ .
Note that

mθ N = mp q + m N + θ
with mp and q coprime.

Then,

Y (θ N ) = N →+∞ log N q + O(1) and Y (mθ N ) = N →+∞ log N q + O(1).
Thus,

Y (θ N ) - Y (mθ N ) m = N →+∞ m -1 m • log N q + O(1) -→ N →+∞ +∞.
It follows that the function

θ ∈ B → Y (θ) - Y (mθ) m
is unbounded in any neighborhood of p/q. This implies our lemma since the set of rational numbers p/q with q and m coprime is dense in R.

Let us now prove estimate (3.1). We will use the continued fraction notation:

[a 0 , a 1 , . . . , a n ] = a 0 + 1

a 1 + 1 . . . + 1 a n . Set θ N = p q + k N + θ
. Recall that a rational has two finite continued fraction expansions, one with n odd, equal to the initial segment of close enough reals x < p/q and another with n even, equal to the initial segment of close enough reals x > p/q. An approximant of a real number is a partial quotient of its continued fraction expansion. It is also called a convergent. Let us recall the following property: if θ ∈ R and |θ -p/q| < 1/2q 2 then p/q is one of the approximants of θ (Theorem 184 p. 196 in [10]). If N is large enough, p/q is an approximant of θ N > p/q: there exists n even and N such that for all N > N : Then (Theorem 150 p. 167 in [10]) p q -q p = (-1) n = 1 because n is even, and (see [18] Section 1.2 p. 12)

p q = [a 0 ,
α 0 α 1 • • • α n-1 = |q θ N -p | = q p -p q q + kq N + θ -→ N →+∞ 1 q , α 0 α 1 • • • α n = |qθ N -p| = kq N + θ -→ N →+∞ 0 and 1 α n = - q θ N -p qθ N -p = N + θ kq 2 - q q = θ kq 2 mod 1 kq 2 .
In particular, α n+1 , which is the fractional part of 1/α n , can take only kq 2 values which all are Brjuno numbers. It follows that as N → +∞,

Y (θ N ) = log 1 α 0 + . . . + α 0 α 1 • • • α n-2 log 1 α n-1 O(1) + α 0 α 1 • • • α n-1 log 1 α n q -1 log N +O(1) + α 0 α 1 • • • α n Y (α n+1 ) o(1) = log N q + O(1). Lemma 37. -∃C > 0, ∀m ∈ N * , ∀θ ∈ R, Y (θ) Y (mθ) + C log m.
Proof. For θ ∈ Q it reads: -∞ -∞, which holds. Let us now assume θ irrational. For m = 1 it reads: Y (θ) Y (θ) which is trivial. So we now also assume that m 2. We will use the Brjuno sum:

B(θ) = n∈N log q n+1 q n
where p n /q n are the approximants of θ. We have the following arithmetical property (c.f. [18], page 14):

B(θ) -Y (θ) is bounded.
We recall that

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(a) if p n /q n are the approximants of α then [START_REF] Risler | Linéarisation des perturbations holomorphes des rotations et applications[END_REF] 1 2q n q n+1 < α -p n q n < 1 q n q n+1 , and also that (b) q n F n where F n is the n-th Fibonacci number [START_REF] Shishikura | On the quasiconformal surgery of rational functions[END_REF] . Last, (c) if |α-p/q| < 1/2q 2 , then p/q is an approximant of α. (Theorem 184 p. 196 in [10])

Now, for every approximant p n /q n of θ, note mp n /q n = p /q with q = q n /(m ∧ q n ). Case 1: p /q is itself an approximant of mθ in which case if we note p /q the next approximant of mθ, then 1 2q q < mθ -p q = m θ -p n q n < m q n q n+1 whence q > q n+1 q n 2mq q n+1 2m and thus log q q log q n+1 q -log 2m q log q n+1 q n -log 2m q .

Case 2: mp n /q n = p /q is not an approximant of mθ, which means that mθ -p q 1 2q 2 , and thus 1 q n q n+1 θ -p n q n 1 2mq 2 whence q n+1 2mq 2 q n 2mq n and thus log q n+1 q n log q n q n + log 2m q n . [START_REF] Risler | Linéarisation des perturbations holomorphes des rotations et applications[END_REF] By [18] p. 12, α-pn qn = (-1) n /qn(q n+1 +α n+1 qn). Since α n+1 ∈]0, 1[ and qn q n+1 , q n+1 < q n+1 + α n+1 qn < 2q n+1 . [START_REF] Shishikura | On the quasiconformal surgery of rational functions[END_REF] It follows by induction from q 0 = 1, q 1 = a 1 1, and qn = anq n-1 + q n-2 , 10.2.2 p. 166 of [10].

