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For any non-erasing free monoid morphism $\sigma: \cal A^* \to \cal B^*$, andfor 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 $p_Y$ areshown 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 allsufficiently large integers $n \in \N$, provided that $\sigma$ is recognizablein $X$. 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 $\sigma$ isrecognizable in $X$.  As a consequence, there is no meaningful way to define the topologicalentropy 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.

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