Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's

Kleinian群中的发散型半群与Hausdorff维度

Divergence-type semigroups in Kleinian groups and Hausdorff dimension

本文研究了Kleinian群中亚线性锥形Myrberg极限集的Hausdorff维度，并通过构造发散型子半群，给出了临界指数与该维度相等的另一种证明。

Exponential growth of norms in semigroups of linear automorphisms and Hausdorff dimension of self-projective IFS

Given a finitely generated semigroup S of the (normed) set of linear maps of a vector space V into itself, we find sufficient conditions for the exponential growth of the number N(k) of elements of the semigroup contained in the sphere of radius k as k->infinity. We relate the growth rate lim log N(k)/log k to the exponent of a zeta function naturally defined on the semigroup and, in case S is a semigroup of volume-preserving automorpisms, to the Hausdorff and box dimensions of the limit set of the induced semigroup of automorphisms on the corresponding projective space.

Some examples of minimal Cantor set for IFSs with overlap

We give some examples of IFSs with overlap on the interval such that the semigroup action they give rise to has a minimal set homeomorphic to the Cantor set.

低维中的吸引子集与Julia集

Attractor sets and Julia sets in low dimensions

本文证明二维或三维共形IFS的吸引子集可表示为拟正则半群的Julia集，并在二维附加条件下由单元素生成，进而与有理映射Julia集拟共形等价。

On the fine structure of stationary measures in systems which contract-on-average

Suppose $\{f_1,...,f_m\}$ is a set of Lipschitz maps of $\mathbb{R}^d$. We form the iterated function system (IFS) by independently choosing the maps so that the map $f_i$ is chosen with probability $p_i$ ($\sum_{i=1}^m p_i=1$). We assume that the IFS contracts on average. We give an upper bound for the Hausdorff dimension of the invariant measure induced on $\mathbb{R}^d$ and as a corollary show that the measure will be singular if the modulus of the entropy $\sum_i p_i \log p_i$ is less than $d$ times the modulus of the Lyapunov exponent of the system. Using a version of Shannon's Theorem fo…