Let Cn be the origin-containing cluster in subcritical percolation on the lattice 1 nZ , viewed as a random variable in the space Ω of compact, connected, origin-containing subsets of R, endowed with the Hausdorff metric δ. When d ≥ 2, and Γ is any open subset of Ω, we prove that lim n→∞ 1 n logP (Cn ∈ Γ) = − inf S∈Γ λ(S) where λ(S) is the one-dimensional Hausdorff measure of S defined using th...

1. I n t r o d u c t i o n . In [4], De Groot and Aarts constructed Hausdorff compactifications of topological spaces to obtain a new intrinsic characterization of complete regularity. These compactifications were called GA compactifications in [5] and [7]. A characterization of complete regularity was earlier given by Fr ink [3], by means of Wallman compactifications, a method which led to the...

We develop a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In the one dimensional case that we consider here, our methods require only C3 regularity of the maps in the IFS. The key idea, which has been known in varying degrees of generality for many years, is to associate to the IFS a parametrized family of positive, linea...

We construct a geometric model for the mapping class group M(S) of a non-exceptional oriented surface S of genus g with m punctures and use it to show that the action of M(S) on the compact Hausdorff space of complete geodesic laminations for S is topologically amenable. As a consequence, the Novikov higher signature conjecture holds for every subgroup of

A new type of cluster sets, called S-cluster sets, of functions and multifunctions between topological spaces is introduced, thereby providing a new technique for studying S-closed spaces. The deliberation includes an explicit expression of S-cluster set of a function. As an application, characterizations of Hausdorff and S-closed topological spaces are achieved via such cluster sets.

In this paper, we introduce a new entropy-like invariant, named Hausdorff metric entropy, for finitely generated semigroups acting on compact metric spaces from a set-valued view and study its properties. We establish the relation between Hausdorff metric entropy and topological entropy of a semigroup defined by Bis. Some examples with positive or zero Hausdorff metric entropy are given. Moreov...

For a locally compact Hausdorff semigroup S, the L representation algebra R(S) was extensively studied by Dunkl and Ramirez. The FourierStieltjes algebra F (S) of a topological semigroup was studied by Lau. The aim of this paper is to investigate these two algebras and study the amenability of them with respect to the structure of S.

This paper is dedicated to the regularity of optimal sets for second eigenvalue Dirichlet Laplacian. Precisely, we prove that if set $\Omega$ minimizes functional \[ \mathcal F_\Lambda(\Omega)=\lambda_2(\Omega)+\Lambda |\Omega|, \] among all subsets a smooth bounded open $D\subset \mathbb{R}^d$, where $\lambda_2(\Omega)$ Laplacian on and $\Lambda>0$ fixed constant, then equivalent union two dis...

We develop a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In the one dimensional case, our methods require only C3 regularity of the maps in the IFS. The key idea, which has been known in varying degrees of generality for many years, is to associate to the IFS a parametrized family of positive, linear, Perron-Frobenius op...