Hausdorff dimension and σ finiteness of p harmonic measures in space when p≥n

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...

Hausdorff Dimension of the Harmonic Measure on Trees

For a large class of Markov operators on trees we prove the formula HD = h=l connecting the Hausdorr dimension of the harmonic measure on the tree boundary, the rate of escape l and the asymptotic entropy h. Applications of this formula include random walks on free groups, conditional random walks, random walks in random environment and random walks on treed equivalence relations. 0. Introducti...

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

The Hausdorff Dimension and Scale-Space Normalisation of Natural Images

Fractal Brownian motions have been introduced as a statistical descriptor of natural images. We analyze the Gaussian scale-space scaling of derivatives of fractal images. On the basis of this analysis we propose a method for estimation of the fractal dimension of images and scale-space normalization used in conjunction with automatic scale selection assuming either constant energy over scale or...