Topological entropy on saddle sets in \(\mathbb{P}^2\)
type: Item
instance of: scholarly article
title: Topological entropy on saddle sets in \(\mathbb{P}^2\)
zbMATH Open document ID: 0983.37054
DOI: 10.1215/S0012-7094-00-10324-9
author: Jeffrey Diller
author: Mattias Jonsson
published in: Duke Mathematical Journal
publication date: 2001-01-30T00:00:00Z
review text: The authors establish concepts of a saddle set and a terminal for a holomorphic map \(f\) from \({\mathbb P}^2\) into itself. A saddle set is an invariant set \(\Lambda\) which is transitive and hyperbolic (with nontrivial expansion and nontrivial contraction), and in some sense it is both maximal and isolated to the latter property. A terminal is an invariant set \(\Lambda\) such that for any negative semi-orbit \(\hat{p} \subset \Lambda\), the iterates of \(f\) act normally on \(W^u_{loc}(\hat{p})\smallsetminus \Lambda\). They prove that for a holomorphic map of degree \(d\) the topological entropy of \(f\) restricted to a saddle set \(\Lambda\) is no greater than \(\log d\) and the equality holds, iff, \(\Lambda\) is terminal.
reviewed by: Jerzy Ombach
Mathematics Subject Classification ID: 37F15
Mathematics Subject Classification ID: 32H50
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 37F10
zbMATH DE Number: 1484734
zbMATH Keywords: holomorphic dynamics
zbMATH Keywords: saddle set
zbMATH Keywords: topological entropy
MaRDI profile type: Publication
cites work: Q3135846
cites work: Polynomial diffeomorphisms of \({\mathbb C}^2\): Currents, equilibrium measure and hyperbolicity
cites work: The Dirichlet problem for a complex Monge-Ampère equation
cites work: Entropy for Group Endomorphisms and Homogeneous Spaces
cites work: Equilibrium states and the ergodic theory of Anosov diffeomorphisms
cites work: Unique ergodicity for horocycle foliations
cites work: Q4869193
cites work: Hyperbolic maps on \(\mathbb{P}^2\)
cites work: Socioemotional Development and Health from Adolescence to Adulthood, L.Pulkkinen, J.Kaprio, R.J.Rose (Eds.). Cambridge University Press, New York (2006). 439pp., $34.99, hardcover, ISBN: 0‐521‐84631‐5
cites work: JULIA SETS OF SKEW PRODUCTS IN C2
cites work: Q4301007
cites work: Ergodic properties of fibered rational maps
cites work: A nonalgebraic attractor in $\mathbf{P}^{2}$
cites work: Q4840149
cites work: Ergodic Attractors
cites work: Currents, flows and diffeomorphisms
cites work: Hyperbolicité des polynômes fibrés
cites work: Fatou sets in complex dynamics on projective spaces
cites work: Q4251727
full work available at URL: s0012-7094-00-10324-9
OpenAlex ID: W1977405179
description: scientific article; zbMATH DE number 1484734