A general theory of self-similarity II: recognition

General self-similarity: an overview

Informal seminar notes explaining the ideas in math.DS/0411344 and math.DS/0411345.

The Similarity Boundary of a Self-Similar Set

We define the similarity boundary of a self-similar set and use it to analyze the properties of self-similar sets in the general setting of any complete metric space. The similarity boundary is an attempt at extending the concept of the topological boundary in a way that is consistent with our intuitive understanding of the term boundary. We also show how with the analysis of the similarity boundary, we can restrict ourselves to the space $K$ and disregard the ambient space.

气泡的自相似性

Self-similarity of bubbles

本文研究与圆周微分同胚相关的分形集气泡（Arnold舌的复类似物），证明了其近似自相似性。

On the self-similarity problem for smooth flows on orientable surfaces

On each compact, connected, orientable surface of genus greater than one we construct a class of flows without self-similarities.

Graphs, groups and self-similarity

We study a family of C*-algebras generalizing both Katsura algebras and certain algebras introduced by Nekrashevych in terms of self-similar groups.

On the self-similarity problem for Gaussian-Kronecker flows

It is shown that a countable symmetric multiplicative subgroup $G=-H\cup H$ with $H\subset\mathbb{R}_+^\ast$ is the group of self-similarities of a Gaussian-Kronecker flow if and only if $H$ is additively $\mathbb{Q}$-independent. In particular, a real number $s\neq\pm1$ is a scale of self-similarity of a Gaussian-Kronecker flow if and only if $s$ is transcendental. We also show that each countable symmetric subgroup of $\mathbb{R}^\ast$ can be realized as the group of self-similarities of a simple spectrum Gaussian flow having the Foias-Stratila property.