The bifurcation measure has maximal entropy
| Autor: | Thomas Gauthier, Gabriel Vigny, Henry de Thelin |
|---|---|
| Přispěvatelé: | Laboratoire Émile Picard (LEP), Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), NASA Goddard Space Flight Center (GSFC), Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 (LAMFA), Université de Picardie Jules Verne (UPJV)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord |
| Rok vydání: | 2018 |
| Předmět: |
Pure mathematics
Endomorphism General Mathematics [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] Holomorphic function 0102 computer and information sciences Dynamical Systems (math.DS) Lambda 01 natural sciences 37B40 28D20 37F45 37F10 Variational principle FOS: Mathematics Entropy (information theory) 0101 mathematics [MATH]Mathematics [math] Mathematics - Dynamical Systems Complex Variables (math.CV) Bifurcation ComputingMilieux_MISCELLANEOUS Mathematics Mathematics - Complex Variables 010102 general mathematics [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV] 16. Peace & justice Complex dynamics 010201 computation theory & mathematics Complex manifold |
| Zdroj: | Israël Journal of Mathematics Israël Journal of Mathematics, Hebrew University Magnes Press, 2020, 235 (1), pp.213-243. ⟨10.1007/s11856-019-1955-6⟩ |
| ISSN: | 0021-2172 1565-8511 |
| DOI: | 10.48550/arxiv.1805.11508 |
| Popis: | Let $\Lambda$ be a complex manifold and let $(f_\lambda)_{\lambda\in \Lambda}$ 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 $\mathbb{P}^k$. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|
Full text from SpringerLink