Let E denote the σ-ideal generated by closed null sets on the reals. We show that the uniformity and the covering of E can be added to Cichoń’s maximum with distinct values. More specifically, it is consistent that ℵ1<add(N)<cov(N)<b<non(E)<non(M)<cov(M)<cov(E)<d<non(N)<cof(N)<2ℵ0 holds

Existence of a measure with maximal entropy (that is, an invariant measure whose measure-theoretic entropy is equal to the topological entropy of the system) is an important subject in dynamical system theory. Weak expansiveness of the dynamical system, including h-expansiveness and asymptotic h-expansiveness, provides a non-trivial sufficient condition for existence of such measures. In this paper we give new characterizations for weak expansiveness of actions by amenable groups, by interpreting equivalently topological conditional entropy of such actions using either topological entropy of subsets in the sense of separated and spanning sets or ideas of Bowen’s dimensional entropy of subsets. Our results extend not only the well-known results about weak expansiveness by Bowen (1972, 1973), Marczyńska (1977) and Misiurewicz (1976) from single transformations to amenable group actions, but also a recent result relating topological entropy to Bowen’s dimensional entropy for amenable group actions which was proved first by Zheng and Chen (2016) using measure-theoretic methods and then re-proved by Dou and Zhang (2018) via purely topological tools.

Inspired by the very successful study of Borel equivalence relations under Borel reducibility in descriptive set theory and equivalence relations on ω under computable reducibility in computability theory, R. Miller defined a family of reducibility notions. Defined on equivalence relations on Baire space or Cantor space, these reducibilities are required to succeed (uniformly) on all finite or countable subsets of the whole space. In this paper, we combine methods from computability theory and descriptive set theory to study equivalence relations under these reductions. In particular, we show existence and non-existence of complete equivalence relations in various settings.

We show that separability and second-countability are first-order properties of topological spaces definable in o-minimal expansions of $(\mathbb R, \lt )$. We do so by introducing first-order characterizations – definable separability and definable secon

In the case of Schrödinger cocycles, by the so-called Thouless formula the harmonicity of the Lyapunov exponent is equivalent to uniform hyperbolicity. In this paper we consider monotonic cocycles and give a sufficient condition for the harmonicity of the

We study the class of analytic binary relations on Polish spaces, compared by the notions of continuous reducibility or injective continuous reducibility. In particular, we characterize when a locally countable Borel relation is ${\bf \Sigma }^{0}_{\xi }$

We classify pointed spaces according to the first fundamental group of their reduced suspension. A pointed space is either of so-called totally path disconnected type or of horseshoe type. These two camps are defined topologically but a characterization is given in terms of fundamental groups. Among totally path disconnected spaces the fundamental group is shown to be a complete invariant for a notion of topological equivalence weaker than that of homeomorphism.

Let X be a full-shift on the alphabet [0, 1] a and let (Y, S) be an arbitrary dynamical system. We prove that any equivariant continuous map from X to Y has conditional metric mean dimension not less than a − mdim(Y, S). This solves a problem posed in [ST23, Problem 1.6].

We continue the study of projective Fraïssé limits of trees initiated by Charatonik and Roe and further continued by Charatonik, Kwiatkowska, Roe and Yang. We construct many generalized Ważewski dendrites as topological realizations of projective Fraïssé limits of families of finite trees with (weakly) coherent epimorphisms. Moreover we use the categorical approach to Fraïssé limits developed by Kubiś to construct all generalized Ważewski dendrites as topological realizations of Fraïssé limits of suitable categories of finite structures. As an application we recover a homogeneity result for countable dense sets of endpoints in generalized Ważewski dendrites.