SUBMITTED ARTICLE : NEWYOCCOZ.TEX Finally, B(θ) = case 1 log q n+1 q n + case 2 log q n+1 q n log q q + log(2m) 1 q + log F n F n + log(2m) 1 F n .
The prime in the sum means the summand needs to be replaced by the smallest non increasing sequence greater or equal to the sequence log F n /F n .

For different values of n, the approximants p /q of mθ are different since p /q = mp n /q n and thus

B(θ) B(mθ) + log(2m) 1 F n + log F n F n + log(2m) 1 F n
Since F n is exponentially increasing, the sums (independent of θ) they are involved in are finite. We get

Y (θ) Y (mθ) + C 1 log m + C 2 Y (mθ) + C 3 log m with C 1 = 2 1 Fn , C 2 = log Fn Fn + 2 1 Fn log(2) + 2 B -Y ∞ and C 3 = (C 1 + C 2 / log 2) < +∞.
Proof. Choose r > 0, such that ∀a ∈ B, f a is defined and injective on D(0, r). By Yoccoz's Theorem B, for all a ∈ B, f a has a (not necessarily maximal) rotation domain contained in D(0, r) and of conformal radius r 0 := r e -Y (θ)-C . It implies that φ fa is convergent on a disc containing D(0, r 0 ), and maps this disc in D(0, r). Using Cauchy's inequalities, φ n (a) r/r n 0 whence n -1 log | φ n (a)| log(r)/n + log(1/r 0 ), which is bounded. By Fatou's lemma, [START_REF] Slodkowski | Holomorphic motions and polynomial hulls[END_REF] the function a → -log R(f a ) is therefore (everywhere) below its average on circles. But we can say more: by the Brelot-Cartan theorem (see [13], Theorem 3.4.3, p. 64), if we note u * (a) = lim sup a →a u(a ), then u * is subharmonic and u = u * except on a polar set.

We however cannot say that u itself is subharmonic (iff u = u * ) because it is not necessarily upper semicontinuous, as the following counterexample shows. Let f 0 = e 2πiθ z + O(z 2 ) be the restriction to D of a map f defined on an open set Ω containing D, and such that its Siegel disk ∆( f ) in Ω goes beyond the edge of D, i.e. is not contained in D (for instance f

(z) = e 2πiθ z on Ω = C). Then ∆(f 0 ) is the biggest f -invariant subdisk of ∆( f ) that is contained in D. Since ∆(f 0 ) ∆( f ), rad ∆(f 0 ) < rad ∆( f ).
Recall that (see Section 1. [START_REF] Slodkowski | Holomorphic motions and polynomial hulls[END_REF] Fatou's lemma states that if fn is a sequence of non-negative measurable functions on a measure space, then lim inf fn lim inf fn. As a corollary, if fn is a sequence of functions on the circle, uniformly bounded from above, then lim sup fn lim sup fn where the integral is for Lebesgue's measure on the circle.

SUBMITTED ARTICLE : NEWYOCCOZ.TEX Thus a → R(f a ) is not upper semi-continuous at a = 0. Now, still assuming θ ∈ B, the upper semicontinuity holds if, instead of considering the map a → -log R(f a ) we consider a → -log rad(∆(f a D )) and if all f a are defined on D and f a -→ f 0 for the compact open topology on D. This is a corollary of the work of Risler [14]. Lower semicontinuity also holds, this time for an elementary reason: see Lemma 25. We have thus proved:

Proposition 39. -Given θ ∈ B, let H θ (D) be the set of analytic functions f : D → C fixing 0 with multiplier e 2πiθ , equipped with the compact open topology. [START_REF] Sullivan | Extending holomorphic motions[END_REF] The map

H θ (D) → (0, 1] f → rad ∆(f ) is continuous.
This is not true for θ / ∈ B: for the rotation z → e 2πiθ z restricted to D, we have ∆ = D, so rad = 1, but for the nearby map e 2πiθ z + εz 2 , we have rad = 0. Proof. We already mentioned the continuity. Now, the same trick [START_REF] Yoccoz | Théorème de Siegel, nombres de Bruno et polynômes quadratiques[END_REF] as before yields subharmonicity with little effort: consider a function g as in the discussion above, i.e. holomorphic on D and with singularities at all points of ∂D. Consider the sequence of families

(a, z) → τ -1 n f a (τ n z) + 1 n z 2 g(z) with τ n = 1 - 1 n .
They all satisfylog R = -log rad ∆ as above, whence all these are (continuous) subharmonic functions of a. By the previous proposition, these functions tend (locally uniformly as A is locally compact) tolog rad ∆(f a ). [START_REF] Sullivan | Extending holomorphic motions[END_REF] uniform convergence on compact subsets of D [START_REF] Yoccoz | Théorème de Siegel, nombres de Bruno et polynômes quadratiques[END_REF] It would be nice to have a more satisfactory (no power series) proof. Also, it could be true that subharmonicity still holds if the domain of definition of f undergoes a holomorphic motion.

