A class of subsets of ℝⁿ is constructed that have certain homogeneity and non-coincidence properties with respect to Hausdorff and box dimensions. For each triple (r,s,t) of numbers in the interval (0,n] with r < s < t, a compact set K is constructed so that for any non-empty subset U relatively open in K, we have $(di{m}_H\left(U\right),{\underline{dim}}_B\left(U\right),{\overline{dim}}_B\left(U\right))=(r,s,t)$. Moreover, $2^{-n}\le H^r\left(K\right)\le 2n^{r/2}$.

We describe the homotopy classes of self-homeomorphisms of solenoids and Knaster continua. In particular, we demonstrate that homeomorphisms within one homotopy class have the same (explicitly given) topological entropy and that they are actually semi-conjugate to an algebraic homeomorphism in the case when the entropy is positive.

We present a very quick and easy proof of the classical Stepanov-Hopf ratio ergodic theorem, deriving it from Birkhoff's ergodic theorem by a simple inducing argument.

We show that almost every function (in the sense of prevalence) in a Sobolev space is multifractal: Its regularity changes from point to point; the sets of points with a given Hölder regularity are fractal sets, and we determine their Hausdorff dimension.

Subadditivity of capacities is defined initially on the compact sets and need not extend to all sets. This paper explores to what extent subadditivity holds. It presents some incidental results that are valid for all subadditive capacities. The main result states that for all hull-additive capacities (a class that contains the strongly subadditive capacities) there is countable subadditivity on a class at least as large as the universally measurable sets (so larger than the analytic sets).

In this paper we introduce the $ℐ$- and ${ℐ}^{*}$-convergence and divergence of nets in $\left(\ell \right)$-groups. We prove some theorems relating different types of convergence/divergence for nets in $\left(\ell \right)$-group setting, in relation with ideals. We consider both order and $\left(D\right)$-convergence. By using basic properties of order sequences, some fundamental properties, Cauchy-type characterizations and comparison results are derived. We prove that ${ℐ}^{*}$-convergence/divergence implies $ℐ$-convergence/divergence for every ideal, admissible for...

We prove that for every Borel ideal, the ideal limits of sequences of continuous functions on a Polish space are of Baire class one if and only if the ideal does not contain a copy of Fin × Fin. In particular, this is true for $F_{\sigma \delta }$ ideals. In the proof we use Borel determinacy for a game introduced by C. Laflamme.

We find an analytic formulation of the notion of Hopf image, in terms of the associated idempotent state. More precisely, if π:A → Mₙ(ℂ) is a finite-dimensional representation of a Hopf C*-algebra, we prove that the idempotent state associated to its Hopf image A' must be the convolution Cesàro limit of the linear functional φ = tr ∘ π. We then discuss some consequences of this result, notably to inner linearity questions.

We consider an important subclass of self-similar, non-gaussian stable processes with stationary increments known as self-similar stable mixed moving averages. As previously shown by the authors, following the seminal approach of Jan Rosiński, these processes can be related to nonsingular flows through their minimal representations. Different types of flows give rise to different classes of self-similar mixed moving averages, and to corresponding general decompositions of these processes. Self-similar...

For any 1-1 measure preserving map T of a probability space we can form the [T,Id] and $[T,T^{-1}]$ automorphisms as well as the corresponding endomorphisms and decreasing sequence of σ-algebras. In this paper we show that if T has zero entropy and the [T,Id] automorphism is isomorphic to a Bernoulli shift then the decreasing sequence of σ-algebras generated by the [T,Id] endomorphism is standard. We also show that if T has zero entropy and the [T²,Id] automorphism is isomorphic to a Bernoulli shift then the...

Let $X=X\left(t\right),t\in {ℝ}^N$ be a Gaussian random field in ${ℝ}^d$ with stationary increments. For any Borel set $E\subset {ℝ}^N$, we provide sufficient conditions for the image X(E) to be a Salem set or to have interior points by studying the asymptotic properties of the Fourier transform of the occupation measure of X and the continuity of the local times of X on E, respectively. Our results extend and improve the previous theorems of Pitt [24] and Kahane [12,13] for fractional Brownian motion.