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

Abstract

This article describes an example of a real projective K3 surface admitting a real automorphism $f$ satisfying $h_{top}(f,X(\mathbb{C}))<2h_{top}(f,X(\mathbb{R}))$. 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.

29 pages, 4 figures, comments welcome! Added arc certifications and replaced the previous by-hand interval estimates with computer-verified interval arithmetic. Accompanying code in https://github.com/ethanhcoo/K3entropy, tagged arXiv v2