Measure Theoretic Entropy of Random Substitution Subshifts

Abstract

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.

Multifractal analysis of measures arising from random substitutions

“…In [15], a result analogous to Theorem A is obtained for entropy in terms of the inflation word entropy, without the compatibility assumption. In fact, to highlight the generality of our results on L q -spectra, as a consequence of Theorem A, we obtain new proofs of this result (as well as a result on topological entropy).…”

“…(a) We obtain the main result of [14] on topological entropy, which states that for subshifts of primitive and compatible random substitutions, the topological entropy can be characterised in terms of the asymptotic growth rate of inflation words. (b) For frequency measures corresponding to primitive and compatible random substitutions, we also obtain the characterisation of (measure theoretic) entropy obtained in [15,Theorem 3.3]-under the additional hypothesis that the substitution is compatible. This is described in the following corollary.…”

“…More generally, frequency measures provide a rich family of shift-invariant ergodic measures which exhibit interesting and unique properties in symbolic dynamics in a natural way. 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).…”

“…Another standard assumption is that of recognisability, which heuristically states that each element of the subshift is the unique image of another element of the subshift. 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].…”

Topological mixing of random substitutions

“…They have received renewed attention in recent years, following the work of Rust and Spindeler [26], in which they initiated the study of random substitutions in the context of topological dynamics and ergodic theory and they posed numerous open questions and routes for further investigation. Since then, random substitution subshifts have been studied in terms of (among others) their frequency measures [18], diffraction spectrum [5,14], periodicity [25], automorphism groups [12], topological and measure theoretic entropy [15,16], and relations to shifts of finite type [17]. In their work, Rust and Spindeler determined by ad hoc methods that the subshift associated with the random period doubling substitution was not topologically mixing and they asked if a more general method could be established to determine when an arbitrary random substitution gives rise to a topologically mixing subshift.…”