Birational maps with transcendental dynamical degree

TL;DR

This paper constructs examples of birational maps in higher-dimensional projective spaces with transcendental dynamical degrees, challenging existing conjectures and combining algebraic dynamics with diophantine approximation techniques.

Contribution

It provides the first known examples of birational maps with transcendental dynamical degrees, contradicting previous conjectures.

Findings

Existence of birational maps with transcendental dynamical degrees in dimensions d ≥ 3

Counterexample to Bellon and Viallet's conjecture

Application of algebraic dynamics and diophantine approximation methods

Abstract

We give examples of birational selfmaps of $\mathbb{P}^d, d \geq 3$, whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics and diophantine approximation.