Forbidden patterns and shift systems

TL;DR

This paper explores forbidden order patterns in one-dimensional dynamical systems and shift systems, revealing how certain patterns are avoided and analyzing their implications for chaos and information theory.

Contribution

It introduces a novel connection between permutation pattern avoidance and dynamical systems, particularly in shift systems, with detailed analysis of forbidden patterns.

Findings

Allowed patterns avoid forbidden root patterns and their shifts

Shift systems exhibit properties like sensitivity and mixing related to forbidden patterns

Results can be transferred to other systems via order-isomorphisms

Abstract

The scope of this paper is two-fold. First, to present to the researchers in combinatorics an interesting implementation of permutations avoiding generalized patterns in the framework of discrete-time dynamical systems. Indeed, the orbits generated by piecewise monotone maps on one-dimensional intervals have forbidden order patterns, i.e., order patterns that do not occur in any orbit. The allowed patterns are then those patterns avoiding the so-called forbidden root patterns and their shifted patterns. The second scope is to study forbidden patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. Due to its simple structure, shift systems are accessible to a more detailed analysis and, at the same time, exhibit all important properties of low-dimensional chaotic dynamical systems (e.g., sensitivity to initial conditions, strong mixing and a dense set of periodic points), allowing to export the results to other dynamical systems via order-isomorphisms.