Directional complexity and entropy for lift mappings

TL;DR

This paper introduces the concepts of directional complexity and entropy for degree 1 circle maps, relating them to symbolic dynamics and topological entropy, with exact formulas and maximal values derived.

Contribution

It provides the first detailed analysis of directional entropy for circle maps, connecting it to symbolic complexity and topological entropy with explicit formulas.

Findings

Directional entropy equals topological entropy at maximum

Exact formulas for directional entropy are derived

Maximal directional entropy matches topological entropy

Abstract

We introduce and study the notion of a directional complexity and entropy for maps of degree 1 on the circle. For piecewise affine Markov maps we use symbolic dynamics to relate this complexity to the symbolic complexity. We apply a combinatorial machinery to obtain exact formulas for the directional entropy, to find the maximal directional entropy, and to show that it equals the topological entropy of the map. Keywords: Rotation interval, Space-time window, Directional complexity, Directional entropy;