Beyond Bowen’s Specification Property

Climenhaga, Vaughn; Thompson, Daniel J.

doi:10.1007/978-3-030-74863-0_1

Cited by 9 publications

(2 citation statements)

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“…For example, it was proved in [15,Theorem 4.8] that for a certain class of random substitutions, the corresponding subshift supports a frequency measure that is the unique ergodic measure of maximal entropy. However, this measure is not a Gibbs measure with respect to the zero potential, and therefore the system does not satisfy the common specification property, which is a well-known strategy for proving intrinsic ergodicity of symbolic dynamical systems (see [5] and the references therein). Moreover, there are examples of random substitutions such that the corresponding subshift supports multiple ergodic measures of maximal entropy [17,Example 6.5].…”

Section: Introductionmentioning

confidence: 99%

Multifractal analysis of measures arising from random substitutions

Mitchell¹,

Rutar²

2023

Preprint

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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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Section: Introductionmentioning

confidence: 99%

Multifractal analysis of measures arising from random substitutions

Mitchell¹,

Rutar²

2023

Preprint

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“…Thompson have introduced an alternate approach based on Bowen's proof of uniqueness for uniformly hyperbolic systems using expansivity and specification [Bow75]. There are non-uniform versions of these properties [CT12, CT16] that have found applications in examples such as geodesic flow in nonpositive curvature [BCFT18] and no conjugate points [CKW19]; see the forthcoming survey [CT20] for a more complete description of this approach.…”

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confidence: 99%

SRB and equilibrium measures via dimension theory

Climenhaga

2020

Preprint

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It is well-known that SRB and equilibrium measures for uniformly hyperbolic flows admit a product structure in terms of measures on stable and unstable leaves with scaling properties given by the potential function. We describe a construction of these leaf measures analogous to the definition of Hausdorff measure, relying on the Pesin-Pitskel' description of topological pressure as a dimensional characteristic using Bowen balls. These leaf measures were constructed for discrete-time systems by the author, Ya. Pesin, and A. Zelerowicz. In the continuous-time setting here, the description of the scaling properties is more complete, and we use a similar procedure with two-sided Bowen balls to directly produce the equilibrium measure itself.For Tolya, who taught me to learn things a second time.

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