Related papers: Sensitivity, proximal extension and higher order a…

We give a number of approximation metatheorems for monotone maximization problems expressible in the first-order logic, in substantially more general settings than the previously known. We obtain * constant-factor approximation algorithm in…

Power-law sensitivity to initial conditions, characterizing the behaviour of dynamical systems at their critical points (where the standard Liapunov exponent vanishes), is studied in connection with the family of nonlinear 1D logistic-like…

A Borel system $(X,S)$ is `almost Borel universal' if any free Borel dynamical system $(Y,T)$ of strictly lower entropy is isomorphic to a Borel subsystem of $(X,S)$, after removing a null set. We obtain and exploit a new sufficient…

For strictly ergodic systems, we introduce the class of CF-Nil($k$) systems: systems for which the maximal measurable and maximal topological $k$-step pronilfactors coincide as measure-preserving systems. Weiss' theorem implies that such…

A sofic approximation to a countable group is a sequence of partial actions on finite sets that asymptotically approximates the action of the group on itself by left-translations. A group is sofic if it admits a sofic approximation. Sofic…

We study minimal idempotents $J^{\mathrm{min}}(X)$ in the Ellis semigroup $E(X)$ associated with a Floyd-Auslander system $(X,T)$. We show that $(X,T)$ is non-tame if and only if $|J^{\mathrm{min}}(X)| > 2^{\aleph_0}$, which happens exactly…

Let (X,T) be a topologically transitive dynamical system. We show that if there is a subsystem (Y,T) of (X,T) such that (X\times Y, T\times T) is transitive, then (X,T) is strongly chaotic in the sense of Li and Yorke. We then show that…

For minimal $\mathbb{Z}^{2}$-topological dynamical systems, we introduce a cube structure and a variation of the regionally proximal relation for $\mathbb{Z}^2$ actions, which allow us to characterize product systems and their factors. We…

We study almost automorphic solutions of the discrete delayed neutral dynamic system% \[ x(t+1)=A(t)x(t)+\Delta Q(t,x(t-g(t)))+G(t,x(t),x(t-g(t))) \] by means of a fixed point theorem due to Krasnoselskii. Using discrete variant of…

In this paper, we develop several structure theorems concerning commuting transformations and minimal $\mathbb{R}$-flows. Specifically, we show that if $(X,S)$, $(X,T)$ are minimal systems with $S$ and $T$ being commutative, then they…

We study dp-minimal infinite profinite groups that are equipped with a uniformly definable fundamental system of open subgroups. We show that these groups have an open subgroup $A$ such that either $A$ is a direct product of countably many…

We introduce the notions of directional dynamical cubes and directional regionally proximal relation defined via these cubes for a minimal $\mathbb{Z}^d$-system $(X,T_1,\ldots,T_d)$. We study the structural properties of systems that…

This article presents three characterizations of the weak factorization systems on finitely complete categories that interpret intensional dependent type theory with Sigma-, Pi-, and Id-types. The first characterization is that the weak…

The sensitivity, $\sigma(G)$, of a finite undirected simple graph $G$ is the smallest maximum degree of an induced subgraph on more than the maximum number of independent vertices. Call an indexed family of graphs $G_n$ with maximum degree…

Given a semigroup $G$ and a bounded function $f: G \to \mathbb{C}$, a topological Furstenberg system of $f$ is a topological dynamical system $\mathbb{X}=(X, (T_g)_{g \in G})$ that encodes the dynamical behaviour of $f$. We show that…

The ability to cast values between related types is a leitmotiv of many flavors of dependent type theory, such as observational type theories, subtyping, or cast calculi for gradual typing. These casts all exhibit a common structural…

A minimal Cantor system is said to be self-induced whenever it is conjugate to one of its induced systems. Substitution subshifts and some odometers are classical examples, and we show that these are the only examples in the equicontinuous…

We construct a family of shift spaces with almost specification and multiple measures of maximal entropy. This answers a question from Climenhaga and Thompson [Israel J. Math. 192 (2012), no. 2, 785--817]. Elaborating on our examples we…

Let $M$ be a compact Riemannian manifold. The set $\text{F}^{r}(M)$ consisting of sequences $(f_{i})_{i\in\mathbb{Z}}$ of $C^{r}$-diffeomorphisms on $M$ can be endowed with the compact topology or with the strong topology. A notion of…

Let $f: M \to M$ be a $C^{1+\alpha}$ map/diffeomorphism of a compact Riemannian manifold $M$ and $\mu$ be an expanding/hyperbolic ergodic $f$-invariant Borel probability measure on $M$. Assume $f$ is average conformal expanding/hyperbolic…