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Exponential mixing of all orders and CLT for generic birational maps of $\mathbb{P}^k$

Henry De Thélin, Gabriel Vigny

Abstract

For Hénon maps, Bianchi and Dinh recently proved the exponential mixing of all orders for the measure of maximal entropy and, as a consequence of the recent work of Björklund and Gorodnik, the CLT for Hölder observables. We extend their results to generic birational maps of $\mathbb{P}^k$. Because of the indeterminacy set, Hölder maps are not stable under iteration, so we need to work with a suitable space of test functions.

Exponential mixing of all orders and CLT for generic birational maps of $\mathbb{P}^k$

Abstract

For Hénon maps, Bianchi and Dinh recently proved the exponential mixing of all orders for the measure of maximal entropy and, as a consequence of the recent work of Björklund and Gorodnik, the CLT for Hölder observables. We extend their results to generic birational maps of . Because of the indeterminacy set, Hölder maps are not stable under iteration, so we need to work with a suitable space of test functions.
Paper Structure (5 sections, 111 equations)

This paper contains 5 sections, 111 equations.

Table of Contents

  1. Introduction
  2. Background on generic birational maps of $\mathbb{P}^k$ and super-potentials theory
  3. A space of test functions
  4. Preliminary estimates
  5. Exponential mixing of all orders and CLT

Theorems & Definitions (10)

  • proof
  • proof
  • proof : Proof of the theorem
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof : Proof of Theorem \ref{['tm_exp']}
  • proof