Fractal dimensions of fractal transformations and Quantization dimensions for bi-Lipschitz mappings

TL;DR

This paper investigates the fractal and quantization dimensions of graphs of fractal transformations and measures supported on them, providing new estimates and characterizations under bi-Lipschitz conditions.

Contribution

It introduces new methods to determine fractal and quantization dimensions for graphs of fractal transformations and invariant measures under bi-Lipschitz mappings.

Findings

Determined the fractal dimension of the graph of a fractal transformation.

Established bounds for the quantization dimension of measures on these graphs.

Provided estimates for invariant measures under weighted iterated function systems.

Abstract

In this paper, we study the fractal dimension of the graph of a fractal transformation and also determine the quantization dimension of a probability measure supported on the graph of the fractal transformation. Moreover, we estimate the quantization dimension of the invariant measures corresponding to a weighted iterated function system consisting of bi-Lipschitz mappings under the strong open set condition.