For a one-to-one self-conformal contractive system ${w_j}_{j=1}^m$ on ${ℝ}^d$ with attractor K and conformality dimension α, Peres et al. showed that the open set condition and strong open set condition are both equivalent to $0<{\mathscr{H}}^{\alpha }\left(K\right)<\infty$. We give a simple proof of this result as well as discuss some further properties related to the separation condition.

In this paper we explore topological factors in between the Kronecker factor and the maximal equicontinuous factor of a system. For this purpose we introduce the concept of sequence entropy $n$-tuple for a measure and we show that the set of sequence entropy tuples for a measure is contained in the set of topological sequence entropy tuples [H- Y]. The reciprocal is not true. In addition, following topological ideas in [BHM], we introduce a weak notion and a strong notion of complexity pair for a...

Let $X$ be a locally convex space, $m:\Sigma \to X$ be a vector measure defined on a $\sigma$-algebra $\Sigma$, and $L^1\left(m\right)$ be the associated (locally convex) space of $m$-integrable functions. Let $\Sigma \left(m\right)$ denote $\{{\chi }_{{}_E};E\in \Sigma \}$, equipped with the relative topology from $L^1\left(m\right)$. For a subalgebra $𝒜\subseteq \Sigma$, let ${𝒜}_{\sigma }$ denote the generated $\sigma$-algebra and ${\overline{𝒜}}_s$ denote the sequential closure of $\chi \left(𝒜\right)=\{{\chi }_{{}_E};E\in 𝒜\}$ in $L^1\left(m\right)$. Sets of the form ${\overline{𝒜}}_s$ arise in criteria determining separability of $L^1\left(m\right)$; see [6]. We consider some natural questions concerning ${\overline{𝒜}}_s$ and, in particular, its relation to $\chi \left({𝒜}_{\sigma }\right)$. It is shown that...

The extension theorem of bounded, weakly compact, convex set valued and weakly countably additive measures is established through a discussion of convexity, compactness and existence of selection of the set valued measures; meanwhile, a characterization is obtained for continuous, weakly compact and convex set valued measures which can be represented by Pettis-Aumann-type integral.

We show that there exists a closed non-$\sigma$-porous set of extended uniqueness. We also give a new proof of Lyons’ theorem, which shows that the class of $H^{\left(n\right)}$-sets is not large in $U_0$.

We study differentiability of topological conjugacies between expanding piecewise $C^{1+ϵ}$ interval maps. If these conjugacies are not C¹, then their derivative vanishes Lebesgue almost everywhere. We show that in this case the Hausdorff dimension of the set of points for which the derivative of the conjugacy does not exist lies strictly between zero and one. Moreover, by employing the thermodynamic formalism, we show that this Hausdorff dimension can be determined explicitly in terms of the Lyapunov spectrum....

We study natural measures on sets of $\beta$-expansions and on slices through self similar sets. In the setting of $\beta$-expansions, these allow us to better understand the measure of maximal entropy for the random $\beta$-transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading...

A Borel subset of the unit square whose vertical and horizontal sections are two-point sets admits a natural group action. We exploit this to discuss some questions about Borel subsets of the unit square on which every function is a sum of functions of the coordinates. Connection with probability measures with prescribed marginals and some function algebra questions is discussed.

We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors). The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect...