Quasiperiodic infinite words : multi-scale case and dynamical properties

Paper withdrawn; will be replaced by revised version containing application to lattice models as well. We study hierarchical properties of Sturmian words. These properties are similar to those of substitution dynamical systems. This approach allows one to carry over to Sturmian dynamical systems methods developed in the context of substitutions. For example, it allows for a very simple proof of a uniform ergodic type theorem for additive functions taking values in a Banach ...

In this paper we introduce and study a new property of infinite words: An infinite word $x\in A^\mathbb{N}$, with values in a finite set $A$, is said to be $k$-self-shuffling $(k\geq 2)$ if $x$ admits factorizations: $x=\prod_{i=0}^\infty U_i^{(1)}\cdots U_i^{(k)}=\prod_{i=0}^\infty U_i^{(1)}=\cdots =\prod_{i=0}^\infty U_i^{(k)}$. In other words, there exists a shuffle of $k$-copies of $x$ which produces $x$. We are particularly interested in the case $k=2$, in which case we ...

A subshift with linear block complexity has at most countably many ergodic measures, and we continue of the study of the relation between such complexity and the invariant measures. By constructing minimal subshifts whose block complexity is arbitrarily close to linear but has uncountably many ergodic measures, we show that this behavior fails as soon as the block complexity is superlinear. With a different construction, we show that there exists a minimal subshift with an er...

Let $G$ be the group $\mathbb{Z}^d$ or the monoid $\mathbb{N}^d$ where $d$ is a positive integer. Let $X$ be a subshift over $G$, i.e., a closed and shift-invariant subset of $A^G$ where $A$ is a finite alphabet. We prove that the topological entropy of $X$ is equal to the Hausdorff dimension of $X$ and has a sharp characterization in terms of the Kolmogorov complexity of finite pieces of the orbits of $X$. In the version of this paper that has been published in Theory of C...

We consider word complexity and topological entropy for random substitution subshifts. In contrast to previous work, we do not assume that the underlying random substitution is compatible. 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. For constant length primitive random substitution...

This paper aims to better understand the link better understand the links between aperiodicity in subshifts and pattern complexity. Our main contribution deals with substitutive subshifts, an equivalent to substitutive tilings in the context of symbolic dynamics. For a class of substitutive subshifts, we prove a quadratic lower bound on their pattern complexity. Together with an already known upper bound, this shows that this class of substitutive subshifts has a pattern comp...

We study ergodic-theoretic properties of coded shift spaces. A coded shift space is defined as a closure of all bi-infinite concatenations of words from a fixed countable generating set. We derive sufficient conditions for the uniqueness of measures of maximal entropy and equilibrium states of H\"{o}lder continuous potentials based on the partition of the coded shift into its sequential set (sequences that are concatenations of generating words) and its residual set (sequence...

This paper studies balancedness for infinite words and subshifts, both for letters and factors. Balancedness is a measure of disorder that amounts to strong convergence properties for frequencies. It measures the difference between the numbers of occurrences of a given word in factors of the same length. We focus on two families of words, namely dendric words and words generated by substitutions. The family of dendric words includes Sturmian and Arnoux-Rauzy words, as well as...

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, subsh...

Let $ \beta $ be a real number less than -1. In this paper, we prove the uniqueness of the measure with maximal entropy of the negative $\beta$-shift. Endowed with the shift, this symbolic dynamical system is coded under certain conditions, but in all cases, it is shown that the measure with maximal entropy is carried by a support coded by a recurrent positive code. One of the difference between the positive and the negative $\beta$-shift is the existence of gaps in the syste...