The Arnold cat map has the interesting property that, after some large but finite number of iterations, a state arbitrarily close to the initial state appears. According to the Poincaré recurrence lemma, a map has such a "recurrence property" if and only if it is a measure-preserving map. Thus Arnold's cat map could be generalized to a family of measure-preserving maps of some space to itself.
In the case of an -dimensional torus , the group of measure-preserving transformations from to is isomorphic to the group of integer matrices with determinant of absolute value 1. Moreover, we can think of this group in terms of isotopy classes of transformations, which makes this group the index-two supergroup for the mapping class group.
This Demonstration shows the two-dimensional version of a generalized Arnold's cat map. It displays the state after each iteration, both in the unit square and on the torus.
This Demonstration provides three pictures as examples: a stack of apples, a flower and a house. You can upload pictures to the initialization section to see the cat map working on other pictures. You can select resolutions from 32 to 256 dots per inch. Powers of 2 are used to minimize the number of iterations required to return to the initial state.