Ergodic and Bernoulli properties of analytic maps of complex projective space (Q2781413)

The main theorem of the paper is the following ergodic theoretic statement: If a measurable dynamical system is isomorphic to a one-sided Bernoulli shift of entropy \(h\), then the product diagonal factor (the product map modulo the relation \((x, y)\sim(y, x)\)) is isomorphic to a one-sided Bernoulli shift with entropy \(2h\).NEWLINENEWLINE It is furthermore shown that Ueda maps [\textit{T.Ueda}, Adv. Ser. Dyn. Syst. 13, 120--138 (1993; Zbl 0924.58059)] are product diagonal factors of some analytic maps of \(\mathbb{P}^1\). Using both results, the author derives examples of analytic maps in \(\mathbb{P}^2\) which are one-sided Bernoulli with respect to the measure of maximal entropy. The main result of the last section provides a description of a family of rational maps (with parabolic orbifold) which are one-sided Bernoulli with respect to their unique measure of maximal entropy.NEWLINENEWLINE The paper contains a detailed description of the background on one-sided generators and the Bernoulli problem for rational map [cf. \textit{R. Mané}, Ergodic Theory Dyn. Syst. 5, 71--88 (1985; Zbl 0605.28011)].