Slicing the Torus and the thermodynamics of self-similar measures with overlaps

TL;DR

This paper explores the connection between the dimension theory of self-similar measures with overlaps, like projections of the Sierpinski triangle, and thermodynamic formalism on the torus, revealing new insights into their structure.

Contribution

It establishes a novel link between the dimension theory of overlapping self-similar measures and thermodynamic formalism on the torus, using rational slices and translational parameters.

Findings

Connected the dimension theory of self-similar measures with thermodynamic formalism.

Established a correspondence between translational parameters and rational slices of the torus.

Provided a new perspective for understanding projections of self-similar measures.

Abstract

Orthogonal projections of the uniform measure on the Sierpinski triangle form a family of self similar measures with overlaps. The main result of this work is to make a connection between the dimension theory of these measures and the thermodynamic formalism of the doubling map restricted to rational slices of the torus. Of note is how we establish a correspondence between the varying translational parameter and varying rational slices. This gives a new direction from which to understand the dimension theory of projections of self similar measures.