Entropy dimension for Deterministic Walks in Random Sceneries

TL;DR

This paper investigates the entropy dimension of deterministic walks in random sceneries, demonstrating that by selecting appropriate rotations, these models can exhibit any desired entropy dimension within [0,1], thus classifying their complexity levels.

Contribution

It introduces a method to realize any entropy dimension in skew products of irrational rotations with Bernoulli systems, expanding understanding of complexity in dynamical systems.

Findings

Models can attain any entropy dimension in [0,1]

Entropy dimension classifies intermediate growth rates

Constructive approach for specific rotation choices

Abstract

Entropy dimension is an entropy-type quantity which takes values in $[0,1]$ and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.