Jonqui\`eres maps and $\mathrm{SL}(2;\mathbb{C})$-cocycles

TL;DR

This paper investigates a family of birational maps on complex projective space, analyzing their dynamical properties and computing Lyapunov exponents of associated $ ext{SL}(2; ext{C})$-cocycles using recent theoretical advances.

Contribution

It introduces a detailed study of Jonquières maps, characterizes their dynamical behavior, and computes Lyapunov exponents for related cocycles leveraging recent $ ext{SL}(2; ext{C})$-cocycle results.

Findings

Trivial centralizer for generic parameters

Zero topological entropy for generic maps

Explicit Lyapunov exponent calculations for cocycles

Abstract

We start the study of the family of birational maps $(f_{\alpha,\beta})$ of $\mathbb{P}^2_\mathbb{C}$ in \cite{Deserti}. For generic $\alpha$ and $\beta$ of modulus 1 the centraliser of $f_{\alpha,\beta}$ is trivial, the topological entropy of $f_{\alpha,\beta}$ is 0, there exist two areas of linearisation: in the first one the closure of the orbit of a point is a torus, in the other one the closure of the orbit of a point is the union of two circles. On $\mathbb{P}^1_\mathbb{C}\times \mathbb{P}^1_\mathbb{C}$ any $f_{\alpha,\beta}$ can be viewed as a cocyle; using recent results about $\mathrm{SL}(2;\mathbb{C})$-cocycles (\cite{Avila}) we determine the \textsc{Lyapunov} exponent of the cocyle associated to $f_{\alpha,\beta}$.