Cyclic permutations realized by signed shifts

TL;DR

This paper characterizes the periodic patterns of signed shifts, a class of one-dimensional dynamical systems, using combinatorial methods, and provides exact formulas for counting these patterns in various cases.

Contribution

It offers a combinatorial characterization of periodic patterns for signed shifts and derives explicit formulas for their enumeration, extending previous results.

Findings

Characterization of periodic patterns via descent sets of cyclic permutations

Exact formulas for counting periodic patterns in shift maps, reverse shifts, and the tent map

New enumeration formulas for pattern-avoiding cycles

Abstract

The periodic (ordinal) patterns of a map are the permutations realized by the relative order of the points in its periodic orbits. We give a combinatorial characterization of the periodic patterns of an arbitrary signed shift, in terms of the structure of the descent set of a certain cyclic permutation associated to the pattern. Signed shifts are an important family of one-dimensional dynamical systems that includes shift maps and the tent map as particular cases. Defined as a function on the set of infinite words on a finite alphabet, a signed shift deletes the first letter and, depending on its value, possibly applies the complementation operation on the remaining word. For shift maps, reverse shift maps, and the tent map, we give exact formulas for their number of periodic patterns. As a byproduct of our work, we recover results of Gessel--Reutenauer and Weiss--Rogers and obtain new enumeration formulas for pattern-avoiding cycles.