A graph-theoretic proof of Cobham’s Dichotomy for automatic sequences

We give a new graph-theoretic proof of a theorem of Cobham which says that the support of an automatic sequence is either sparse, that is, grows polylogarithmically, or grows at least like $N^{\alpha }$ for some $\alpha >0$. The proof uses the notions of tied vertices and cycle arborescences. With the ideas of the proof we can also give a new interpretation of the rank of a sparse sequence as the height of its cycle arborescence. In the non-sparse case we are able to show that the support has asymptotic behavior of the form $N^{B}log{\left(N\right)}^{r-1}$, where $B$ turns out to be the logarithm of an integer root of a Perron number.