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arXiv · 1410.5519

Growth degree classification for finitely generated semigroups of integer matrices

Jason P. Bell·Michael Coons·Kevin G. Hare
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Abstract

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 $\lim_{n\to\infty} \log m_n(\mathcal{A})/\log n\in\mathbb{Z}_{\geqslant 0}\cup\{\infty\}.$ This has applications to the growth of regular sequences as defined by Allouche and Shallit.

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Jason P. Bell, Michael Coons, Kevin G. Hare. 2014-10-21. Growth degree classification for finitely generated semigroups of integer matrices. https://arxiv.org/abs/1410.5519

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