Topological mixing of random substitutions

Miro, Eden Delight; Rust, Dan; Sadun, Lorenzo; Tadeo, Gwendolyn S.

doi:10.1007/s11856-022-2406-3

“…The key feature of compatibility is that the one can define a deterministic substitution matrix , such that the Perron–Frobenius eigenvalue is the asymptotic growth rate of lengths of words under repeated substitution, and the corresponding right eigenvector encodes the asymptotic frequency with which the individual letters appear. Compatibility is a common assumption: for example, it is assumed in the main results of [ BSS18 , Goh20 , MRS+23 , Rus20 ].…”

Section: Introductionmentioning

confidence: 99%

“…Recognisability precludes the existence of periodic points [ Rus20 ] and is one of the assumptions required to to establish intrinsic ergodicity in [ GMR+23 ]. It is also assumed in the main results of [ FRS22+ , MRS+23 ].…”

Section: Introductionmentioning

confidence: 99%

Multifractal Analysis of Measures Arising from Random Substitutions

Mitchell,

Rutar

2024

Commun. Math. Phys.

1

0

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We study regularity properties of frequency measures arising from random substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of each letter is chosen independently from a fixed finite set. In particular, for a natural class of such measures, we derive a closed-form analytic formula for the $$L^q$$ L q -spectrum and prove that the multifractal formalism holds. This provides an interesting new class of measures satisfying the multifractal formalism. More generally, we establish results concerning the $$L^q$$ L q -spectrum of a broad class of frequency measures. We introduce a new notion called the inflation word$$L^q$$ L q -spectrum of a random substitution and show that this coincides with the $$L^q$$ L q -spectrum of the corresponding frequency measure for all $$q \ge 0$$ q ≥ 0 . As an application, we obtain closed-form formulas under separation conditions and recover known results for topological and measure theoretic entropy.

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“…The key feature of compatibility is that the one can define a deterministic substitution matrix, such that the Perron-Frobenius eigenvalue is the asymptotic growth rate of lengths of words, and the corresponding right eigenvector encodes the asymptotic frequency with which the individual letters appear. Compatibility is a common assumption: for example, it is assumed in the main results of [3,14,21,27].…”

Section: Introductionmentioning

confidence: 99%

“…Recognisability precludes the existence of periodic points [27] and is one of the assumptions required to to establish intrinsic ergodicity in [15]. It is also assumed in the main results of [12,21]. 1.3.…”

Section: Introductionmentioning

confidence: 99%

Multifractal analysis of measures arising from random substitutions

Mitchell¹,

Rutar²

2023

Preprint

0

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We study regularity properties of frequency measures arising from random substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of each letter is chosen independently from a fixed finite set. In particular, for a natural class of such measures, we derive a closed-form analytic formula for the L q -spectrum and prove that the multifractal formalism holds. This provides an interesting new class of measures satisfying the multifractal formalism. More generally, we establish results concerning the L q -spectrum of a broad class of frequency measures. We introduce a new notion called the inflation word L q -spectrum of a random substitution and show that this coincides with the L q -spectrum of the corresponding frequency measure for all q ≥ 0. As an application, we obtain closedform formulas under separation conditions and recover known results for topological and measure theoretic entropy. CONTENTS 1. Introduction 1.1. Entropy and L q -spectra 1.2. Random substitutions 1.3. Statement and discussion of main results 1.4. Discussion and further work 2. Preliminaries 2.1. Symbolic notation 2.2. Dynamics, entropy and dimension 2.3. L q -spectra and smoothness 2.4. Multifractal spectrum and multifractal formalism 2.5. Random substitutions and frequency measures 2.6. Primitive random substitutions 2.7. Compatible random substitutions 2.8. Frequency measures 2.9. Separation conditions and recognisability 3. L q -spectra of frequency measures 3.1. Inflation word L q -spectra 3.2. L q -spectra for non-negative q

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Measure Theoretic Entropy of Random Substitution Subshifts

Gohlke

¹

,

Mitchell

²

,

Rust

³

et al. 2022

Ann. Henri Poincaré

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Subshifts of deterministic substitutions are ubiquitous objects in dynamical systems and aperiodic order (the mathematical theory of quasicrystals). Two of their most striking features are that they have low complexity (zero topological entropy) and are uniquely ergodic. Random substitutions are a generalisation of deterministic substitutions where the substituted image of a letter is determined by a Markov process. In stark contrast to their deterministic counterparts, subshifts of random substitutions often have positive topological entropy, and support uncountably many ergodic measures. The underlying Markov process singles out one of the ergodic measures, called the frequency measure. Here, we develop new techniques for computing and studying the entropy of these frequency measures. As an application of our results, we obtain closed form formulas for the entropy of frequency measures for a wide range of random substitution subshifts and show that in many cases there exists a frequency measure of maximal entropy. Further, for a class of random substitution subshifts, we prove that this measure is the unique measure of maximal entropy. These subshifts do not satisfy Bowen’s specification property or the weaker specification property of Climenhaga and Thompson and hence provide an interesting new class of intrinsically ergodic subshifts.

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Topological mixing of random substitutions

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Cited by 5 publications

References 26 publications

Multifractal Analysis of Measures Arising from Random Substitutions

Multifractal Analysis of Measures Arising from Random Substitutions

Multifractal analysis of measures arising from random substitutions

Measure Theoretic Entropy of Random Substitution Subshifts

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