We prove that the σ-ideal I(E) (of closed smooth sets with respect to a non-smooth Borel equivalence relation E) does not have the covering property. In fact, the same holds for any σ-ideal containing the closed transversals with respect to an equivalence relation generated by a countable group of homeomorphisms. As a consequence we show that I(E) does not have a Borel basis.
We construct strictly ergodic 0-1 Toeplitz flows with pure point spectrum and irrational eigenvalues. It is also shown that the property of being regular is not a measure-theoretic invariant for strictly ergodic Toeplitz flows.
The properties of topological dynamical systems which are disjoint from all minimal systems of zero entropy, , are investigated. Unlike the measurable case, it is known that topological -systems make up a proper subset of the systems which are disjoint from . We show that has an invariant measure with full support, and if in addition is transitive, then is weakly mixing. A transitive diagonal system with only one minimal point is constructed. As a consequence, there exists a thickly syndetic...
We investigate the connections between Ramsey properties of Fraïssé classes and the universal minimal flow of the automorphism group of their Fraïssé limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class has finite Ramsey degree for embeddings, then this degree equals the size of . We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if is a relational Ramsey class and is amenable, then admits a unique invariant...
The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces and a sequence of continuous maps , , is defined. If all the spaces are compact real intervals and all the maps are piecewise monotone then, under some additional assumptions, a formula for the entropy of the system is obtained in terms of the number of pieces of monotonicity of . As an application we construct a large class of smooth triangular maps of the square of type and positive...
Let X be an uncountable compact metrizable space of topological dimension zero. Given any a ∈[0,∞] there is a homeomorphism on X whose topological entropy is a.
We prove that if f, g are smooth unimodal maps of the interval with negative Schwarzian derivative, conjugated by a homeomorphism of the interval, and f is Collet-Eckmann, then so is g.
A continuous map of the interval is chaotic iff there is an increasing sequence of nonnegative integers such that the topological sequence entropy of relative to , , is positive ([FS]). On the other hand, for any increasing sequence of nonnegative integers there is a chaotic map of the interval such that ([H]). We prove that the same results hold for maps of the circle. We also prove some preliminary results concerning topological sequence entropy for maps of general compact metric...