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On partial rigidity of $\mathcal{S}$-adic subshifts

Sebastián Donoso, Alejandro Maass, Tristán Radić

TL;DR

The paper develops a constructive framework for partial rigidity in minimal ${ t S}$-adic subshifts by leveraging Kakutani-Rokhlin partitions. It proves a general if-and-only-if criterion for $oldsymbol{eta}$-rigidity via sequences of complete words and shows how to compute the partial rigidity rate ${oldsymbol eta}$; it also provides a concrete formula in terms of cylinder measures for constant-length substitutions. As applications, it computes the partial rigidity rate for the Thue–Morse subshift as $2/3$ and demonstrates how to realize any rate in $[0,1]$ by suitable ${ t S}$-adic constructions, including products of substitutions. The results yield a versatile, computable method to produce ergodic systems with prescribed rigidity behavior, including examples with multiple ergodic measures and rigid behavior under chosen rigidity sequences, thereby enriching the understanding of rigidity versus mixing in zero-entropy symbolic systems.

Abstract

We develop combinatorial tools to study partial rigidity within the class of minimal $\mathcal{S}$-adic subshifts. By leveraging the combinatorial data of well-chosen Kakutani-Rokhlin partitions, we establish a necessary and sufficient condition for partial rigidity. Additionally, we provide an explicit expression to compute the partial rigidity rate and an associated partial rigidity sequence. As applications, we compute the partial rigidity rate for a variety of constant length substitution subshifts, such as the Thue-Morse subshift, where we determine a partial rigidity rate of 2/3. We also exhibit non-rigid substitution subshifts with partial rigidity rates arbitrarily close to 1 and as a consequence, using products of the aforementioned substitutions, we obtain that any number in $[0, 1]$ is the partial rigidity rate of a system.

On partial rigidity of $\mathcal{S}$-adic subshifts

TL;DR

The paper develops a constructive framework for partial rigidity in minimal -adic subshifts by leveraging Kakutani-Rokhlin partitions. It proves a general if-and-only-if criterion for -rigidity via sequences of complete words and shows how to compute the partial rigidity rate ; it also provides a concrete formula in terms of cylinder measures for constant-length substitutions. As applications, it computes the partial rigidity rate for the Thue–Morse subshift as and demonstrates how to realize any rate in by suitable -adic constructions, including products of substitutions. The results yield a versatile, computable method to produce ergodic systems with prescribed rigidity behavior, including examples with multiple ergodic measures and rigid behavior under chosen rigidity sequences, thereby enriching the understanding of rigidity versus mixing in zero-entropy symbolic systems.

Abstract

We develop combinatorial tools to study partial rigidity within the class of minimal -adic subshifts. By leveraging the combinatorial data of well-chosen Kakutani-Rokhlin partitions, we establish a necessary and sufficient condition for partial rigidity. Additionally, we provide an explicit expression to compute the partial rigidity rate and an associated partial rigidity sequence. As applications, we compute the partial rigidity rate for a variety of constant length substitution subshifts, such as the Thue-Morse subshift, where we determine a partial rigidity rate of 2/3. We also exhibit non-rigid substitution subshifts with partial rigidity rates arbitrarily close to 1 and as a consequence, using products of the aforementioned substitutions, we obtain that any number in is the partial rigidity rate of a system.
Paper Structure (22 sections, 41 theorems, 116 equations)

This paper contains 22 sections, 41 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Measure preserving and topological systems
  4. Cantor and symbolic systems.
  5. Kakutani-Rokhlin partitions
  6. Partial rigidity in ergodic measure preserving systems
  7. Partial rigidity rate
  8. Partial rigidity in ${\mathcal{S}}$-adic subshifts
  9. Kakutani-Rokhlin partitions in ${\mathcal{S}}$-adic subshifts
  10. Sufficient conditions for partial rigidity
  11. Conditions regarding towers heights
  12. Conditions regarding the order
  13. Conditions regarding return words
  14. Condition regarding the directive sequence
  15. Rigidity in ${\mathcal{S}}$-adic subshifts
  16. ...and 7 more sections

Key Result

Theorem 1

Let $(X, {\mathcal{X}}, \mu, T)$ be an ergodic system with $\mu$ a non-atomic measure and let $({\mathscr P}^{(n)})_{n \in {\mathbb N}}$ be a sequence of Kakutani-Rokhlin partitions satisfying standard assumptions called (KR1)-(KR4). Then, the following properties are equivalent:

Theorems & Definitions (102)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • ...and 92 more