Invariant Jordan curves of Sierpiski carpet rational maps

In this paper, we prove that if $R\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ is a postcritically finite rational map with Julia set homeomorphic to the Sierpi\'nski carpet, then there is an integer $n_0$, such that, for any $n\ge n_0$, there exists an $R^n$-invariant Jordan curve $\Gamma$ containing the postcritical set of $R$.Comment: 16 pages, 1 figu

Construction of pp-energy and associated energy measures on the Sierpi\'{n}ski carpet

We establish the existence of a scaling limit $\mathcal{E}_p$ of discrete $p$-energies on the graphs approximating the planar Sierpi\'{n}ski carpet for $p > \dim_{\text{ARC}}(\textsf{SC})$, where $\dim_{\text{ARC}}(\textsf{SC})$ is the Ahlfors regular conformal dimension of the Sierpi\'{n}ski carpet. Furthermore, the function space $\mathcal{F}_{p}$ defined as the collection of functions with finite $p$-energies is shown to be a reflexive and separable Banach space that is dense in the set of continuous functions with respect to the supremum norm. In particular, $(\mathcal{E}_2, \mathcal{F}_2)$ recovers the canonical regular Dirichlet form constructed by Barlow and Bass or Kusuoka and Zhou. We also provide $\mathcal{E}_{p}$-energy measures associated with the constructed $p$-energy and investigate its basic properties like self-similarity and chain rule.Comment: 80 pages, 9 figures; fixed several typos and some mistakes, edited some proofs. Sections 4 and 5 from the previous version have been merged into one section. Theorem 2.20 and subsection 6.3 are ne

Beurling slow and regular variation

We give a new theory of Beurling regular variation ( Part II). This includes the previously known theory of Beurling slow variation ( Part I) to which we contribute by extending Bloom's theorem. Beurling slow variation arose in the classical theory of Karamata slow and regular variation. We show that the Beurling theory includes the Karamata theory

Beurling slow and regular variation

We give a new theory of Beurling regular variation ( Part II). This includes the previously known theory of Beurling slow variation ( Part I) to which we contribute by extending Bloom's theorem. Beurling slow variation arose in the classical theory of Karamata slow and regular variation. We show that the Beurling theory includes the Karamata theory

Contracting boundaries of amalgamated free products of CAT(0) groups with applications for right-angled Coxeter groups

In this thesis, we study topological spaces associated to CAT(0) groups called contracting boundaries. We examine contracting boundaries of amalgamated free products of CAT(0) groups and focus on situations where totally disconnected contracting boundaries are involved. We use our insights to investigate which right-angled Coxeter groups have totally disconnected contracting boundaries