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Entropy Density of Ergodic Nonadapted Measures for Markov Interval Maps

Łukasz Krzywoń

Abstract

Given a uniformly expanding transitive Markov interval map, we show that within the set of ergodic measures the set of nonadapted ergodic measures is residual in with respect to the topology induced by the $\overline{d}$-metric. This set of measures is also shown to be path connected in many cases.

Entropy Density of Ergodic Nonadapted Measures for Markov Interval Maps

Abstract

Given a uniformly expanding transitive Markov interval map, we show that within the set of ergodic measures the set of nonadapted ergodic measures is residual in with respect to the topology induced by the -metric. This set of measures is also shown to be path connected in many cases.
Paper Structure (14 sections, 7 theorems, 38 equations, 4 figures)

This paper contains 14 sections, 7 theorems, 38 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Adapted and Nonadapted Measures for Interval Maps
  3. Connection to Dispersing Billiards
  4. Preliminary Definitions
  5. Definitions
  6. Coding
  7. Constructions of Measures
  8. Construction A
  9. Construction B
  10. Proofs
  11. Proof of Theorem \ref{['thm:one']}
  12. Proof of Statement (1) and (2) of Theorem \ref{['thm:one']}
  13. Proof of Statement (3) of Theorem \ref{['thm:one']}
  14. We also have:

Key Result

Theorem 1.1

Let $f \colon I=[0,1] \to I$ be a piecewise $C^1$, uniformly expanding, transitive Markov map with a periodic point, $c$. Then, the following hold.

Figures (4)

  • Figure 2.1: Examples of types of periodic discontinuities
  • Figure 3.1: Construction A
  • Figure 3.2: The map $T_\textbf{y}$ on an interval $I \in \mathcal{J}_\textbf{y} \subset \mathcal{I}$.
  • Figure 3.3: Construction B

Theorems & Definitions (23)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 13 more