A Real K3 Automorphism with Most of Its Entropy in the Real Part

TL;DR

This paper constructs a specific real K3 surface with an automorphism where the entropy on the real part dominates that on the complex part, and develops a shadowing lemma for real surface diffeomorphisms.

Contribution

It presents a novel example of a real K3 surface with an automorphism exhibiting high real entropy and introduces a shadowing lemma for real surface diffeomorphisms.

Findings

Example of a real K3 surface with entropy dominance on the real part

Development of a shadowing lemma for $C^2$ real surface diffeomorphisms

Application of computational tools for entropy and periodic point estimation

Abstract

This article describes an example of a real projective K3 surface admitting a real automorphism $f$ satisfying $h_{top}(f, X(\mathbb{C})) < 2 h_{top}(f, X(\mathbb{R}))$. The example presented is a $(2,2,2)$-surface in $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ given by the vanishing set of $(1 + x^2)(1 + y^2)(1 + z^2) + 10xyz - 2$, first considered by McMullen. Along the way, we develop an ad hoc shadowing lemma for $C^2$ (real) surface diffeomorphisms, and apply it to estimate the location of a periodic point in $X(\mathbb{R})$. This result uses the GNU MPFR arbitrary precision arithmetic library in C and the Flipper computer program.