Hausdorff measure of linear sets

On Intersections of Cantor Sets: Hausdorff Measure

We establish formulas for bounds on the Haudorff measure of the intersection of certain Cantor sets with their translates. As a consequence we obtain a formula for the Hausdorff dimensions of these intersections.

HAUSDORFF MEASURE OF p-CANTOR SETS

In this paper we analyze Cantor type sets constructed by the removal of open intervals whose lengths are the terms of the p-sequence, {k−p}∞k=1. We prove that these Cantor sets are s-sets, by providing sharp estimates of their Hausdorff measure and dimension. Sets of similar structure arise when studying the set of extremal points of the boundaries of the so-called random stable zonotopes.

ON THE HAUSDORFF h-MEASURE OF CANTOR SETS

We estimate the Hausdorff measure and dimension of Cantor sets in terms of a sequence given by the lengths of the bounded complementary intervals. The results provide the relation between the decay rate of this sequence and the dimension of the associated Cantor set. It is well-known that not every Cantor set on the line is an s-set for some 0 ≤ s ≤ 1. However, if the sequence associated to the...

The Hausdorff Dimension and Exact Hausdorff Measure of Random Recursive Sets with Overlapping

Invariant Sets with Zero Measure and Full Hausdorff Dimension

For a subshift of finite type and a fixed Hölder continuous function, the zero measure invariant set of points where the Birkhoff averages do not exist is either empty or carries full Hausdorff dimension. Similar statements hold for conformal repellers and two-dimensional horseshoes, and the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages...