Work

On the Hausdorff Dimension of Bernoulli Convolutions

Year: 2018

Type: article

Abstract: We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution $\nu _{\beta }$ to arbitrary given accuracy whenever $\beta $ is algebraic. In particular, if the Garsia entropy $H(\beta )$ is not equal to $\log (\beta )$ then we have a finite time algorithm to determine whether or not $\operatorname{dim_H} (\nu _{\beta })=1$.

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rny209

Authors: Shigeki Akiyama, De-Jun Feng, Tom Kempton, Tomas Persson

Institutions: University of Tsukuba, Chinese University of Hong Kong, University of Manchester, Lund University

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Japan Society for the Promotion of Science

Awards: Relation between structure of spaces and analysis, Self-inducing structure of arithmetic algorithms