arXiv subjects

Henry De Thélin

Publications and source records attributed to Henry De Thélin.

5 recordsLinked to original sources

arXiv2024

Exponential mixing of all orders and CLT for generic birational maps of $\mathbb{P}^k$

For Hénon maps, Bianchi and Dinh recently proved the exponential mixing of all orders for the measure of maximal entropy and, as a consequence of the recent work of Björklund and Gorodnik, the CLT for Hölder observables. We extend their results to generic birational maps of $\mathbb{P}^k$. Because of the indeterminacy set, Hölder maps are not stable under iteration, so we need to work with a suitable space of test functions.

math.DS↗

arXiv2020

Parametric Lyapunov exponents

In an algebraic family of rational maps of $\mathbb{P}^1$, we show that, for almost every parameter for the trace of the bifurcation current of a marked critical value, the critical value is Collet-Eckmann. This extends previous results of Graczyk and Świcatek in the unicritical family, using Makarov theorem. Our methods are based instead on ideas of laminar currents theory.

math.DS↗

arXiv2018

Entropy of meromorphic maps acting on analytic sets

Let $f : X\to X$ be a dominating meromorphic map on a compact Kähler manifold $X$ of dimension $k$. We extend the notion of topological entropy $h^l_{\mathrm{top}}(f)$ for the action of $f$ on (local) analytic sets of dimension $0\leq l \leq k$. For an ergodic probability measure $ν$, we extend similarly the notion of measure-theoretic entropy $h_ν^l(f)$. Under mild hypothesis, we compute $h^l_{\mathrm{top}}(f)$ in term of the dynamical degrees of $f$. In the particular case of endomorphisms of $\mathbb{P}^2$ of degree $d$, we show that $h^1_{\mathrm{top}}(f)= \log d$ for a large class of maps but we give examples where $h^1_{\mathrm{top}}(f)\neq \log d$.

math.CV↗

arXiv2018

The bifurcation measure has maximal entropy

Let $Λ$ be a complex manifold and let $(f_λ)_{λ\in Λ}$ be a holomorphic family of rational maps of degree $d\geq 2$ of $\mathbb{P}^1$. We define a natural notion of entropy of bifurcation, mimicking the classical definition of entropy, by the parametric growth rate of critical orbits. We also define a notion a measure-theoretic bifurcation entropy for which we prove a variational principle: the measure of bifurcation is a measure of maximal entropy. We rely crucially on a generalization of Yomdin's bound of the volume of the image of a dynamical ball. Applying our technics to complex dynamics in several variables, we notably define and compute the entropy of the trace measure of the Green currents of a holomorphic endomorphism of $\mathbb{P}^k$.

math.DS↗

arXiv2010

Dynamics of automorphisms on compact Kähler manifolds

We study holomorphic automorphisms on compact Kähler manifolds having simple actions on the Hodge cohomology ring. We show for such automorphisms that the main dynamical Green currents admit complex laminar structures (woven currents) and the Green measure is the unique invariant probability measure of maximal entropy.

math.DS↗