Thermodynamic Formalism and Dimension of Overlapping Fractal Measures

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This project will study the dimension theory of self-similar measures with overlaps. In particular, we will study dimension drop in simple classes of overlapping self-similar measures. This is perhaps the biggest open problem in fractal geometry, and is one in which there has been substantial recent progress (most notably by Hochman, Shmerkin, Varju and Breuillard). Much of this recent progress has been made through ergodic theory. This project will continue to work at the interface of fractal geometry and ergodic theory.The student will focus on the case of self similar sets with three or more contractions, in which the contraction rate is 1/2 (projections of the Sierpinski gasket) or a reciprocal of a Pisot number. The initial goal is to link the question to problems in thermodynamic formalism involving the doubling map on the torus. The novelty of the project will lie both in making new connections between fractal geometry and thermodynamic formalism, and in solving new problems in thermodynamic formalism.