Ergodic properties of fibered rational maps
type: Item
instance of: scholarly article
title: Ergodic properties of fibered rational maps
zbMATH Open document ID: 1021.37019
DOI: 10.1007/BF02384321
author: Mattias Jonsson
published in: Arkiv för Matematik
publication date: 2003-06-16T00:00:00Z
review text: Let \(X\) be a compact metric space, let \(g \colon X \to X\) be a continuous mapping, and let \(\widehat \mathbb{C}\) denote the Riemann sphere. A rational map of degree \(d\) fibered over \(g\) is a continuous mapping \(f \colon X \times \widehat \mathbb{C}\to X \times \widehat \mathbb{C}\) of the form \[ f(x,z) = (g(x),Q_x(z)) , \] where \(Q_x\) is a rational function of degree \(d\), depending continuously on \(x \in X\). In this paper, the author investigates the ergodic properties of fibered rational maps. The special case when \(X\) is a point recovers the class of (nonfibered) rational maps of \(\widehat \mathbb{C}\). The study of the ergodic properties of such maps was initiated by \textit{H. Brolin} [Ark. Mat. 6, 103-144 (1965; Zbl 0127.03401)] (in the case of polynomials), and further developed by \textit{A. Freire}, \textit{A. Lopez} and \textit{R. Mané }[Bol. Soc. Bras. Mat. 14, 45-62 (1983; Zbl 0568.58027)], \textit{M. Lyubich} [Ergodic Theory Dyn. Syst. 3, 351-385 (1983; Zbl 0537.58035] and \textit{R. Mané} [Bol. Soc. Bras. Mat. 14, 27-43 (1983; Zbl 0568.58028]. The purpose of this paper is to generalize the above results to the fibered setting. In particular, the author computes the topological entropy of such maps and constructs a measure of maximal relative entropy. The measure is shown to be the unique one with this property. One motivation for the study of fibered rational maps is that they can be used to understand the dynamics of certain holomorphic mappings in two complex dimensions. The first situation that the author addresses is when \(Y\) is a ruled surface, that is a smooth projective variety with the structure of a \(\mathbb{P}^1\)-bundle \(\pi \colon Y \to X\) over a compact Riemann surface \(X\). He also considers holomorphic mappings of the complex projective plane \(\mathbb{P}^2\).
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37A25
zbMATH DE Number: 1928894
zbMATH Keywords: ergodic properties
zbMATH Keywords: topological entropy
zbMATH Keywords: measure of maximal relative entropy
zbMATH Keywords: fibered rational maps
zbMATH Keywords: holomorphic mappings
zbMATH Keywords: complex projetive plane
MaRDI profile type: Publication
cites work: Entropy for Group Endomorphisms and Homogeneous Spaces
cites work: Invariant sets under iteration of rational functions
cites work: An invariant measure for rational maps
cites work: Q3750690
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: Dynamics of polynomial skew products on \(\mathbb{C}^2\)
cites work: Q3741845
cites work: A Relativised Variational Principle for Continuous Transformations
cites work: Q3322903
cites work: On the uniqueness of the maximizing measure for rational maps
cites work: Hyperbolicité des polynômes fibrés
cites work: Fatou sets in complex dynamics on projective spaces
cites work: Q4844406
full work available at URL: https://doi.org/10.1007/bf02384321
OpenAlex ID: W1979511225
description: scientific article; zbMATH DE number 1928894