Forbidden ordinal patterns in higher dimensional dynamics

TL;DR

This paper investigates the existence and properties of forbidden ordinal patterns in higher-dimensional dynamical systems, showing that expansive maps with finite entropy necessarily have such patterns and exploring their robustness.

Contribution

It extends the concept of forbidden ordinal patterns to n-dimensional maps, establishing their necessity in expansive systems with finite entropy and analyzing their statistical properties.

Findings

Expansive interval maps with finite topological entropy have forbidden patterns.

Forbidden patterns are robust against observational white noise.

Theoretical results are illustrated with 2D examples using counting and Chao's estimators.

Abstract

Forbidden ordinal patterns are ordinal patterns (or `rank blocks') that cannot appear in the orbits generated by a map taking values on a linearly ordered space, in which case we say that the map has forbidden patterns. Once a map has a forbidden pattern of a given length $L_{0}$, it has forbidden patterns of any length $L\ge L_{0}$ and their number grows superexponentially with $L$. Using recent results on topological permutation entropy, we study in this paper the existence and some basic properties of forbidden ordinal patterns for self maps on n-dimensional intervals. Our most applicable conclusion is that expansive interval maps with finite topological entropy have necessarily forbidden patterns, although we conjecture that this is also the case under more general conditions. The theoretical results are nicely illustrated for n=2 both using the naive counting estimator for forbidden patterns and Chao's estimator for the number of classes in a population. The robustness of forbidden ordinal patterns against observational white noise is also illustrated.