Characterizations and Enumerations of Patterns of Signed Shifts

TL;DR

This paper characterizes and counts permutation patterns generated by signed shift maps, extending understanding of their combinatorial structure and entropy, with implications for dynamical systems and symbolic dynamics.

Contribution

It provides a complete characterization of permutation patterns of signed shifts and exact enumeration for negative shifts, advancing the combinatorial understanding of these maps.

Findings

Complete characterization of permutation patterns of signed shifts

Exact enumeration of patterns for negative shifts

Improved bounds and entropy calculation for the tent map

Abstract

Signed shifts are generalizations of the shift map in which, interpreted as a map from the unit interval to itself sending x to the fractional part of Nx, some slopes are allowed to be negative. Permutations realized by the relative order of the elements in the orbits of these maps have been studied recently by Amigo, Archer and Elizalde. In this paper, we give a complete characterization of the permutations (also called patterns) realized by signed shifts. In the case of the negative shift, which is the signed shift having only negative slopes, we use the characterization to give an exact enumeration of these patterns. Finally, we improve the best known bounds for the number of patterns realized by the tent map, and calculate the topological entropy of signed shifts using these combinatorial methods.