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Classification of Conditional Measures Along Certain Invariant One-Dimensional Foliations

Marcielis Espitia, Gabriel Ponce, Régis Varão

Abstract

Let $f:M\to M$ be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure $μ$ defined on the completion of the Borel $σ$-algebra and $\mathcal F$ a $f$-invariant one dimensional continuous foliation of $M$ by $C^1$-leaves. Then, if $f$ preserves a continuous $\mathcal{F}$-arc length system, then we only have three possibilities for the conditional measures of $μ$ along $\mathcal F$, namely: - they are atomic for almost every leaf, or - for almost every leaf they are equivalent to the measure $λ_x$ induced by the invariant arc-length system over $\mathcal F$, or - for almost every leaf their support is a nowhere dense, perfect subset of the leaf. Furthermore, we show that restricted to ergodic partially hyperbolic diffeomorphism with one-dimensional topological neutral center direction, we are able to eliminate the third case obtaining a dichotomy.

Classification of Conditional Measures Along Certain Invariant One-Dimensional Foliations

Abstract

Let be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure defined on the completion of the Borel -algebra and a -invariant one dimensional continuous foliation of by -leaves. Then, if preserves a continuous -arc length system, then we only have three possibilities for the conditional measures of along , namely: - they are atomic for almost every leaf, or - for almost every leaf they are equivalent to the measure induced by the invariant arc-length system over , or - for almost every leaf their support is a nowhere dense, perfect subset of the leaf. Furthermore, we show that restricted to ergodic partially hyperbolic diffeomorphism with one-dimensional topological neutral center direction, we are able to eliminate the third case obtaining a dichotomy.

Paper Structure

This paper contains 14 sections, 22 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Organization of the paper
  4. Basics on conditional measures
  5. Invariant arc-length and invariant metric systems
  6. Properties of non-atomic disintegrations over a continuous one-dimensional foliation
  7. Proof Theorem \ref{['theorem:A']}
  8. Technical Lemmas for the case $\Delta = \infty$
  9. Technical Lemmas for the case $\Delta < \infty$
  10. Case 1: $\Delta<\infty$ and $\mu(D)=0$.
  11. Case 2: $\Delta=\infty$ and $\mu(D^{\infty})=0$.
  12. Case 3: $\Delta = \infty$ and $\mu(D^{\infty})=1$.
  13. Case 4: $\Delta < \infty$ and $\mu(D)=1$.
  14. Proof Theorem \ref{['theorem:casobonatti']}

Key Result

Theorem 1

Let $f:M\to M$ be a homeomorphism over a compact Riemannian manifold, $\mathcal{F}$ be a $f$-invariant one dimensional continuous foliation of $M$ by $C^1$-leaves and $\{l_x\}$ a $\mathcal{F}$-arc length system. If $f$ is ergodic with respect to a $f$-invariant probability measure $\mu$ then one of

Theorems & Definitions (53)

  • Theorem 1
  • Theorem 2
  • Proposition 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Definition 3.5
  • Definition 3.6
  • Proposition 3.7
  • ...and 43 more