How do topological entropy and factor complexity behave under monoid morphisms and free group basis changes ?

TL;DR

This paper investigates how topological entropy and factor complexity of subshifts behave under free monoid morphisms and basis changes, revealing that entropy is generally not preserved and proposing implications for currents on free groups.

Contribution

It establishes bounds on complexity functions under recognizable monoid morphisms and shows entropy invariance fails in general, impacting the understanding of currents in free groups.

Findings

Complexity functions are bounded and comparable under recognizable morphisms.

Topological entropy is not preserved under general morphisms.

A meaningful entropy concept for currents on free groups is limited to certain cases.

Abstract

For any non-erasing free monoid morphism $\sigma: \cal A^* \to \cal B^*$, and for any subshift $X \subset \cal A^\Z$ and its image subshift $Y = \sigma(X) \subset \cal B^\Z$, the associated complexity functions $p_X$ and $p_Y$ are shown to satisfy: there exist constants $c, d, C > 0$ such that $$c \cdot p_X(d \cdot n) \,\, \leq \,\, p_Y(n) \,\, \leq \,\, C \cdot p_X(n)$$ holds for all sufficiently large integers $n \in \N$, provided that $\sigma$ is recognizable in $X$. If $\sigma$ is in addition letter-to-letter, then $p_Y$ belongs to $\Theta(p_X)$ (and conversely). Otherwise, however, there are examples where $p_X$ is not in $\cal O(p_Y)$. It follows that in general the value $h_X$ of the topological entropy of $X$ is not preserved when applying a morphism $\sigma$ to $X$, even if $\sigma$ is recognizable in $X$. As a consequence, there is no meaningful way to define the topological entropy of a current on a free group $F_N$; only the distinction of currents $\mu$ with topological entropy $h_{\tiny\supp(\mu)} = 0$ and $h_{\tiny\supp(\mu)} > 0$ is well defined.