Speaker
Shunsuke Usuki, Keio University
Abstract
In 2019, Pablo Shmerkin resolved Furstenberg’s \(\times 2,\times 3\) intersection conjecture (simultaneously with Meng Wu), by introducing the concept of dynamically-driven self-similar measures and studying their \(L^q\) dimensions. In this talk, we introduce Markov-driven self-similar measures which generalize this concept. By analyzing their \(L^q\) dimensions, we tackle slicing and dimension problems related to non-homogeneous self-similar IFSs which were previously out of reach.
A broad class of fractal measures, including self-similar measures, self-conformal measures and their products, projections and convolutions, can be represented as Markov-driven self-similar measures, allowing us to study their \(L^q\) dimensions within the framework of Markov models. In particular, under the strong exponential separation condition and a suitable Diophantine condition, we obtain the expected formulas for the \(L^q\) dimensions of convolutions of two non-homogeneous self-similar measures on the line, non-homogeneous planar self-similar measures and their projections. As a corollary, we establish the Furstenberg-type slicing results for two non-homogeneous self-similar sets on the line satisfying these separation and Diophantine conditions.
This is joint work with Pablo Shmerkin.