Absolute-winning properties of equicontinuously-twisted badly approximable points in continued fractions and beta-transformations

Abstract: It is well know that in a $\beta$-transformation system for an integer $\beta>0$, the set ${x: \liminf_{n\to\infty}|T^nx-y_n|>0}$ has full Hausdorff dimension for all sequences $(y_n)$ in $[0,1)$ and in the Gauss map system ${x: \liminf_{n\to\infty}|T^nx-0|>0}$ also has full Hausdorff dimension. In this talk, I will introduce a dynamical approach to understanding these sets, and the new technique will allow us to strengthen the results so that the “targets’’ can be generalized to any equicontinuous sequence of functions, enabling the targets to vary by trajectories. In particular, notably this will imply the full dimension of non-recurrent points, bridging the problems of shrinking targets and Poincare recurrence.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar