Geometry of measures in random systems with complete connections

TL;DR

This paper investigates the geometric properties and dimensions of measures in complex random systems, including countable IFS with overlaps and Smale endomorphisms, establishing exact dimensionality and calculating Hausdorff dimensions.

Contribution

It introduces new results on the exact dimensionality and Hausdorff dimension of stationary measures in countable conformal IFS with overlaps and explores measure families on fractals within these systems.

Findings

Stationary measures are proven to be exact dimensional.

Hausdorff dimensions of these measures are explicitly determined.

Construction of measure families on subfractals related to system geometry.

Abstract

We study new relations between countable iterated function systems (IFS) with overlaps, Smale endomorphisms and random systems with complete connections. We prove that stationary measures for countable conformal IFS with overlaps and placedependent probabilities, are exact dimensional; moreover we determine their Hausdorff dimension. Next, we construct a family of fractals in the limit set of a countable IFS with overlaps S, and study the dimension for certain measures supported on these subfractals. In particular, we obtain families of measures on these subfractals which are related to the geometry of the system.