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Mixing and ergodicity of compositions of inner functions

Gustavo Rodrigues Ferreira, Artur Nicolau

Abstract

We study ergodic and mixing properties of non-autonomous dynamics on the unit circle generated by inner functions fixing the origin.

Mixing and ergodicity of compositions of inner functions

Abstract

We study ergodic and mixing properties of non-autonomous dynamics on the unit circle generated by inner functions fixing the origin.
Paper Structure (9 sections, 16 theorems, 62 equations)

This paper contains 9 sections, 16 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Inner functions and the space $L^2 (\partial \mathbb{D})$
  4. Equidistribution of sequences on $\partial\mathbb{D}$
  5. A general characterisation of ergodicity
  6. Ergodicity in different scenarios
  7. Mixing in the usual sense
  8. Examples and counterexamples
  9. Recurrence

Key Result

Theorem A

Let $g_n\colon\mathbb{D}\to\mathbb{D}$ be inner functions fixing the origin, and let $G_n\mathrel{\vcenter{\hbox{\scriptsize.}\hbox{\scriptsize.}}} = g_n\circ\cdots \circ g_1$, $n\in\mathbb{N}$.

Theorems & Definitions (31)

  • Theorem A: BEFRS19 and Fer23
  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Lemma 2.1: Lowner's Lemma
  • Definition 2.2
  • Lemma 2.3: Weyl's Criterion
  • ...and 21 more