Thomas Jordan

Dimension of ergodic measures projected onto self-similar sets

Joint work with Ariel Rapaport. Self-similar sets satisfying the open set condition are now extremely well understood. In particular the dimension of any ergodic measure projected from the shift space can be found as a ratio of entropy and Lyapunov exponent. The general case where this separation is not satisfied has recently seen significant process with the work of Hochman. In particular Hochman has shown that on the line for a system where the exponential separation condition is satisfied the dimension of a self-similar measure (projection of a Bernoulli measure from the shift space) is still given by the ratio of entropy and Lyapunov exponent. This exponential separation condition holds for a much wider range of sets than the open set condition (I will give examples to show this). We will show how Hochman’s results can be extended from self-similar measures to any projection of an ergodic measure. Our methods rely on an extension of Hochman’s results on self-similar measures by Shmerkin which gives sharp bounds on the Holder exponents of these measures.