On the stable dynamical spectrum of complex surfaces

Simon Brandhorst

doi:10.1007/s00208-020-01978-7

Abstract

We characterize Salem numbers which have some power arising as dynamical degree of an automorphism on a complex (projective) 2-Torus, K3 or Enriques surface.

Highlights

To a bimeromorphic transformation F : X its dynamical degreeX of a Kähler surface one can associate λ(F) = lim sup(||(Fn)∗||)1/n, n→∞where F∗ denotes the action on H 2(X, Z) and || · || is any norm on End(H 2(X, Z))
Where F∗ denotes the action on H 2(X, Z) and || · || is any norm on End(H 2(X, Z))
The question this paper is dealing with is: which Salem numbers are dynamical degrees of surface automorphisms and which ones are coming from automorphisms of projective surfaces?

Summary

Introduction

Where F∗ denotes the action on H 2(X , Z) and || · || is any norm on End(H 2(X , Z)). The dynamical degree is a bimeromorphic invariant of (X , F) which measures the dynamical complexity of F. It describes the asymptotic degree growth of defining equations for F. If F is an automorphism, the dynamical degree λ(F) is given by the spectral radius of F∗. It is either 1 or a Salem number, that is, an algebraic integer λ > 1 which is Galois conjugate to 1/λ and all whose other conjugates lie on the unit circle. The question this paper is dealing with is: which Salem numbers are dynamical degrees of surface automorphisms and which ones are coming from automorphisms of projective surfaces?

Present Address

Lattices

Embeddings and gluing

Twists

Surfaces and their automorphisms