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Hyperbolic absolutely continuous invariant measures for C^r one-dimensional maps

Alexandre Delplanque

Abstract

For r > 1, we show, using the Ledrappier-Young entropy characterization of SRB measures for non-invertible maps, that if a C^r map f of the interval or the circle has its Lyapunov exponent greater than 1/r log ||f ' || $\infty$ on a set E of positive Lebesgue measure, then it admits hyperbolic ergodic invariant measures that are absolutely continuous with respect to the Lebesgue measure. We also show that the basins of these measures cover E Lebesgue-almost everywhere.

Hyperbolic absolutely continuous invariant measures for C^r one-dimensional maps

Abstract

For r > 1, we show, using the Ledrappier-Young entropy characterization of SRB measures for non-invertible maps, that if a C^r map f of the interval or the circle has its Lyapunov exponent greater than 1/r log ||f ' || on a set E of positive Lebesgue measure, then it admits hyperbolic ergodic invariant measures that are absolutely continuous with respect to the Lebesgue measure. We also show that the basins of these measures cover E Lebesgue-almost everywhere.

Paper Structure

This paper contains 18 sections, 158 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Reparametrization Lemma
  3. Bounded reparametrizations
  4. Reparametrization Lemma
  5. The case of circle maps
  6. Positive density of geometric times
  7. Geometric times are hyperbolic times
  8. The empirical measure
  9. Definition of the empirical measure
  10. Convergence of the empirical measure
  11. Properties of the limit measure
  12. Entropy of the empirical measure
  13. First entropy estimates
  14. Definition of the partition $\mathcal{P}_q$
  15. Gibbs inequality
  16. ...and 3 more sections

Figures (4)

  • Figure 1: Iterating Yomdin's division process to bound the distortion of iterates of $f$
  • Figure 2: A tree where children of a node $\sigma \circ \theta$ are of the form $\sigma \circ \theta \circ \varphi$ where $\varphi$ is an affine contraction
  • Figure 3: A visualization of the sets $E_n^{M,m}$ and $E_n^{M',m}$
  • Figure 4: We prove that $y_0$ is in some $\left ( \sigma \circ \theta_{i^{a_{j+1}}} \right )_*$ which when iterated by $g^{a_{j+1}}$ contains $B = \mathcal{B}_{g^{a_{j+1}} y_0}$, so it had to contain $V$ when iterated by only $g^{b_j}$

Theorems & Definitions (23)

  • proof
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  • ...and 13 more