Dynamical degrees of (pseudo)-automorphisms fixing cubic hypersurfaces

TL;DR

This paper constructs groups of pseudo-automorphisms fixing cubic hypersurfaces in rational varieties, revealing their dynamical properties and answering a question about birational maps with high dynamical degree.

Contribution

It introduces a method to construct pseudo-automorphisms fixing cubic hypersurfaces and analyzes their dynamical degrees, expanding understanding of their structure and properties.

Findings

Most elements have dynamical degree >1

The constructed groups are free products of involutions

The Picard group is not big in higher dimensions

Abstract

We give a way to construct group of pseudo-automorphisms of rational varieties of any dimension that fix pointwise the image of a cubic hypersurface of $P^n. These group are free products of involutions, and most of their elements have dynamical degree >1. Moreover, the Picard group of the varieties obtained is not big, if the dimension is at least 3. We also answer a question of E. Bedford on the existence of birational maps of the plane that cannot be lifted to automorphisms of dynamical degree >1, even if we compose them with an automorphism of the plane.