Entropy of real rational surface automorphisms

TL;DR

This paper investigates the relationship between real and complex dynamics of rational surface automorphisms, focusing on entropy and the influence of invariant cubic curves, providing examples with varying entropy behaviors.

Contribution

It introduces methods to analyze how real automorphisms act on homology and presents examples illustrating entropy preservation and decrease when restricting to real parts.

Findings

Entropy of complex automorphisms can equal that of real restrictions.

Examples where real restriction reduces entropy compared to complex automorphisms.

An automorphism with positive entropy and all periodic cycles real.

Abstract

We compare real and complex dynamics for automorphisms of rational surfaces that are obtained by lifting \chg{some} quadratic birational maps of the plane. In particular, we show how to exploit the existence of an invariant cubic curve to understand how the real part of an automorphism acts on homology. We apply this understanding to give examples where the entropy of the full (complex) automorphism is the same as its real restriction. Conversely and by different methods, we exhibit different examples where the entropy is strictly decreased by restricting to the real part of the surface. Finally, we give an example of a rational surface automorphism with positive entropy whose periodic cycles are all real.