arXiv subjects
Publications and source records attributed to Juan Rivera-Letelier.
The pourpose of these notes is to give an elementary description of one of the simplest Berkovich's spaces: that associated to the projective line of a complete and algebraically closed ultrametric field. \ L'objectif de ces notes est de donner une description élémentaire de l'un des espaces de Berkovich le plus simple : celui associé à la droite projective, où le corps ultramétrique de base sera supposé complet et algébriquement clos.
We study geometric and statistical properties of complex rational maps satisfying the Topological Collet-Eckmann Condition. We show that every such a rational map possesses a unique conformal probability measure of minimal exponent, and that this measure is non-atomic, ergodic and that its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we show that there is a unique invariant probability measure that is absolutely continuous with respect to this conformal measure, and we show that this measure is exponentially mixing (it has exponential decay of correlations) and that it satisfies the Central Limit Theorem. We also show that for a complex rational map f the existence of such an invariant measure characterizes the Topological Collet-Eckmann Condition, and that this measure is the unique equilibrium state with potential - HD(J(f)) ln |f'|.
We introduce and study a non uniform hyperbolicity condition for complex rational maps, that does not involve a growth condition. We call this condition Backward Contraction. We show this condition is weaker than the Collet-Eckmann condition, and than the summability condition with exponent~1. Our main result is a connecting lemma for Backward Contracting rational maps, roughly saying that we can perturb a rational map to connect each critical orbit in the Julia set with an orbit that does not accumulate on critical points. The proof of this result is based on Thurston's algorithm and some rigidity properties of quasi-conformal maps. We also prove that the Lebesgue measure of the Julia set of a Backward Contracting rational map is zero, when it is not the whole Riemann sphere. The basic tool of this article are sets having a Markov property for backward iterates, that are holomorphic analogues of nice intervals in real one-dimensional dynamics.
We introduce a new class of adelic heights on the projective line. We estimate their essential minimum and prove a result of equidistribution (at every place) for points of small height with estimates on the speed of convergence. To each rational function R in one variable and defined over a number field K, is associated a normalized height on the algebraic closure of K. We show that these dynamically defined heights are adelic in our sense, and deduce from this equidistribution results for preimages of points under R at every place of K. Our approach follows that of Bilu, and relies on potential theory in the complex plane, as well as in the Berkovich space associated to the projective line over C_p, for each prime p.
A classification of the periodic components of the Fatou set of $p$-adic rational maps. Each such periodic component is either an immediate attracting basin or an open affinoid, where the dynamics is quasi-periodic (the $p$-adic analogues of Siegel discs and Herman rings.) The main tool is a fixed point theorem for connected subsets of Berkovich's projective line (or p-adic hyperbolic space).
We prove an analog of the famous equidistribution theorem of Brolin for rational mappings in one variable defined over the p-adic field C_p. We construct a mixing invariant probability measure which describes the asymptotic distribution of iterated preimages of a given point. This measure is supported on the Berkovich space associated to the projective line over C_p. We show that its support is precisely the Julia set as defined by Rivera-Letelier. Our results are based on the construction of a Laplace operator on real trees with arbitrary number of branching as done by Favre-Jonsson.
It is conjectured that a rational map whose coefficients are algebraic over $\Q_p$ has no wandering components of the Fatou set. R. Benedetto has shown that any counter example to this conjecture must have a wild recurrent critical point. We provide here the first examples of rational maps whose coefficients are algebraic over $\Q_p$ and that have a (wild) recurrent critical point. In fact, we show that there is such a rational map in every one parameter family of rational maps that is defined over a finite extension of $\Q_p$ and that has a Misiurewicz bifurcation.
We study the dynamics of skew product endomorphisms acting on the cylinder $\cyl$, of the form $$ \tht \mapsto (\ell θ, \la θ+ τ(θ)), $$ where $ \ell \geq 2$ is an integer, $\la \in (0,1)$ and $τ: \T \to \R$ is a continuous function. We are interested on {\it topological} properties of the global attractor $\Omegalt$ of this map. Given $\ell$ and a Lipschitz function $τ$, we show that the attractor set $\Omegalt$ is homeomorphic to a closed topological annulus for all $\la$ sufficiently close to 1. Moreover, we prove that $\Omegalt$ is a Jordan curve for at most finitely many $\la \in (0,1)$. These results rely on a detailed study of iterated ``cohomological'' equations of the form $τ= \cL_{\la_1} μ_1$, $μ_1 = \cL_{\la_2} μ_2, >...$, where $\cL_\la μ= μ\circ \ml - \la μ$ and $\ml: \T \to \T$ denotes the multiplication by $\ell$ map. We show the following finiteness result: each Lipschitz function $τ$ can be written in a canonical way as, $$ τ= \cL_{\la_1} \circ ... \circ \cL_{\la_m} μ, $$ where $m \ge 0$, $λ_1, ..., λ_m \in (0, 1]$ and the Lipschitz function $μ$ satisfies $μ\neq \cL_\la ρ$ for every continuous function $ρ$ and every $\la \in (0,1]$.