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

Alexandra Antoniouk; Karsten Keller; Sergiy Maksymenko

doi:10.3934/dcds.2014.34.1793

Article Contents

2014, Volume 34, Issue 5: 1793-1809. Doi: 10.3934/dcds.2014.34.1793

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

1.
Institute of Mathematics of NAS of Ukraine, Tereshchenkivs'ka str., 3, 01601 Kyiv
2.
Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, 23562 Lübeck

Received: April 2013

Revised: July 2013

Published: May 2014

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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.

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Mathematics Subject Classification: Primary: 37A35, 54E52; Secondary: 58A20.

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