Let $T:ℝ/\mathbb{Z}\to ℝ/\mathbb{Z}$ be a von Neumann-Kakutani $q$- adic adding machine transformation and let $\varphi \in C^1\left(\right[0,1\left]\right)$. Put$${\varphi }_n\left(x\right):=\varphi \left(x\right)+\varphi \left(Tx\right)+...+\varphi (T^{n-1}x),x\in ℝ/\mathbb{Z},n\in \mathbb{N}.$$We study three questions:1. When will $\left({\varphi }_n\right(x){)}_{n\ge 1}$ be bounded?2. What can be said about limit points of $\left({\varphi }_n\right(x){)}_{n\ge 1}?$3. When will the skew product $(x,y)\mapsto (Tx,y+\varphi (x\left)\right)$ be ergodic on $ℝ/\mathbb{Z}\times ℝ?$

Compact group extensions of 2-fold simple actions of locally compact second countable amenable groups are considered. It is shown what the elements of the centralizer of such a system look like. It is also proved that each factor of such a system is determined by a compact subgroup in the centralizer of a normal factor.

We prove that in Polish, abelian, non-locally-compact groups the family of Haar null sets of Christensen does not fulfil the countable chain condition, that is, there exists an uncountable family of pairwise disjoint universally measurable sets which are not Haar null. (Dougherty, answering an old question of Christensen, showed earlier that this was the case for some Polish, abelian, non-locally-compact groups.) Thus we obtain the following characterization of locally compact, abelian groups: Let...

In this note we show the characteristic function of every indecomposable set F in the plane is BVequivalent to the characteristic function a closed set See Formula in PDF See Formula in PDF . We show by example this is false in dimension three and above. As a corollary to this result we show that for everyϵ > 0 a set of finite perimeter Scan be approximated by a closed subset See Formula in PDF See Formula in PDF with finitely many indecomposable components and with the property that See...

 Let $F_i=1,...,N$ be affine mappings of ${ℝ}^n$. It is well known that if there exists j ≤ 1 such that for every ${\sigma }_1,...,{\sigma }_j\in 1,...,N$ the composition (1) $F_{\sigma 1}\circ ...\circ F_{{\sigma }_j}$ is a contraction, then for any infinite sequence ${\sigma }_1,{\sigma }_2,...\in 1,...,N$ and any $z\in {ℝ}^n$, the sequence (2)$F_{\sigma 1}\circ ...\circ F_{{\sigma }_n}\left(z\right)$ is convergent and the limit is independent of z. We prove the following converse result: If (2) is convergent for any $z\in {ℝ}^n$ and any $\sigma ={\sigma }_1,{\sigma }_2,...$ belonging to some subshift Σ of N symbols (and the limit is independent of z), then there exists j ≥ 1 such that for every $\sigma ={\sigma }_1,{\sigma }_2,...\in \Sigma$ the composition (1) is a contraction. This result...

Let $S_i:{ℝ}^d\to {ℝ}^d$ for i = 1,..., N be contracting similarities, let $(p₁,...,p_N,p)$ be a probability vector and let ν be a probability measure on ${ℝ}^d$ with compact support. It is well known that there exists a unique inhomogeneous self-similar probability measure μ on ${ℝ}^d$ such that $\mu ={\sum }_{i=1}^N{p}_i\mu \circ S_i^{-1}+p\nu$. We give satisfactory estimates for the lower and upper bounds of the $L^q$ spectra of inhomogeneous self-similar measures. The case in which there are a countable number of contracting similarities and probabilities is considered. In particular, we...