This thesis combines computability theory and various notions of fractal dimension, mainly Hausdorff dimension. An algorithmic approach to Hausdorff measures makes it possible to define the Hausdorff dimension of individual points instead of sets in a metric space. This idea was first realized by Lutz (2000b). Working in the Cantor space 2ω of all infinite binary sequences, we study the theory ...

We show that if a real x ∈ 2 is Hausdorff H-random, where h is an order function, then it is also random for a continuous probability measure μ such that the μ-measure of the basic open cylinders shrinks according to h. We use this result to derive a characterization of effective Hausdorff dimension similar to Frostman’s Theorem.

[1] introduced fractal geometric entropies and dimensions for Voiculescu’s microstate spaces ([3], [4]). One can associate to a finite set of selfadjoint elements X in a tracial von Neumann algebra and an α > 0 an extended real number H(X) ∈ [−∞,∞]. H(X) is a kind of asymptotic logarithmic α-Hausdorff measure of the microstate spaces of X. One can also define a free Hausdorff dimension of X, de...

In this paper we establish a formal connection between the average decay of the Fourier transform of functions with respect to a given measure and the Hausdorff behavior of that measure. We also present a generalization of the classical restriction theorem of Stein and Tomas replacing the sphere with sets of prefixed Hausdorff dimension n− 1 + α, with 0 < α <

This thesis combines computability theory and various notions of fractal dimension, mainly Hausdorff dimension. An algorithmic approach to Hausdorff measures makes it possible to define the Hausdorff dimension of individual points instead of sets in a metric space. This idea was first realized by Lutz (2000b). Working in the Cantor space 2ω of all infinite binary sequences, we study the theory ...

We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent h equal to the Hausdorff dimension of the Julia set. Both h-dimensional Hausdorff and packing measures are finite and positive on the Julia set and ...

The meromorphic maps fλ(z) = λ(1−exp(−2z))−1, λ > 0, of the complex plane are thoroughly investigated. With each map fλ associated is its projection Fλ on the infinite cylinder Q. This map and the set Jr(Fλ) consisting of those points in the cylinder Q whose ω-limit set under Fλ is not contained in the set {0,−∞} will form the primary objects of our interest in this article. Let hλ = HD(Jr(Fλ))...

The theory of Hausdorff dimension provides a general notion of the size of a set in a metric space. We define Hausdorff measure and dimension, enumerate some techniques for computing Hausdorff dimension, and provide applications to self-similar sets and Brownian motion. Our approach follows that of Stein [4] and Peres [3].

In this paper, first of all the distance measure entitled generalized Hausdorff distance is defined between two generalized fuzzy numbers that has been introduced by Chen [3]. Then using another distance and combining it with the generalized Hausdorff distance, a fuzzy distance measure is introduced with the help of fuzzy distance measure proposed in [1] to generalize fuzzy numbers. The concept...