Growth degree classification for finitely generated semigroups of integer matrices.

Let $${\mathcal {A}}$$ be a finite set of $$d\times d$$ matrices with integer entries and let $$m_n({\mathcal {A}})$$ be the maximum norm of a product of $$n$$ elements of $${\mathcal {A}}$$ . In this paper, we classify gaps in the growth of $$m_n({\mathcal {A}})$$ ; specifically, we prove that $$\lim _{n\rightarrow \infty } \log m_n({\mathcal {A}})/\log n\in \mathbb {Z}_{\geqslant 0}\cup \{\infty \}.$$ This has applications to the growth of regular sequences as defined by Allouche and Shallit. [ABSTRACT FROM AUTHOR]