Kolmogorov-Sinai entropy via separation properties of order-generated sigma-algebras

TL;DR

This paper generalizes a characterization of Kolmogorov-Sinai entropy using order-generated sigma-algebras, extending previous work by relaxing separation conditions and analyzing smooth systems on manifolds.

Contribution

It relaxes the separation condition in the entropy characterization and explores the abundance of smooth vectors in systems on manifolds.

Findings

Generalized the entropy characterization to broader sigma-algebras.

Showed the set of smooth vectors is large in Lebesgue measure on manifolds.

Extended the applicability of order-based entropy analysis.

Abstract

In a recent paper, K.Keller has given a characterization of the Kolmogorov-Sinai entropy of a discrete-time measure-preserving dynamical system on the base of an increasing sequence of special partitions. These partitions are constructed from order relations obtained via a given real-valued random vector, which can be interpreted as a collection of observables on the system and is assumed to separate points of it. In the present paper we relax the separation condition in order to generalize the given characterization of Kolmogorov-Sinai entropy, providing a statement on equivalence of sigma-algebras. On its base we show that in the case that a dynamical system is living on an m-dimensional smooth manifold and the underlying measure is Lebesgue absolute continuous, the set of smooth random vectors of dimension n>m with given characterization of Kolmogorov-Sinai entropy is large in a certain sense.