Nonempty Intersections of Middle $\alpha$ Cantor Sets

Random Intersections of Thick Cantor Sets

Let C1, C2 be Cantor sets embedded in the real line, and let τ1, τ2 be their respective thicknesses. If τ1τ2 > 1, then it is well known that the difference set C1 − C2 is a disjoint union of closed intervals. B. Williams showed that for some t ∈ int(C1−C2), it may be that C1∩ (C2 + t) is as small as a single point. However, the author previously showed that generically, the other extreme is tru...

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.

Self-similar Measures and Intersections of Cantor Sets

It is natural to expect that the arithmetic sum of two Cantor sets should have positive Lebesgue measure if the sum of their dimensions exceeds 1, but there are many known counterexamples, e.g. when both sets are the middle-α Cantor set and α ∈ ( 1 3 , 1 2 ). We show that for any compact set K and for a.e. α ∈ (0, 1), the arithmetic sum of K and the middle-α Cantor set does indeed have positive...

There are no C-stable intersections of regular Cantor sets

Hyperbolic Sets with Nonempty Interior

In this paper we study hyperbolic sets with nonempty interior. We prove the folklore theorem that every transitive hyperbolic set with interior is Anosov. We also show that on a compact surface every locally maximal hyperbolic set with nonempty interior is Anosov. Finally, we give examples of hyperbolic sets with nonempty interior for a non-Anosov diffeomorphism.