Slicing the Torus and the thermodynamics of self-similar measures with overlaps
Peej Ingarfield
TL;DR
This work investigates the dimension theory of projections of the Sierpinski triangle’s uniform measure along lines of rational slope by linking it to the thermodynamic formalism of the doubling map restricted to rational torus slices. The authors define the projected measures $\mu_{\theta}$ and prove a dimension drop $\dim(\mu_{\theta})<1$ for every rational angle $\theta=\tan^{-1}(p/q)$, establishing a bound in terms of a topological pressure $P(l_{pq},T|_{l_{pq}},\phi)$ via a $\phi$ that is independent of $p/q$. The approach blends Fourier-analytic and automata methods to show $\mu_{\theta}$ is singular yet equivalent to a dynamically invariant measure, thereby connecting the drop to the pressure along rational lines. A central contribution is the construction of a potential $\phi$ on $[0,1)^2$ and a corresponding weak Gibbs framework that yields a concrete, uniformly applicable bound on the dimension across all rational slices, offering a new perspective on overlaps in self-similar measures and a path toward generalizations to broader contraction families.
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.