We investigate the connections between Ramsey properties of Fraïssé classes and the universal minimal flow $M(G_{)}$ of the automorphism group $G$ 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 $M(G_{)}$. We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if is a relational Ramsey class and $G$ is amenable, then $M(G_{)}$ admits a unique invariant...

The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces ${\left(X_i\right)}_{i=1}^{\infty }$ and a sequence of continuous maps ${\left(f_i\right)}_{i=1}^{\infty }$, $f_i:X_i\to X_{i+1}$, 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 $f_n○...○f_2○f_1$. As an application we construct a large class of smooth triangular maps of the square of type $2^{\infty }$ 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.

In this paper, we deal with the product of spaces which are either $𝒢$-spaces or ${𝒢}_p$-spaces, for some $p\in {\omega }^{*}$. These spaces are defined in terms of a two-person infinite game over a topological space. All countably compact spaces are $𝒢$-spaces, and every ${𝒢}_p$-space is a $𝒢$-space, for every $p\in {\omega }^{*}$. We prove that if $\{X_{\mu }:\mu <{\omega }_1\}$ is a set of spaces whose product $X={\prod }_{\mu <{\omega }_1}{X}_{\mu }$ is a $𝒢$-space, then there is $A\in {\left[{\omega }_1\right]}^{\le \omega }$ such that $X_{\mu }$ is countably compact for every $\mu \in {\omega }_1\setminus A$. As a consequence, $X^{{\omega }_1}$ is a $𝒢$-space iff $X^{{\omega }_1}$ is countably compact, and if $X^{2^{𝔠}}$ is a $𝒢$-space, then all...

Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover $$𝒰$$ of X there is a sequence of maps (f n: X → X)nεgw such that each f n is $$𝒰$$ -near to the identity map of X and the family f n(X)n∈ω is locally finite...

In his classical paper [Ann. of Math. 45 (1944)] P. R. Halmos shows that weak mixing is generic in the measure preserving transformations. Later, in his book, Lectures on Ergodic Theory, he gave a more streamlined proof of this fact based on a fundamental lemma due to V. A. Rokhlin. For this reason the name of Rokhlin has been attached to a variety of results, old and new, relating to the density of conjugacy classes in topological groups. In this paper we will survey some of the new developments...

Rough sets, developed by Pawlak, are an important model of incomplete or partially known information. In this article, which is essentially a continuation of [11], we characterize rough sets in terms of topological closure and interior, as the approximations have the properties of the Kuratowski operators. We decided to merge topological spaces with tolerance approximation spaces. As a testbed for our developed approach, we restated the results of Isomichi [13] (formalized in Mizar in [14]) and...

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.