# American Institute of Mathematical Sciences

May  2014, 34(5): 1793-1809. doi: 10.3934/dcds.2014.34.1793

## 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, Ukraine, Ukraine 2 Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, 23562 Lübeck, Germany

Received  April 2013 Revised  July 2013 Published  October 2013

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.
Citation: Alexandra Antoniouk, Karsten Keller, Sergiy Maksymenko. Kolmogorov-Sinai entropy via separation properties of order-generated $\sigma$-algebras. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1793-1809. doi: 10.3934/dcds.2014.34.1793
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##### References:
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