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Classifying Hyperbolic Ergodic Stationary Measures on Compact Complex Surfaces with Large Automorphism Groups

Megan Roda

Abstract

Let $X$ be a compact complex surface. Consider a finitely supported probability measure $μ$ on $\text{Aut}(X)$ such that $Γ_μ = \langle \text{Supp}(μ)\rangle<\text{Aut}(X)$ is non-elementary. We do not assume that $Γ_μ$ contains any parabolic elements. In this paper, we study and classify hyperbolic, ergodic $μ$-stationary probability measures.

Classifying Hyperbolic Ergodic Stationary Measures on Compact Complex Surfaces with Large Automorphism Groups

Abstract

Let be a compact complex surface. Consider a finitely supported probability measure on such that is non-elementary. We do not assume that contains any parabolic elements. In this paper, we study and classify hyperbolic, ergodic -stationary probability measures.

Paper Structure

This paper contains 65 sections, 112 theorems, 424 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main theorem and background
  3. Strategy
  4. Preliminaries
  5. Setup of the random dynamical system
  6. Oseledets' theorem and the global stable and unstable manifolds
  7. Suspension flow
  8. Conditional measures and the results of BEF
  9. Compatible families and quantitative non-integrability
  10. Proof of Theorem \ref{['bigthmeasy']}
  11. Alternative 1
  12. Alternative 2
  13. Alternative 3
  14. Tools required for the proof of Theorem \ref{['bigthmeasy']}
  15. Furstenberg theory
  16. ...and 50 more sections

Key Result

Theorem 1.1.1

Let $X$ be a compact complex surface, $\mu$ a finitely supported probability measure on the group of biholomorphisms $\text{Aut}(X)$ such that $\Gamma_{\mu}= \langle \text{Supp}(\mu) \rangle$ is non-elementary, and $\nu$ a hyperbolic ergodic $\mu$-stationary probability measure on $X$. Assume there

Figures (1)

  • Figure 1: This figure shows the basic configuration of the points used in the proof of Proposition \ref{['U-']}. The magnitude of time that we flow between the points is labelled as well.

Theorems & Definitions (244)

  • Theorem 1.1.1
  • Lemma 2.1.1: Proposition 1.2, K, LQ
  • Remark 2.1.2
  • Remark 2.1.3
  • Remark 2.2.1
  • Definition 1: LQ, Section IV.2
  • Proposition 2.4.1: LQ, Proposition 2.1, CD, Section 7.6
  • Definition 2: Random compatible family of subgroups
  • Remark 2.5.1
  • Definition 3: Random QNI
  • ...and 234 more