Sublinear measures, Menger curvature, and Hausdorff dimension

Hausdorff Dimension of Measures via Poincaré Recurrence

We study the quantitative behavior of Poincaré recurrence. In particular, for an equilibrium measure on a locally maximal hyperbolic set of a C diffeomorphism f , we show that the recurrence rate to each point coincides almost everywhere with the Hausdorff dimension d of the measure, that is, inf{k > 0 : fx ∈ B(x, r)} ∼ r. This result is a non-trivial generalization of work of Boshernitzan conc...

Menger curvature and rectifiability

E3 c(x, y, z)dH(x)dH(y)dH(z) where H1 is the 1-dimensional Hausdorff measure in Rn, c(x, y, z) is the inverse of the radius of the circumcircle of the triangle (x, y, z), that is, following the terminology of [6], the Menger curvature of the triple (x, y, z). A Borel set E ⊂ Rn is said to be “purely unrectifiable” if for any Lipschitz function γ : R → Rn, H1(E ∩ γ(R)) = 0 whereas it is said to ...

Menger Curvature and Rectifiability 833

where H1 is the 1-dimensional Hausdorff measure in Rn, c(x, y, z) is the inverse of the radius of the circumcircle of the triangle (x, y, z), that is, following the terminology of [6], the Menger curvature of the triple (x, y, z). A Borel set E ⊂ Rn is said to be “purely unrectifiable” if for any Lipschitz function γ : R → Rn, H1(E ∩ γ(R)) = 0 whereas it is said to be rectifiable if there exist...

Hausdorff Dimension and Conformal Measures of Feigenbaum Julia Sets

1.1. Statement of the results. One of the first questions usually asked about a fractal subset of R is whether it has the maximal possible Hausdorff dimension, n. It certainly happens if the set has positive Lebesgue measure. On the other hand, it is easy to construct fractal sets of zero measure but of dimension n. Moreover, this phenomenon is often observable for fractal sets produced by conf...

Hausdorff Dimension and Hausdorff Measures of Julia Sets of Elliptic Functions