Quantization dimension for Gibbs-like measures on cookie-cutter sets

TL;DR

This paper determines the quantization dimension for Gibbs-like measures on cookie-cutter sets, linking it to thermodynamic formalism and multifractal analysis, and proves positivity of the lower quantization coefficient.

Contribution

It introduces a method to compute the quantization dimension for Gibbs-like measures on cookie-cutter sets and relates it to the temperature function in thermodynamic formalism.

Findings

Existence of a unique quantization dimension function $\,\kappa_r$ for the measure.

Functional relationship between $\,\kappa_r$ and the temperature function.

Positivity of the $\,\kappa_r$-dimensional lower quantization coefficient.

Abstract

In this paper using Banach limit we have determined a Gibbs-like measure $\mu_h$ supported by a cookie-cutter set $E$ which is generated by a single cookie-cutter mapping $f$. For such a measure $\mu_h$ and $r\in (0, +\infty)$ we have shown that there exists a unique $\kappa_r \in (0, +\infty)$ such that $\kappa_r$ is the quantization dimension function of the probability measure $\mu_h$, and established its functional relationship with the temperature function of the thermodynamic formalism. The temperature function is commonly used to perform the multifractal analysis, in our context of the measure $\mu_h$. In addition, we have proved that the $\kappa_r$-dimensional lower quantization coefficient of order $r$ of the probability measure is positive.