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The recurrence spectrum for dynamical systems beyond specification

Hiroki Takahasi

Abstract

We introduce {\it (W')-specification} in terms of language decompositions of subshifts, and show that any recurrence set of a subshift with this property has full Hausdorff dimension. Our main result applies to a wide class of subshifts without specification, such as all $S$-gap shifts, some coded shifts, and the coding space of any transitive piecewise monotonic interval map with positive entropy. Further, for a wide class of piecewise expanding interval maps we show that any recurrence set has full Hausdorff dimension.

The recurrence spectrum for dynamical systems beyond specification

Abstract

We introduce {\it (W')-specification} in terms of language decompositions of subshifts, and show that any recurrence set of a subshift with this property has full Hausdorff dimension. Our main result applies to a wide class of subshifts without specification, such as all -gap shifts, some coded shifts, and the coding space of any transitive piecewise monotonic interval map with positive entropy. Further, for a wide class of piecewise expanding interval maps we show that any recurrence set has full Hausdorff dimension.

Paper Structure

This paper contains 18 sections, 18 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Recurrence spectrum for subshifts
  3. Coding spaces of interval maps
  4. Coded shifts
  5. Subshifts in C18
  6. Recurrence spectrum for piecewise expanding interval maps
  7. Dimension of recurrence sets for subshifts
  8. Various notions of specification
  9. Computational lemma
  10. Construction of seed sets
  11. Construction of subsets of the recurrence set
  12. Construction of Moran fractals
  13. Proof of theorem \ref{['mainthm']}
  14. Dimension of recurrence sets for interval maps
  15. Markov diagram
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\Sigma$ be a subshift that has a language decomposition $\mathcal{L}(\Sigma)=\mathcal{C}^{\rm p}\mathcal{G}\mathcal{C}^{\rm s}$ such that $\mathcal{G}$ has (W')-specification and $h(\mathcal{C}^{\rm p}\cup\mathcal{C}^{\rm s})<h_{\rm top}(\Sigma)$. For all $a,b\in[0,\infty]$ with $a\leq b$, we h

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1: (W)-specification CT12CT13
  • Definition 2.2: (W')-specification
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Proposition 2.5
  • Remark 2.6
  • ...and 23 more