The ghosts of the cat

Abstract

Let $A\in SL(2,{\Bbb Z})$ be given and $n\in{\Bbb N}$. $A$ induces a map $A_n$ on ${\Bbb Z}^2_n$ by $(x,y)\mapsto(a_{11}x+a_{12}y,a_{21}x+a_{22}y)$ mod $n$. $A_n$ can be thought of as a discrete approximation of the similarly defined map on the unit square. It is known that the $A_n$ have a surprisingly small period: there always is an $r\le3n$ such that $A^r_n$ is the identity on ${\Bbb Z}^2_n$.

One can visualize the properties of the iterates of $A_n$ by considering the sets $P,A_nP,A^2_nP,\ldots$ for a fixed subset $P$ of ${\Bbb Z}^2_n$. (Here the cat comes into play: $P$ has often been chosen to be the digitalized picture of a cat, in particular when treating the special case of the Anosov map $A={2\ 1\atopwithdelims() 1\ 1}$.) It turns out that in the sequence $(A^k_nP)_k$ mostly ‘chaotic’ pictures occur as one would expect. However, some kind of regular behaviour can also be observed for certain powers $A^k_n$ before the end of the period.

It is the aim of this paper to describe these phenomena and to explain why and how they occur. In order to quantify the chaotic behaviour of the maps on ${\Bbb Z}^2_n$ under consideration, we introduce suitable functions.

Our methods are elementary. Only basic results from the geometry of numbers will be used.