Quantization dimension for self-similar measures of overlapping construction

TL;DR

This paper investigates the quantization dimension of self-similar measures with overlaps, specifically those satisfying the Weak Separation Property, and relates it to thermodynamic formalism.

Contribution

It extends the computation of quantization dimension to overlapping self-similar measures satisfying WSP, beyond the well-separated case under OSC.

Findings

Quantization dimension is computed for overlapping self-similar measures.

The quantization dimension relates to the temperature function of thermodynamic formalism.

The work bridges fractal measure theory and thermodynamic formalism.

Abstract

Quantization dimension has been computed for many invariant measures of dynamically defined fractals having well separated cylinders, that is, in the cases when the so-called Open Set Condition (OSC) holds. To attack the same problem in case of heavy overlaps between the cylinders, we consider a family of self-similar measures, for which the underlying Iterated Function System satisfies the so-called Weak Separation Property (WSP) but does not satisfy the OSC since complete overlaps occur in between the cylinders. The work in this paper also shows that the quantization dimension determined for the set of overlap self-similar construction satisfying the WSP has a relationship with the temperature function of the thermodynamic formalism.