Please Note: This seminar will be given online.
Probability Seminar Series
Pieter Allaart, Professor
University of North Texas
Link to join seminar: Hosted on Webex.
On univoque and strongly univoque sets
For a number beta in the interval (1,2), the univoque set U_beta is the set of numbers which have exactly one expansion in base beta. It has been well studied in the literature, and can be viewed as the set of points whose orbits avoid the hole in an open dynamical system. In 2011, Jordan, Shmerkin and Solomyak introduced a subset U_\beta' of the univoque set which we'll call the strongly univoque set, and used it to study the multifractal spectrum of Bernoulli convolutions. In this talk we will see another, rather unexpected, application of the strongly univoque set, to the infinite derivatives of Okamoto's self-affine function. We show that U_beta and U_beta' have the same Hausdorff dimension, and characterize the Hausdorff dimension of the difference U_beta\U_beta'.