A canonical partition of the periodic orbits of chaotic maps

Abstract:

We show that the periodic orbits of an area-contracting horseshoe map can be partitioned into subsets of orbits of minimum period $k,\;2k,\;4k,\;8k \ldots$, for some positive integer $k$. This partition is natural in the following sense: for any parametrized area-contracting map which forms a horseshoe, the orbits in one subset of the partition are contained in a single component of orbits in the full parameter space. Furthermore, prior to the formation of the horseshoe, this component contains attracting orbits of minimum period ${2^m}k$, for each nonnegative integer $m$.