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Stable local dynamics: expansion, quasi-conformality and ergodicity

Abbas Fakhari, Meysam Nassiri, Hesam Rajabzadeh

TL;DR

This work addresses the problem of stable ergodicity for semigroups of diffeomorphisms on smooth manifolds by introducing a local, robust mechanism called the quasi-conformal blender, which enforces ergodicity through controlled quasi-conformal orbit-branches. By developing a fiber-bundle framework and a covering property, the authors extend Sullivan’s one-dimensional exponential expansion strategy to all dimensions, obtaining stably ergodic finite generating sets on any closed manifold and establishing robust minimality for sphere actions near the identity. Key technical pillars include expanding sequences with geometry control, bounded distortion, and infiltrated quasi-conformality, which together yield local ergodicity and then global stable ergodicity and minimality results. The findings have broad implications for smooth ergodic theory, including uniqueness of absolutely continuous stationary measures and new foliations with stable ergodicity properties, and open up several directions for further exploration of generators, quasi-conformality, and stationary measures.

Abstract

In this paper, we study stable ergodicity of the action of groups of diffeomorphisms on smooth manifolds. Such actions are known to exist only on one-dimensional manifolds. The aim of this paper is to introduce a geometric method to overcome this restriction and to construct higher dimensional examples. In particular, we show that every closed manifold admits stably ergodic finitely generated group actions by diffeomorphisms of class $C^{1+α}$. We also prove the stable ergodicity of certain algebraic actions, including the natural action of a generic pair of matrices near the identity on a sphere of arbitrary dimension. These are consequences of the quasi-conformal blender, a local and stable mechanism/phenomenon introduced in this paper, which encapsulates our method for proving stable local ergodicity by providing quasi-conformal orbits with fine controlled geometry. The quasi-conformal blender is developed in the context of pseudo-semigroup actions of locally defined smooth diffeomorphisms, which allows for applications in diverse settings.

Stable local dynamics: expansion, quasi-conformality and ergodicity

TL;DR

This work addresses the problem of stable ergodicity for semigroups of diffeomorphisms on smooth manifolds by introducing a local, robust mechanism called the quasi-conformal blender, which enforces ergodicity through controlled quasi-conformal orbit-branches. By developing a fiber-bundle framework and a covering property, the authors extend Sullivan’s one-dimensional exponential expansion strategy to all dimensions, obtaining stably ergodic finite generating sets on any closed manifold and establishing robust minimality for sphere actions near the identity. Key technical pillars include expanding sequences with geometry control, bounded distortion, and infiltrated quasi-conformality, which together yield local ergodicity and then global stable ergodicity and minimality results. The findings have broad implications for smooth ergodic theory, including uniqueness of absolutely continuous stationary measures and new foliations with stable ergodicity properties, and open up several directions for further exploration of generators, quasi-conformality, and stationary measures.

Abstract

In this paper, we study stable ergodicity of the action of groups of diffeomorphisms on smooth manifolds. Such actions are known to exist only on one-dimensional manifolds. The aim of this paper is to introduce a geometric method to overcome this restriction and to construct higher dimensional examples. In particular, we show that every closed manifold admits stably ergodic finitely generated group actions by diffeomorphisms of class . We also prove the stable ergodicity of certain algebraic actions, including the natural action of a generic pair of matrices near the identity on a sphere of arbitrary dimension. These are consequences of the quasi-conformal blender, a local and stable mechanism/phenomenon introduced in this paper, which encapsulates our method for proving stable local ergodicity by providing quasi-conformal orbits with fine controlled geometry. The quasi-conformal blender is developed in the context of pseudo-semigroup actions of locally defined smooth diffeomorphisms, which allows for applications in diverse settings.

Paper Structure

This paper contains 32 sections, 32 theorems, 111 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Quasi-conformal blenders
  3. Preliminary definitions and notations
  4. Localized dynamics
  5. Expanding and contracting maps
  6. Quasi-conformality
  7. Quasi-conformal dynamics
  8. Stable quasi-conformality
  9. Reinterpretation of the covering property
  10. Covering condition for $\mathrm{SL}(d,\mathbb{R})$ actions
  11. Behaviours of typical branches.
  12. Existence of bounded branches
  13. Sufficient conditions for covering: explicit construction
  14. Sufficient conditions for covering: geometric construction
  15. Expanding sequences
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $G$ be a group of $\mathcal{C}^{1+\alpha}$ circle diffeomorphisms, with $\alpha> 0$. Assume that for all $x \in \mathbb{S}^{1}$, there exists some $g\in G$ such that $g'(x) > 1$. If the action of $G$ is minimal, then it is ergodic.

Figures (5)

  • Figure 1: Covering condition (\ref{['eq:cover-W']}): diversity of maps corresponding to $x\in V$, i.e. maps in the subfamily $\mathcal{F}_x$.
  • Figure 2: A quasi-conformal sequence of matrices
  • Figure 3: Sequence of maps satisfying (H0)-(H3) in Lemma \ref{['lem:iteration-balls-Rd']}, where the iterations of small ball remain almost round.
  • Figure 4: Families of expanding maps satisfying the covering condition in Theorem \ref{['thm:blender']}.
  • Figure 5: Local constructions of $f$ and $h$.

Theorems & Definitions (88)

  • Theorem 1.1: Sullivan
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Definition 1.2: Quasi-conformal blender
  • Theorem 6
  • Definition 2.1
  • Definition 2.2
  • ...and 78 more