Hausdorff dimension of frequency sets in beta-expansions

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...

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...

Calculating Hausdorff Dimension of Julia Sets and Kleinian Limit Sets

Packing Dimension, Hausdorff Dimension and Cartesian Product Sets

We show that the dimension adim introduced by R. Kaufman (1987) coincides with the packing dimension Dim, but the dimension aDim introduced by Hu and Taylor (1994) is different from the Hausdorff dimension. These results answer questions raised by Hu and Taylor. AMS Classification numbers: Primary 28A78, 28A80.

The Hausdorff dimension of recurrent sets in symbolic spaces

Let ( , σ ) be the one-sided shift space onm symbols. For any x = (xi)i 1 ∈ and positive integer n, define Rn(x) = inf{j n : x1x2 · · · xn = xj+1xj+2 · · · xj+n}. We prove that for each pair of numbersα, β ∈ [0,∞] withα β, the following recurrent set Eα,β = { x ∈ : lim inf n→∞ logRn(x) n = α, lim sup n→∞ logRn(x) n = β } has Hausdorff dimension one. AMS classification scheme number: 28A80