Local dimensions for the random beta-transformation

TL;DR

This paper investigates the local dimensions of the invariant measure for the random beta-transformation, linking it to Bernoulli convolutions and providing insights into their fractal structure for specific beta values.

Contribution

It establishes a connection between the local dimensions of the invariant measure of the random beta-transformation and Bernoulli convolutions, offering new results for special beta values.

Findings

Determines local dimensions for specific beta values.

Links invariant measure properties to Bernoulli convolutions.

Provides new insights into fractal structure of measures.

Abstract

The random beta-transformation K is isomorphic to a full shift. This relation gives an invariant measure for K that yields the Bernoulli convolution by projection. We study the local dimension of the invariant measure for K for special values of beta and use the projection to obtain results on the local dimension of the Bernoulli convolution.