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Proposition 42. -Assume that f a : U a → C is an analytic family of maps of the form f (z) = e 2πiθ z + O(z 2 ), so that the rotation number is independent of a. Assume they all have a Siegel disk ∆(f a ), and that ∆(f a ) = C. Assume that the boundary in C of the Siegel disk ∂∆(f a ) undergoes a holomorphic motion (we do not require ∆(f a ) U a and we do not require the motion to commute with the dynamics). Then the function a → -log rad ∆(f a ) is harmonic. This is kind of surprising: let A denote the fact that a simply connected domain undergoes a holomorphic motion (of its boundary), and B denote the fact that this domain is a Siegel disk of an analytically varying family of analytic maps (with fixed rotation number) in D. Then A =⇒ -log rad is subharmonic, B =⇒ -log rad is subharmonic, (A and B) =⇒ -log rad is harmonic. . . Is it fair that when a number has two reasons to be negative, then it is null?

A.4. Other radii of interest

We have R(f ) = the radius of convergence of φ f and rad ∆(f ) = the biggest radius R below which φ f maps in ∆(f ).

Here are a few other "natural" radii that one could study

A.5. Thanks

We would like to thank the Referees for their very useful suggestions, that greatly enhance the readability of the present work, and allowed us to correct mistakes in the original manuscript.

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Yoccoz's work linking arithmetical properties of the rotation number with the size of Siegel disks Definition 5 .-

 5 For θ ∈ R \ Q let

Corollary 15 .

 15 -Under the same assumptions as in Theorem 12, let C B = h ∈ C θ(h) ∈ B . Then the function Υ : C B → R defined by Υ(h) = log rad ∆(h) + Y θ(h) is uniformly bounded. Corollary 16. -Let d 2 be an integer, and C be the boundary of the central hyperbolic component of the family of unicritical polynomials z d +c, i.e. the set of c ∈ C for which the polynomial z d + c has an indifferent fixed point (this is the only indifferent cycle by the Fatou-Shishikura inequality). Then the function Υ is bounded on C B .

  unbounded on any interval. It follows that the conclusions in Corollary 15 do not hold for this family: Corollary 19. -For the family

  is normally convergent on every compact subset of U and such that |h(z)| < ε on K and such that for all n, f = f 0 + h is unbounded on the segment [u n , v n ]. If such a function f had a path extension g, the same g would yield by analytic continuation of equalities a path extension of f along [u n , v n ] for some n. Contradiction. Lemma 25. -If f and f n are holomorphic functions defined on a common open subset U of C containing 0, with f (z) = e 2πiθ z + O(z 2 ) and f n (z) = e 2πiθn z + O(z 2 ), if moreover f n has a Siegel disk for all n and f n tends to f on every compact subset of U , then rad ∆(f ) lim sup n→+∞ rad ∆(f n ).

  f a : D(0, 1) → C a∈C and g b : D 0, 1/|b| → C b∈C defined by:

Lemma 30 .-

 30 If |b| < 2/19, the map g b has a quadratic-like restriction g b : U b → V with U b = z ∈ D(0, 4) g b (z) ∈ D(0, 492/121) and V = D(0, 492/121).

  g b is the unique conformal map from D(0, rad ∆(g b )) → ∆(g b ) that maps 0 to 0 and has derivative 1 there. For each b, the sequence w n (b) converges to a point in the boundary of the Siegel disk ∆(g b ). Thus, u n (b) rad ∆(g b ) converges to rad ∆(g b ). Recall that R(g b ) rad ∆(g b ) and that the map ψ g b and φ g b must coincide on D(0, rad ∆(g b )) (see Section 1.1). As a corollary, the map (b, w) → ψ g b (w) is analytic: indeed, the coefficients φ n (g b ) are polynomials of the coefficients of g b , thus vary analytically with b. Thus the map (

  Definition 33. -Let avg |z|=r m(z) denote the average of the function m(z) on the circle |z| = r (with respect to the Lebesgue measure on the circle). As an immediate consequence of Proposition 32, we have the following equality: log rad ∆(Q θ ) = avg |b|=1/10 log rad ∆(g b ). And since for |b| = 1/10 the map g b is polynomial-like, rad ∆(g b ) = R(g b ).

