arXiv subjects
Publications and source records attributed to Tom Kempton.
We describe a family $ϕ_λ$ of dynamical systems on the unit interval which preserve Bernoulli convolutions. We show that if there are parameter ranges for which these systems are piecewise convex, then the corresponding Bernoulli convolution will be absolutely continuous with bounded density. We study the systems $ϕ_λ$ and give some numerical evidence to suggest values of $λ$ for which $ϕ_λ$ may be piecewise convex.
We study natural measures on sets of beta-expansions and on slices through self similar sets. In the setting of beta-expansions, these allow us to better understand the measure of maximal entropy for the random beta-transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading to conditions under which almost every slice through a self similar set has positive Hausdorff measure, generalising long known results about almost everywhere values of the Hausdorff dimension.
We construct a Lebesgue measure preserving natural extension of the random beta-transformation. This allows us to give a formula for the density of the absolutely continuous invariant probability measure, answering a question of Dajani and de Vries, and also to evaluate some estimates on the typical branching rate of the set of beta-expansions of a real number.
It is well known that the Bernoulli convolution $ν_β$ associated to the golden mean has Hausdorff dimension less than 1, i.e. that there exists a set $A$ with $ν_β(A)=1$ and $dim_H(A)<1$. We construct such a set $A$ explicitly and discuss how our approach might be generalised to prove the singularity of other Bernoulli convolutions
We study the typical growth rate of the number of words of length n which can be extended to beta-expansions of x. In the general case we give a lower bound for the growth rate, while in the case that the Bernoulli convolution associated to parameter beta is absolutely continuous we are able to give the growth rate precisely. This gives new necessary and sufficient conditions for the absolute continuity of Bernoulli convolutions.