相关论文: Conformal Fractals for Normal Subgroups of Free Gr…

We find new bounds on the conformal dimension of small cancellation groups. These are used to show that a random few relator group has conformal dimension 2+o(1) asymptotically almost surely (a.a.s.). In fact, if the number of relators…

Michael Barnsley introduced a family of fractals sets which are repellers of piecewise affine systems. The study of these fractals was motivated by certain problems that arose in fractal image compression but the results we obtained can be…

We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups proved by Green, Tao and the first author. We also give a…

In this paper, we develop techniques to study the Hausdorff dimensions of non-conical and Myrberg limit sets for groups acting on negatively curved spaces. We establish maximality of the Hausdorff dimension of the non-conical limit set of…

We derive, from conformal invariance and quantum gravity, the multifractal spectrum f(alpha,c) of the harmonic measure (or electrostatic potential, or diffusion field) near any conformally invariant fractal in two dimensions, corresponding…

We construct functions $f \colon [0,1] \to [0,1]$ whose graph as a subset of $\mathbb{R}^2$ has Hausdorff dimension greater than any given value $\alpha \in (1,2)$ but conformal dimension $1$. These functions have the property that a…

Let $\Phi = \{\phi_e\}_{e\in E}$ be a finitely irreducible conformal graph directed Markov system (CGDMS) with symbolic representation $E_A^{\infty}$ and limit set $J$. Under a mild condition on the system, we give a multifractal analysis…

I show that $L^{p}-L^{q}$ estimates for the Kakeya maximal function yield lower bounds for the conformal dimension of Kakeya sets, and upper bounds for how much quasisymmetries can increase the Hausdorff dimension of line segments inside…

In this paper we introduce and develop the theory of non-autonomous graph directed Markov systems which is a generalization of the theory of conformal graph directed Markov systems of Mauldin and Urba\'nski, first presented in their book,…

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of…

This paper provides a new model to compute the fractal dimension of a subset on a generalized-fractal space. Recall that fractal structures are a perfect place where a new definition of fractal dimension can be given, so we perform a…

Given a fractal $\mathcal{I}$ whose Hausdorff dimension matches with the upper-box dimension, we propose a new method which consists in selecting inside $\mathcal{I}$ some subsets (called quasi-Cantor sets) of almost same dimension and with…

Certain subsets of limit sets of geometrically finite Fuchsian groups with parabolic elements are considered. It is known that Jarn\'{\i}k limit sets determine a "weak multifractal spectrum" of the Patterson measure in this situation. This…

We consider a generalisation of the self-affine iterated function systems of Lalley and Gatzouras by allowing for a countable infinity of non-conformal contractions. It is shown that the Hausdorff dimension of the limit set is equal to the…

In 2004, Bishop proved that for Kleinian groups acting on hyperbolic space, the Hausdorff dimension of the limit set is completely determined by two extremal dynamical behaviors: recurrent geodesics and geodesics escaping linearly to…

We study the behavior of Sobolev mappings defined on the Heisenberg groups with respect to a foliation by left cosets of a horizontal homogeneous subgroup. We quantitatively estimate, in terms of Euclidean Hausdorff dimension, the size of…

The main result of this paper is to show that if $\H$ is a normal subgroup of a Kleinian group $G$ such that $G/\H$ contains a coset which is represented by some loxodromic element, then the Hausdorff dimension of the transient limit set of…

In this article we provide simple and provable bounds on the size and shape of the locus of discrete subgroups of $\mathsf{PSL}(2,\mathbb{C})\cong \operatorname{Isom}^+(\mathbb{H}^3)$ which split as a free product of cyclic groups…

Let $\Gamma$ be a non-elementary, non-convex-cocompact Kleinian group acting on $\mathbb{H}^{d}$. We show that the Hausdorff dimension of the sublinearly conical Myrberg limit set of $\Gamma$ is equal to the critical exponent of $\Gamma$.…

This is the second part of the works on Hausdorff dimensions of Schottky groups. It has been conjectured that the Hausdorff dimensions of nonclassical Schottky groups are strictly bounded from below. In this second part of our works we…