Dynamical degrees of birational transformations of projective surfaces

TL;DR

This paper investigates the set of dynamical degrees of birational transformations on projective surfaces, revealing their algebraic and topological properties and their relation to conjugacy classes.

Contribution

It characterizes the set of all dynamical degrees for birational transformations of projective surfaces, especially the complex projective plane, and explores their structural properties.

Findings

The set of dynamical degrees is closed and well ordered.

Dynamical degrees are algebraic numbers.

Relationship between dynamical degrees and conjugacy classes.

Abstract

The dynamical degree $\lambda(f)$ of a birational transformation $f$ measures the exponential growth rate of the degree of the formulae that define the $n$-th iterate of $f$. We study the set of all dynamical degrees of all birational transformations of projective surfaces, and the relationship between the value of $\lambda(f)$ and the structure of the conjugacy class of $f$. For instance, the set of all dynamical degrees of birational transformations of the complex projective plane is a closed and well ordered set of algebraic numbers.