Estimating the Hausdorff measure of Denjoy maps by recurrence

We will look at a new method of giving lower bounds on the Hausdorff measure (and dimension) of some fractal sets. The idea is to compute/estimate the consecutive closest return times of a typical point to itself and make use of some known relations between the return times and the density of invariant measure w.r.t. the Hausdorff one.

Then we will apply this method to a C^{1+\delta} diffeomorphism of a circle that is not conjugate to an irrational rotation (Denjoy map). Such a map always has a minimal Cantor set - we will give a bound on its Hausdorff measure from below.