Hausdorff dimension of uniformly non flat sets with topology

Fractal sets and Hausdorff dimension

We consider Farey series of rational numbers in terms of fractal sets labeled by the Hausdorff dimension with values defined in the interval 1 < h < 2 and associated with fractal curves. Our results come from the observation that the fractional quantum Hall effect-FQHE occurs in pairs of dual topological quantum numbers, the filling factors. These quantum numbers obey some properties of the Far...

Hausdorff Dimension and Hausdorff Measure for Non-integer based Cantor-type Sets

We consider digits-deleted sets or Cantor-type sets with β-expansions. We calculate the Hausdorff dimension d of these sets and show that d is continuous with respect to β. The d-dimentional Hausdorff measure of these sets is finite and positive.

A Hausdorff dimension for finite sets

The classical Hausdorff dimension, denoted dimH , of finite or countable sets is zero. We define an analog for finite sets, called finite Hausdorff dimension and denoted dimfH , which is non-trivial. It turns out that a finite bound for dimfH (F ) guarantees that every point of F has ”nearby” neighbors. This property is important for many computer algorithms of great practical value, that obtai...

Hausdorff Dimension of Limit Sets for Parabolic Ifs with Overlaps

We study parabolic iterated function systems with overlaps on the real line. We show that if a d-parameter family of such systems satisfies a transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of 1 and the least zero of the pressure function. Moreover, the local dimension of the exceptional set of parameters is estimated. If t...

Calculating Hausdorff Dimension of Julia Sets and Kleinian Limit Sets