Quasiperiodic infinite words : multi-scale case and dynamical properties

We consider the class ${\cal P}_1$ of all infinite words $x\in A^\omega$ over a finite alphabet $A$ admitting a prefixal factorization, i.e., a factorization $x= U_0 U_1U_2 \cdots $ where each $U_i$ is a non-empty prefix of $x.$ With each $x\in {\cal P}_1$ one naturally associates a "derived" infinite word $\delta(x)$ which may or may not admit a prefixal factorization. We are interested in the class ${\cal P}_{\infty}$ of all words $x$ of ${\cal P}_1$ such that $\delta^n(x) ...

We give a lower bound on the growth of a subshift based on a simple condition on the set of forbidden patterns defining that subshift. Aubrun et Al. showed a similar result based on the Lov\'asz Local Lemma for subshift over any countable group and Bernshteyn extended their approach to deduce, amongst other things, some lower bound on the exponential growth of the subshift. However, our result has a simpler proof, is easier to use for applications, and provides better bounds ...

In our earlier paper [A square root map on Sturmian words, Electron. J. Combin. 24.1 (2017)], we introduced a symbolic square root map. Every optimal squareful infinite word $s$ contains exactly six minimal squares and can be written as a product of these squares: $s = X_1^2 X_2^2 \cdots$. The square root $\sqrt{s}$ of $s$ is the infinite word $X_1 X_2 \cdots$ obtained by deleting half of each square. We proved that the square root map preserves the languages of Sturmian word...

This paper studies subsets of one-sided shift spaces on a finite alphabet. Such subsets arise in symbolic dynamics, in fractal constructions, and in number theory. We study a family of decimation operations, which extract subsequences of symbol sequences in infinite arithmetic progressions, and show they are closed under composition. We also study a family of $n$-ary interleaving operations, one for each $n \ge 1$. Given subsets $X_0, X_1, ..., X_{n-1}$ of the shift space, th...

If an infinite non-periodic word is uniformly recurrent or is of bounded repetition, then the limit of its periodicity complexity is infinity. Moreover, there are uniformly recurrent words with the periodicity complexity arbitrarily high at infinitely many positions.

Using results relating the complexity of a two dimensional subshift to its periodicity, we obtain an application to the well-known conjecture of Furstenberg on a Borel probability measure on $[0,1)$ which is invariant under both $x\mapsto px \pmod 1$ and $x\mapsto qx \pmod 1$, showing that any potential counterexample has a nontrivial lower bound on its complexity.

This study investigates the natural or intrinsic measure of a symbolic dynamical system $\Sigma$. The measure $\mu([i_{1},i_{2},...,i_{n}])$ of a pattern $[i_{1},i_{2},...,i_{n}]$ in $\Sigma$ is an asymptotic ratio of $[i_{1},i_{2},...,i_{n}]$, which arises in all patterns of length $n$ within very long patterns, such that in a typical long pattern, the pattern $[i_{1},i_{2},...,i_{n}]$ appears with frequency $\mu([i_{1},i_{2},...,i_{n}])$. When $\Sigma=\Sigma(A)$ is a shift ...

We consider shift spaces in which elements of the alphabet may overlap nontransitively. We define a notion of entropy for such spaces, give several techniques for computing lower bounds for it, and show that it is equal to a limit of entropies of (standard) full shifts. When a shift space with overlaps arises as a model for a discrete dynamical system with a finite set of overlapping neighborhoods, the entropy gives a lower bound for the topological entropy of the dynamical s...

Let $\Sigma_{A}$ be a topologically mixing shift of finite type, let $\sigma:\Sigma_{A}\to\Sigma_{A}$ be the usual left-shift, and let $\mu$ be the Gibbs measure for a H\"{o}lder continuous potential that is not cohomologous to a constant. In this paper we study recurrence rates for the dynamical system $(\Sigma_{A},\sigma)$ that hold $\mu$-almost surely. In particular, given a function $\psi:\mathbb{N}\to \mathbb{N}$ we are interested in the following set $$R_{\psi}=\{{\text...

In this article, we compare the dynamics of the shift map and its induced counterpart on the hyperspace of the shift space. We show that many of the properties of induced shift map can be easily demonstrated by appropriate sequences of symbols. We compare the dynamics of the shift system $(\Omega, \sigma)$ with its induced counterpart $(\mathcal{K}(\Omega),\overline{\sigma})$, where $\mathcal{K}(\Omega)$ is the hyperspace of all nonempty compact subsets of $\Omega$. Recently,...