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

Abstract

For any non-erasing free monoid morphism $\sigma :{\mathcal{A}}^{\ast }\to {\mathcal{B}}^{\ast }$, and for any subshift $X\subset {\mathcal{A}}^{\mathbb{Z}}$ and its image subshift $Y=\sigma (X)\subset {\mathcal{B}}^{\mathbb{Z}}$, the associated complexity functions $p_X$ and $p_Y$ are shown to satisfy: there exist constants $c,d,C>0$ such that $$c\cdot p_X(d\cdot n)\le p_Y(n)\le C\cdot p_X(n)$$ holds for all sufficiently large integers $n\in \mathbb{N}$, 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 $\mathcal{O}(p_{\mathcal{Y}})$. 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 $h_{\tiny\supp(μ)} = 0$ and $h_{\tiny\supp(μ)} > 0$ is well defined.