Related papers: Self-similar sets, simple augmented trees, and the…
The topological and metrical equivalence of fractals is an important topic in analysis. In this paper, we use a class of finite state automata, called $\Sigma$-automaton, to construct psuedo-metric spaces, and then apply them to the study…
A densely-defined symmetric linear map from/to a real Hilbert space extends to a self-adjoint map. Extension is expressed via Riesz representation. For a case including Friedrichs extension of a strongly monotone map, self-adjoint extension…
Hyperbolicity is a property of a graph that may be viewed as being a "soft" version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic…
The present paper investigates the limit $G$-space $\mathcal{J}_{G}$ generated by the self-similar action of automatic groups on a regular rooted tree. The limit space $\mathcal{J}_{G}$ is the Gromov-Hausdorff limit of the family of…
The Gromov--Hausdorff distance measures the difference in shape between compact metric spaces. While even approximating the distance up to any practical factor poses an NP-hard problem, its relaxations have proven useful for the problems in…
We consider iterated function systems $\mathrm{IFS}(T_1,\dots,T_k)$ consisting of continuous self maps of a compact metric space $X$. We introduce the subset $S_{\mathrm{t}}$ of {\emph{weakly hyperbolic sequences}} $\xi=\xi_0\ldots\xi_n…
We unify and advance a host of works on iterated function systems of holomorphic self-maps of hyperbolic Riemann surfaces. Our foremost result is a generalisation to left iterated function systems of an unpublished and little known theorem…
Let $\Sigma$ be a hypersurface in an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We study the isometric extension problem for isometric immersions $f:\Sigma\to\mathbb R^n$, where $\mathbb R^n$ is equipped with the Euclidean…
By introducing new deformations on symbolic Cantor sets and ultrametric spaces, we prove that doubling ultrametric spaces admit bilipschitz embedding into Cantor sets. If in addition the spaces are uniformly perfect, we show that they are…
Given a finite collection $\{X_i\}_{i\in I}$ of metric spaces, each of which has finite Nagata dimension and Lipschitz free space isomorphic to $L^1$, we prove that their union has Lipschitz free space isomorphic to $L^1$. The short proof…
For any intrinsic Gromov hyperbolic space we establish a Gehring-Hayman type theorem for conformally deformed spaces. As an application, we prove that any complete intrinsic hyperbolic space with atleast two points in the Gromov boundary…
We develop a formalism for relative Gromov-Witten invariants of Li that is analogous to the Symplectic Field Theory of Eliashberg, Givental, and Hofer. This formalism allows us to express natural degeneration formulae in terms of generating…
We introduce the tree distance, a new distance measure on graphs. The tree distance can be computed in polynomial time with standard methods from convex optimization. It is based on the notion of fractional isomorphism, a characterization…
Gromov hyperbolicity of a metric space measures the distance of the space from a perfect tree-like structure. The measure has a "worst-case" aspect to it, in the sense that it detects a region in the space which sees the maximum deviation…
Partially motivated by the study of I. Binder, N. Makarov, and S. Smirnov [BMS03] on dimension spectra of polynomial Cantor sets, we initiate the investigation on some general harmonic measures, inspired by Sullivan's dictionary, for…
We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk…
Parabolic (resp. hyperbolic) self-embeddings of trees are those which do not fix a non-empty finite subtree and preserve precisely one (resp. two) end(s). We prove that a locally finite tree having a parabolic self-embedding is mutually…
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…
Commability is the finest equivalence relation between locally compact groups such that $G$ and $H$ are equivalent whenever there is a continuous proper homomorphism $G \to H$ with cocompact image. Answering a question of Cornulier, we show…
This is an expository article on visual metrics on boundaries of hyperbolic metric spaces. We discuss the construction of visual metrics, quasisymmetries and their invariants, Hausdorff and conformal dimension, and constructions and…