Constant slope, entropy and horseshoes for a map on a tame graph

We study continuous countably (strictly) monotone maps defined on a tame graph, i.e., a special Peano continuum for which the set containing branchpoints and endpoints has a countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map $f$ of a tame graph $G$ is conjugate to a constant slope map $g$ of a countably affine tame graph. In particular, we show that in the case of a Markov map $f$ that corresponds to recurrent transition matrix, the condition is satisfied for constant slope $e^{h_{\mathrm{top}}(f)}$, where $h_{\mathrm{top}}(f)$ is the topological entropy of $f$. Moreover, we show that in our class the topological entropy $h_{\mathrm{top}}(f)$ is achievable through horseshoes of the map $f$.