Quantization Dimension for Infinite Conformal Iterated Function Systems

Quantization Dimension for Conformal Iterated Function Systems

On the Geometric Structure of the Limit Set of Conformal Iterated Function Systems

We consider infinite conformal function systems on Rd. We study the geometric structure of the limit set of such systems. Suppose this limit set intersects some l-dimensional C1-submanifold with positive Hausdorff t-dimensional measure, where 0 < l < d and t is the Hausdorff dimension of the limit set. We then show that the closure of the limit set belongs to some l-dimensional affine subspace ...

Finer Geometric Rigidity of Limit Sets of Conformal Ifs

We consider infinite conformal iterated function systems in the phase space Rd with d ≥ 3. Let J be the limit set of such a system. Under a mild technical assumption, which is always satisfied if the system is finite, we prove that either the Hausdorff dimension of J exceeds the topological dimension k of the closure of J or else the closure of J is a proper compact subset of either a geometric...

Graph Directed Markov Systems on Hilbert Spaces

We deal with contracting finite and countably infinite iterated function systems acting on Polish spaces, and we introduce conformal Graph Directed Markov Systems on Polish spaces. Sufficient conditions are provided for the closure of limit sets to be compact, connected, or locally connected. Conformal measures, topological pressure, and Bowen’s formula (determining the Hausdorff dimension of l...

A remark on densities of hyperbolic dimensions for conformal iterated function systems with applications to conformal dynamics and fractal number theory

In this note we investigate radial limit sets of arbitrary regular conformal interated function systems. We show that for each of these systems there exists a variety of finite hyperbolic subsystems such that the spectrum made of the Hausdorff dimensions of the limit sets of these subsystems is dense in the interval between 0 and the Hausdorff dimension of the given conformal iterated function ...