Combinatorical and Algebraic Structures Seminar

Session details

Date: 20.11.2018
Speaker: Edita Pelantová, FJFI, České vysoké učení technické v Praze
Title: On substitutions closed under derivation
Abstract: Occurrences of a factor $w$ in an infinite uniformly recurrent sequence ${\bf u}$ can be encoded by an infinite sequence over a finite alphabet. This sequence is usually denoted ${\bf d_{\bf u}}(w)$ and called the derivative sequence to $w$ in ${\bf u}$. If $w$ is a prefix of a fixed point ${\bf u}$ of a primitive substitution $\varphi$, then by Durand's result from 1998, the derivative sequence ${\bf d_{\bf u}}(w)$ is fixed by a primitive substitution $\psi$ as well. For a non-prefix factor $w$, the derivative sequence ${\bf d_{\bf u}}(w)$ is fixed by a substitution only exceptionally. To study this phenomenon, we introduce the new notion: A finite set $M$ of substitutions is said to be closed under derivation if the derivative sequence ${\bf d_{\bf u}}(w)$ to any factor $w$ of any fixed point ${\bf u}$ of $\varphi \in M$ is fixed by a substitution $\psi \in M$. The first example of such a set $M$ was provided in 2017 by Yu-Ke Huang and Zhi-Ying Wen. Their set $M$ contains two substitutions, one of them is the so-called period doubling substitution. In our talk we characterize the Sturmian substitutions which belong to a set $M$ closed under derivation.

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