Folding and Metric Entropies for Extended Shifts

TL;DR

This paper computes the metric and folding entropies for a class of non-invertible symbolic dynamical systems that generalize Bernoulli shifts, providing insights into their complexity and encoding properties.

Contribution

It introduces a framework for calculating entropies in non-invertible symbolic systems that extend classical Bernoulli shifts, including systems modeling baker's transformations.

Findings

Calculated metric and folding entropies for the systems.

Extended understanding of entropy in non-invertible symbolic dynamics.

Applicable to encoding baker's transformations and skew products.

Abstract

In this paper we calculate the metric and folding entropies for a family of non-invertible symbolic dynamical systems $(\Sigma_{m_-,m_+}, \sigma_\phi)$ which generalizes the standard bilateral Bernoulli shifts. The space $\Sigma_{m_-,m_+}$ consists of symbolic sequences over two distinct finite alphabets, with dynamics governed by a shift map $\sigma_\phi$ incorporating a non-invertible function $\phi$ that maps one of the alphabets to the other one. These systems are, for instance, particularly useful for encoding the many-to-one baker's transformation endomorphisms, and they can also be seen as a skew product with a unilateral Bernoulli shift on the base.