A metric space S is called a quasisphere if there is a quasisymmetric homeomorphism f : S → S. We consider the elliptic harmonic measure, i.e., the push forward of 2-dimensional Lebesgue measure by f . It is shown that for certain self similar quasispheres S (snowspheres) the dimension of the elliptic harmonic measure is strictly less than the Hausdorff dimension of S. This result is obtained b...

The paper investigates fixed points and attractors of infinite iterated function systems in Cantor space. By means of the theory of formal languages simple examples of the non-coincidence of fixed point and attractor (closure of the fixed point) are given.

The existence of a weak survival region is established for the anisotropic symmetric contact process on a homogeneous tree T2d of degree 2d ≥ 4 : For parameter values in a certain connected region of positive Lebesgue measure, the population survives forever with positive probability but ultimately vacates every finite subset of the tree with probability one. In this phase, infection trails mus...

For d ∈ {1,2,3}, let (B t ; t ≥ 0) be a d-dimensional standard Brownian motion. We study the d-Brownian span set Span(d) := {t − s;Bd s = B t for some 0 ≤ s ≤ t}. We prove that almost surely the random set Span(d) is σ -compact and dense in R+. In addition, we show that Span(1) = R+ almost surely; the Lebesgue measure of Span(2) is 0 almost surely and its Hausdorff dimension is 1 almost surely;...

We derive universal Diophantine properties for the Patterson measure μG associated with a convex cocompact Kleinian group G acting on (n + 1) -dimensional hyperbolic space. We show that μG is always a S -friendly measure, for every (G, μG) neglectable set S , and deduce that if G is of non-Fuchsian type then μG is an absolutely friendly measure in the sense of [7]. Consequently, by a result of ...

We prove that limit-periodic Dirac operators generically have spectra of zero Lebesgue measure and a dense set them Hausdorff dimension. The proof combines ideas Avila from Schrödinger setting with new commutation argument for generating open spectral gaps. This overcomes an obstacle previously observed in the literature; namely, Schrödinger-type settings, translation corresponds to small L?-pe...

Let y = h(x) be defined for 0 < x < oo and assume values in 0 ^ y ^ + co. Let S be any linear set of points and p an arbitrary positive number. Cover S by a countable number of open intervals h , I2, • • • of lengths xx, x2 , • • • each of which is less than p, and denote by mp(S; h) the lower bound of h(xi) + h(x2) + • • • for all such coverings of S. Then m(S; h) = limp|0mp(/S; h) is called t...

We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a d-parameter family of such semigroups satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the Julia set is the minimum of 2 and the zero of the pressure function. Moreover, the Hausdorff dimension of the exceptional set of ...

In this note we characterize cr-finite Riesz measures that allow one to approximate measurable functions by continuous functions in the sense of Lusin's theorem. We call such measures Lusin measures and show that not all cr-finite measures are Lusin measures. It is shown that if a topological space X is either normal or countably paracompact, then every measure on A' is a Lusin measure. A count...