Jean-Christophe Yoccoz

Julia and John

Using a recent result of Mañé [Ma] we give a classification of polynomials whose Fatou components are John domains.

UN THEOREME D'INDICE POUR LES HOMEOMORPHISMES DU PLAN AU VOISINAGE D'UN POINT FIXE

Let f be a local homeomorphism of the plane with a fixed point z which is a locally maximal invariant set and which is neither a sink nor a source. We prove that there are two integers q > 1 and r > 1 such that the sequence i(fk, Z) of the indices at z of the iterates of f satisfy i(fk, z) = 1 - rq if k is a multiple of q and i(fk, z) = 1 otherwise. As a corollary we deduce that there is no minimal homeomorphism on the infinite annulus or more generally on the two-dimensional sphere minus a finite set of points. We also construct for a local homeomorphism f as above a topological invariant which is a cyclically ordered set with an automorphism on it; this allows us in particular to define a rotation number for f (rational of denominator q).Let f be a local homeomorphism of the plane with a fixed point z which is a locally maximal invariant set and which is neither a sink nor a source. We prove that there are two integers q > 1 and r > 1 such that the sequence i(fk, Z) of the indices at z of the iterates of f satisfy i(fk, z) = 1 - rq if k is a multiple of q and i(fk, z) = 1 otherwise. As a corollary we deduce that there is no minimal homeomorphism on the infinite annulus or more generally on the two-dimensional sphere minus a finite set of points. We also construct for a local homeomorphism f as above a topological invariant which is a cyclically ordered set with an automorphism on it; this allows us in particular to define a rotation number for f (rational of denominator q).

Introduction to Hyperbolic Dynamics

A diffeomorphism of a smooth (compact) manifold may exhibit a globally (uniformly) hyperbolic behaviour, like Morse-Smale, Anosov, or Axiom A diffeomorphisms. But, even when the global behaviour is not hyperbolic, it occurs very frequently that the set of points whose orbits are constrained to stay in some appropriate open subset of the manifold is compact and has a hyperbolic structure. Such hyperbolic compact invariant sets then provide a good starting point for a global understanding of the dynamics.

Complex Brjuno functions

1.1. The real Brjuno function. Let α ∈ R Q and let (pn/qn)n≥0 be the sequence of the convergents of its continued fraction expansion. A Brjuno number is an irrational number α such that ∑∞ n=0 log qn+1 qn < +∞. The importance of Brjuno numbers comes from the study of one–dimensional analytic small divisors problems. In the case of germs of holomorphic diffeomorphisms of one complex variable with an indifferent fixed point, extending a previous result of Siegel [S], Brjuno proved ([Br1], [Br2]) that all germs with linear part λ = e are linearizable if α is a Brjuno number. Conversely the third author proved that this condition is also necessary [Yo1]. Similar results hold for the local conjugacy of analytic diffeomorphisms of the circle ([KH], [Yo2], [Yo3]) and for some area–preserving maps ([Ma], [Da1]), including the standard family ([Da2], [BG1], [BG2]). The set of Brjuno numbers is invariant under the action of the modular group PGL (2,Z) and it can be characterized as the set where the Brjuno function B : R Q→ R ∪ {+∞} is finite. This arithmetical function is Z–periodic and satisfies the remarkable functional equation