More on the dynamics of the symbolic square root map

TL;DR

This paper explores the dynamics of a symbolic square root map on optimal squareful words, extending previous work on Sturmian words, and characterizes periodic points and limit sets, revealing both similarities and differences in behavior.

Contribution

It provides a detailed analysis of the square root map's behavior on new subshifts of optimal squareful words, expanding understanding beyond Sturmian words.

Findings

Characterization of periodic points

Description of the limit set

Differences in dynamics compared to Sturmian case

Abstract

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 words (which are optimal squareful words). The dynamics of the square root map on a Sturmian subshift are well understood. In our earlier work, we introduced another type of subshift of optimal squareful words which together with the square root map form a dynamical system. In this paper, we study these dynamical systems in more detail and compare their properties to the Sturmian case. The main results are characterizations of periodic points and the limit set. The results show that while there is some similarity it is possible for the square root map to exhibit quite different behavior compared to the Sturmian case.