Complexity of dynamical systems arising from random substitutions in one dimension

Mitchell, Andrew ORCID: 0009-0000-8249-7791 (2023). Complexity of dynamical systems arising from random substitutions in one dimension. University of Birmingham. Ph.D.

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Abstract

This thesis is based on three papers the author wrote while a PhD student, which concern different notions of complexity for dynamical systems arising from random substitutions.

Before presenting our main results, we first provide an introduction to random substitutions. In Chapter 2, we give the main definitions that we work with throughout, and prove several basic properties of random substitutions and their associated subshifts. We define the frequency measure corresponding to a random substitution, and prove a key result concerning such measures which will be of fundamental importance in our work.

Chapter 3 is based on a solo-author paper and concerns word complexity and topological entropy of random substitution subshifts. In contrast to previous work, we do not assume that the underlying random substitution is compatible. In our main results, we show that the subshift of a primitive random substitution has zero topological entropy if and only if it can be obtained as the subshift of a deterministic substitution -- answering in the affirmative an open question of Rust and Spindeler -- and provide a systematic approach to calculating the topological entropy for subshifts of constant length random substitutions. We also consider word complexity for constant length random substitutions and show that, without primitivity, the complexity function can exhibit features not possible in the deterministic or primitive random settings.

Chapters 4 and 5 are based on joint work with P. Gohlke, D. Rust and T. Samuel. These chapters focus on measure theoretic entropy and its relationship to topological entropy. In Chapter 4, we introduce a new measure of complexity for primitive random substitutions called measure theoretic inflation word entropy and show that this coincides with the measure theoretic entropy of the subshift with respect to the corresponding frequency measure. This allows the measure theoretic entropy to be explicitly calculated in many cases. In Chapter 5, we provide sufficient conditions under which a random substitution subshift supports a frequency measure of maximal entropy and, under more restrictive conditions, show that this measure is the unique measure of maximal entropy. Notably, we show that random substitutions can give rise to intrinsically ergodic subshifts that do not satisfy Bowen's specification property or the weaker specification property of Climenhaga and Thompson, thus providing an interesting new class of intrinsically ergodic subshifts. We conclude this chapter by showing that the random period doubling substitution is intrinsically ergodic.

Finally, Chapter 6 is based on joint work with A. Rutar. Here, we consider multifractal properties of frequency measures. Specifically, we study the multifractal spectrum and $L^q$-spectrum of frequency measures corresponding to primitive and compatible random substitutions. 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 $q \geq 0$. Under an additional assumption (recognisability) we show that the two notions coincide for all real q. Further, under these assumptions, we show that the multifractal formalism holds. The techniques we develop allow the $L^q$-spectrum and multifractal spectrum to be obtained for many frequency measures.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):