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2 Bulletin of the Polish Academy of Sciences. Mathematics

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Extreme relations for topological flows

Kamiński B., Siemaszko A., Szymański J.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 1

17--24

We introduce the concept of an extreme relation for a topological flow as an analogue of the extreme measurable partition for a measure-preserving transformation considered by Rokhlin and Sinai, and we show that every topological flow has such a relation for any invariant measure. From this result, it follows, among other things, that any deterministic flow has zero topological entropy and any flow which is a K-system with respect to an invariant measure with full support is a topological K-flow.

The determinism and the Kolmogorov property in topological dynamics

Kamiński B., Siemaszko A., Szymański J.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 4

401-417

There are introduced the concepts of deterministic, exact and Kolmogorov flows which are topological analogues of the well known measure-theoretic dynamical systems with the same names. It is shown that all distal flows are deterministic and that the only deterministic subshifts are those with a finite phase space. Deterministic flows have zero entropy. The class of Kolmogorov flows contains flows acting on zero-dimensional phase spaces being measure-theoretic Kolmogorov systems with respect to measures with full supports. All minimal Kolmogorov flows are weakly mixing.