Typical dynamics of plane rational maps with equal degrees

TL;DR

This paper investigates the ergodic properties of plane rational maps with equal degrees, showing that generic perturbations have exactly two ergodic measures of maximal entropy, neither supported on algebraic curves, and lack invariant foliations.

Contribution

It demonstrates that for a broad class of such maps, generic perturbations exhibit two ergodic measures of maximal entropy and are fully two-dimensional, extending understanding of their complex dynamics.

Findings

Existence of exactly two ergodic measures of maximal entropy for generic perturbations.

Neither measure is supported on an algebraic curve.

Perturbed maps do not preserve any singular holomorphic foliation.

Abstract

Let $f:\mathbb{CP}^2\dashrightarrow\mathbb{CP^2}$ be a rational map with algebraic and topological degrees both equal to $d\geq 2$. Little is known in general about the ergodic properties of such maps. We show here, however, that for an open set of automorphisms $T:\mathbb{CP}^2\to\mathbb{CP}^2$, the perturbed map $T\circ f$ admits exactly two ergodic measures of maximal entropy $\log d$, one of saddle and one of repelling type. Neither measure is supported in an algebraic curve, and $T\circ f$ is `fully two dimensional' in the sense that it does not preserve any singular holomorphic foliation. Absence of an invariant foliation extends to all $T$ outside a countable union of algebraic subsets. Finally, we illustrate all of our results in a more concrete particular instance connected with a two dimensional version of the well-known quadratic Chebyshev map.