Ergodic properties of fibered rational maps
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 \(\mathbb{P}^2\).
scientific article; zbMATH DE number 1928894
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