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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.