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Speedups of linearly recurrent subshifts

Henk Bruin

Abstract

A speedup, like a time change in discrete time dynamics, is a way of moving faster through the orbits of a dynamical system. Linearly recurrence is a stronger form of minimality for subshifts, shared by e.g.\ all primitive substitution shifts and Sturmian shifts associated with rotation numbers of bounded type. We prove that the homeomorphic speedup of a linearly recurrent two-sided subshift is again linearly recurrent.

Speedups of linearly recurrent subshifts

Abstract

A speedup, like a time change in discrete time dynamics, is a way of moving faster through the orbits of a dynamical system. Linearly recurrence is a stronger form of minimality for subshifts, shared by e.g.\ all primitive substitution shifts and Sturmian shifts associated with rotation numbers of bounded type. We prove that the homeomorphic speedup of a linearly recurrent two-sided subshift is again linearly recurrent.
Paper Structure (6 sections, 10 theorems, 19 equations)

This paper contains 6 sections, 10 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and miscellaneous results
  3. Subshifts of finite type and sofic shifts
  4. Word-complexity of speedups
  5. Finite non-abelian group extensions
  6. Linear recurrence of speedups

Key Result

Theorem 1.1

Let $S = \sigma^p$ be the homeomorphic transitive speedup of a two-sided subshift $(X,\sigma)$. Then $\sigma$ is linearly recurrent if and only if $S$ is linearly recurrent.

Theorems & Definitions (12)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Remark 3.1
  • Lemma 3.1
  • Proposition 3.1
  • Corollary 3.1
  • Lemma 3.2
  • Proposition 3.2
  • ...and 2 more