  ANNALES DE L'INSTITUT FOURIER set of indifferent periodic points of P : 0 ∈ I P . Let C P be the set of critical points of P . Set G P (z) = w∈Z P ∪I P ∪C P (z -w) 1+deg P (w) where deg P (w) = the local degree of P at w. For b ∈ C let g P,b = P + bG P . First, by compactness of C, by the bound N , and by the definition of G P , we see that the set G P P ∈ C ⊂ C[X] is bounded in the sense that it has bounded degree and bounded coefficients. Therefore, there exists r > 0 and R > 0, independent of P , such that |b| < r =⇒ g P,b (z) has a polynomial-like restriction g P,b of degree d from the component of g -1 P,b D(0, R) contained in D(0, R) to D(0, R). Lemma 35. -Fix any P ∈ C. As b varies in D(0, r), the polynomiallike map g P,b cannot undergo a parabolic bifurcation.

|b|=r/ 2

 2 log R(g P,b).

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  |a|=2/r log R(f P,a ). Now since f P and Q θ(P ) are related by a linear conjugacy: log R(f P ) = log R(Q θ(P ) ) -log |G P (0)| 2 and since f P,a and g P,b are related by a linear conjugacy: log R(f P,a ) = log R(g P,b ) + log |b|. Putting it all together, we get log R(P ) -log r/2 + log R(Q θ(P ) ) -log |G P (0)| 2 .

0 R

 0 1) rad ∆( f ) R( f ) = R(f 0 ). Thus: rad ∆(f 0 ) < R(f 0 ). Let f a = f 0 + az 2 g(z) for a ∈ C,where g(z) is any analytic function on D that is singular on all of ∂D (no path extension). Then by Lemma 23, R(f a ) = rad ∆(f a ). By Lemma 25, lim sup a→ = 0 rad ∆(f a ) rad ∆(f 0 ). Thus lim sup a→ = (f a ) < R(f 0 ).

(

Proposition 40 .-

 40 If θ ∈ B and A is a one complex dimensional parameter space and(a, z) ∈ A × D → f a (z) = e 2πiθ z + O(z 2 )is analytic, then the map a → -log rad ∆(f a ) is continuous and subharmonic.

A

  = the biggest radius R on which φ f is injective, B = the biggest radius R on which φ f has no critical point, C = the biggest radius R on which φ f has a meromorphic extension φ f , D = the biggest radius C on which φ f is injective, E = the biggest radius C on which φ f has no critical point.

  a 1 , . . . , a n ] and θ N = [a 0 , a 1 , . . . , a n + α n ] with α n ∈ ]0, 1[ .

	Set
	p q	= [a 0 , a 1 , . . . , a n-1 ]
	and for m < n, set
	α m = [0, a m+1 , . . . , a n + α n ].
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(*) This research was partially funded by the grant At the Boundary of Chaos of the Agence Nationale de la Recherche, ANR-08-JCJC-0002.

Corollary 4. -Let Q θ (z) = 2Q θ (z/2) (this affine

Appendix A. Remarks

This section does not claim to bring new results. It is just a discussion of probably known and hopefully useful facts.

A.1. Subharmonicity

If θ is a Brjuno number, then for all analytic family f a (z) = e 2πiθ z + O(z 2 ) of analytic maps, a ∈ A (notice that the rotation number does not vary with a), the function a → -log R(f a ) is the lim sup of subharmonic functions (see Definition 20): Proof. Let V a be the image of U a by the inversion z → 1/(z -c a ). The set V a is unbounded and undergoes a holomorphic motion of its boundary. The conformal radius of U a is the inverse of (see [12] Corollary 11.1) the capacity radius of C \ V a , which is itself expressible by an energy minimization (19) as follows (see [13]):

where µ varies in the set of non-atomic probability measures on ∂V a (together with its Borel σ-algebra) and E(µ) (the energy) is defined by

Since µ is non-atomic, the mass of the diagonal (u, v) u = v is null. The integrandlog u -v is bounded from below bylog diam(∂V a ), and the measure µ ⊗ µ is finite, therefore E(µ) is well defined and belongs to (-∞, +∞]. Choose a basepoint a 0 and let ξ a (z) : ∂V a0 → ∂V a be the holomorphic motion. Then, for all probability measure µ on ∂V a0 ,

Again, the integrandlog ξ a (u) -ξ a (v) has a lower bound, uniform over (u, v), when a remains in a compact set, becauselog diam(∂V a ) has. Thus using Harnack's inequality, we get that E (ξ a ) * µ is either a harmonic function of a or the constant function +∞. Now, -log r(a) = -inf E = sup -E so it is the supremum of a set of harmonic functions that are locally bounded from above. This yields a continuous subharmonic function.

A.3. Harmonicity

Let us recall Proposition 31, whose proof was communicated to us by Saeed Zakeri. (19) As a variant of this, one could instead use the transfinite diameter.

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