Tomasz Downarowicz

The choquet simplex of invariant measures for minimal flows

The set of invariant measures of a compact dynamical system is well known to be a nonempty compact metrizable Choquet simplex. It is shown that all such simplices are realized already for the class of minimal flows. Moreover, sufficient is the class of 0–1 Toeplitz flows. Previously, it is proved that the set of invariant measures of the regular Toeplitz flows contains homeomorphic copies of all metric compacta.

Positive topological entropy implies chaos DC2

Using methods of entropy in ergodic theory, we prove as in the title.

Finite-rank Bratteli–Vershik diagrams are expansive

The representation of Cantor minimal systems by Bratteli–Vershik diagrams has been extensively used to study particular aspects of their dynamics. A main role has been played by the symbolic factors induced by the way vertices of a fixed level of the diagram are visited by the dynamics. The main result of this paper states that Cantor minimal systems that can be represented by Bratteli–Vershik diagrams with a uniformly bounded number of vertices at each level (called finite-rank systems) are either expansive or topologically conjugate to an odometer. More precisely, when expansive, they are topologically conjugate to one of their symbolic factors.

POSSIBLE ENTROPY FUNCTIONS

We characterize the class of functions which occur as the entropy function defined on the set of invariant measures of a (minimal) topological dynamical system. Namely, these are all non-negative affine functionsh, defined on metrizable Choquet simplices, which are non-decreasing limits of upper semi-continuous functions. Ifh is itself upper semi-continuous then it can be realized as the entropy function in an expansive dynamical system. The constructions are done effectively using minimal almost 1-1 extensions over a rotation of a group ofp-adic integers (in the expansive case, the construction leads to Toeplitz flows).

Minimal models for noninvertible and not uniquely ergodic systems

Let (Y, S) be a (not necessarilly invertible) topological dynamical system on a zero-dimensional metric spaceY without periodic points. Then there exists a minimal system (X, T) with the same simplex of invariant measures as (Y, S). More precisely, there exists a Borel isomorphism between full sets inY andX such that the adjoint map on measures is a homeomorphism between the corresponding sets of invariant measures in the weak topology. As an application we construct a minimal system carrying isomorphic copies of all nonatomic invariant measures.

Entropy of a symbolic extension of a dynamical system

Residual entropy of a topological system is defined as the infimum increase of entropy necessary to build a symbolic extension of this system. If no symbolic extension exists then residual entropy is set at infinity. In this paper we provide a direct formula for the residual entropy of a system on a totally disconnected compact space in terms of basic notions of conditional entropies viewed as functions of invariant measures. This formula allows us to evaluate residual entropy in many examples as well as to construct new examples with arbitrarily preset topological and residual entropies. The appendix contains a condition equivalent to asymptotic h -expansiveness.

The royal couple conceals their mutual relationship: A noncoalescent toeplitz flow

There exists a regular Toeplitz sequence over a finite alphabet, such that its orbit-closure in the shift system is not topologically coalescent. The notion of a Toeplitz array is introduced.

Merit factors and Morse sequences

Abstract We show that Turyns conjecture, arising from the Theory of Error Correcting Codes, has an equivalent formulation in Dynamical Systems Theory. In particular, Turyns conjecture is true if all binary Morse flows have singular spectra. Our proof uses intermediate estimates for merit factors of products of words, and is purely combinatorial.

Measure-theoretic chaos

We define new isomorphism-invariants for ergodic measure-preserving systems on standard probability spaces, called measure-theoretic chaos and measure-theoretic + chaos. These notions are analogs of the topological chaoses DC2 and its slightly stronger version (which we denote by DC1 1 ). We prove that: 1. If a topological system is measure-theoretically (measure-theoretically + ) chaotic with respect to at least one of its ergodic measures then it is topologically DC2 (DC1 1 ) chaotic. 2. Every ergodic system with positive Kolmogorov–Sinai entropy is measure-theoretically + chaotic (even in a bit stronger uniform sense). We provide an example showing that the latter statement cannot be reversed, i.e., of a system of entropy zero with uniform measure-theoretic + chaos.

Almost totally disconnected minimal systems

A space X is said to be almost totally disconnected if the set of its degenerate components is dense in X . We prove that an almost totally disconnected compact metric space admits a minimal map if and only if either it is a finite set or it has no isolated point. As a consequence we obtain a characterization of minimal sets on dendrites and local dendrites.