arXiv subjects

Paulina Cecchi Bernales

Publications and source records attributed to Paulina Cecchi Bernales.

4 recordsLinked to original sources

arXiv2025

Coboundaries and eigenvalues of finitary S-adic systems

An S-adic system is a symbolic dynamical system generated by iterating an infinite sequence of substitutions or morphisms, called a directive sequence. A finitary S-adic dynamical system is one where the directive sequence consists of morphisms selected from a finite set. We study eigenvalues and coboundaries for finitary recognizable S-adic dynamical systems, i.e., those where points can be uniquely desubstituted using the given sequence of morphisms. To do this we identify the notions of straightness and essential words, and use them to define a coboundary, inspired by of Host's formalism, which allows us to express necessary and sufficient conditions that a complex number must satisfy in order to be a continuous or measurable eigenvalue. We then apply our results to finitary directive sequences of substitutions of constant length, and show how to create constant-length $S$-adic shifts with non-trivial coboundaries. We show that in this case all continuous eigenvalues are rational and we give a complete description of the rationals that can be an eigenvalue, indicating how this leads to a Cobham-style result for these systems.

math.DS↗

arXiv2023

Invariant measures of Toeplitz subshifts on non-amenable groups

Let $G$ be a countable residually finite group (for instance $\mathbb{F}_2$) and let $\overleftarrow{G}$ be a totally disconnected metric compactification of $G$ equipped with the action of $G$ by left multiplication. For every $r\geq 1$ we construct a Toeplitz $G$-subshift $(X,σ,G)$, which is an almost one-to-one extension of $\overleftarrow{G}$, having $r$ ergodic measures $ν_1, \cdots,ν_r$ such that for every $1\leq i\leq r$ the measure-theoretic dynamical system $(X,σ,G,ν_i)$ is isomorphic to $\overleftarrow{G}$ endowed with the Haar measure. The construction we propose is general (for amenable and non-amenable residually finite groups), however, we point out the differences and obstructions that could appear when the acting group is not amenable.

math.DS↗

arXiv2022

Strong Orbit Equivalence and Superlinear Complexity

We show that within any strong orbit equivalent class, there exist minimal subshifts with arbitrarily low superlinear complexity. This is done by proving that for any simple dimension group with unit $(G,G^+,u)$ and any sequence of positive numbers $(p_n)_{n\in\mathbb{N}}$ such that $\lim n/p_n=0$, there exist a minimal subshift whose dimension group is order isomorphic to $(G,G^+,u)$ and whose complexity function grows slower than $p_n$. As a consequence, we get that any Choquet simplex can be realized as the set of invariant measures of a minimal Toeplitz subshift whose complexity grows slower than $p_n$.

math.DS↗

arXiv2018

Balancedness and coboundaries in symbolic systems

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 codings of regular interval exchanges. We prove that dendric words are balanced on letters if and only if they are balanced on words. In the substitutive case, we stress the role played by the existence of coboundaries taking rational values and show simple criteria when frequencies take rational values for exhibiting imbalancedness.

cs.DM↗