Periods of discretized linear Anosov maps

Abstract

Integer $m\times m$ matrices $A$ with determinant $1$ define diffeomorphisms of the $m$-dimensional torus $T^m=({\Bbb R}/{\Bbb Z})^m$ into itself. Likewise, they define bijective self-maps of the discretized tori $({\Bbb Z}/n{\Bbb Z})^m=({\Bbb z}_n)^m$. We present estimates of the surprisingly low order (or period) $\Per_A(n)$ of the iteration $A^r, r=1,2,3,\ldots,$ on the discretized torus $({\Bbb z}_n)^m$. We obtain $\Per_A(n)\leq 3n$ for dimension $m=2$. In the special case of the Anosov map $A={2\ 1 \atopwithdelims() 1\ 1}$, this result is due to Dyson and Talk [DT92]. For arbitrary dimensions $m>2$ we obtain $$\Per_A(n)\leq {\rm constant}\cdot n^{m-1},$$ provided $n$ is a power of a prime number. For general $n$, number theoretic problems arise.