## More on the Dynamics of the Symbolic Square Root Map

Jarkko Peltomäki, Markus Whiteland, More on the Dynamics of the Symbolic Square Root Map. In: Srečko Brlek, Francesco Dolce, Christophe Reutenauer, Élise Vandomme (Eds.), Combinatorics on Words, Lecture Notes in Computer Science 10432, 97–108, Springer, 2017.

http://dx.doi.org/10.1007/978-3-319-66396-8

### Abstract:

In our 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.

### BibTeX entry:

@INPROCEEDINGS{inpPeWh17a,  title = {More on the Dynamics of the Symbolic Square Root Map},  booktitle = {Combinatorics on Words},  author = {Peltomäki, Jarkko and Whiteland, Markus},  volume = {10432},  series = {Lecture Notes in Computer Science},  editor = {Brlek, Srečko and Dolce, Francesco and Reutenauer, Christophe and Vandomme, Élise},  publisher = {Springer},  pages = {97–108},  year = {2017},  ISSN = {0302-9743},}

Belongs to TUCS Research Unit(s): FUNDIM, Fundamentals of Computing and Discrete Mathematics

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