Abstract
The periodic patterns of a map are the permutations realized by the relative order of the points in its periodic orbits. We give a combinatorial description of the periodic patterns of an arbitrary signed shift, in terms of the structure of the descent set of a certain transformation of the pattern. Signed shifts are an important family of one-dimensional dynamical systems. For particular types of signed shifts, namely shift maps, reverse shift maps, and the tent map, we give exact enumeration formulas for their periodic patterns. As a byproduct of our work, we recover some results of Gessel and Reutenauer and obtain new results on the enumeration of pattern-avoiding cycles.
Highlights
1.1 Background and motivationPermutations realized by the orbits of a map on a one-dimensional interval have received a significant amount of attention in the last five years [2]
These are the permutations given by the relative order of the elements of the sequence obtained by successively iterating the map, starting from any point in the interval
One the one hand, understanding these permutations provides a powerful tool to distinguish random from deterministic time series, based on the remarkable fact [6] that every piecewise monotone map has forbidden patterns, i.e., permutations that are not realized by any orbit
Summary
Permutations realized by the orbits of a map on a one-dimensional interval have received a significant amount of attention in the last five years [2] These are the permutations given by the relative order of the elements of the sequence obtained by successively iterating the map, starting from any point in the interval. Our main result is a characterization of the periodic patterns of an (almost) arbitrary signed shift, given in Theorem 2.1. An interesting consequence of our study of periodic patterns is that we obtain new results regarding the enumeration of cyclic permutations that avoid certain patterns.
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