Beta(p,q)-Cantor sets: Determinism and randomness

Usually randomness appears as a sophisticated extension of deterministic models, that are then presented as expectation of some class of random models (this approach is exceedingly well managed in the classical Barucha-Reid's treatise on random functions and stochastic processes). The works [1], [2], [3] and [5] summarize previous studies by the authors, using stochastic definitions of extensions of Cantor's fractal to put forward appropriate deterministic models, that in a precise sense are the expectation of a structured class of models, and investigated bifurcations, Allee' effect, and the Hausdorff dimension. Beta(p,q) models, with either p = 1 or q = 1, or the classical Verhulsts model (p = q = 2), proportionate interesting computable models for which computations both of Hausdorff dimension and probabilities can be explicitly evaluated, either analytically or using the Monte Carlo method.

The present extension, axed on arbitrary symbolic dynamical systems, further develops new fundamental classes of geometric constructions, and exploits the interplay of determinism and randomness on the richness of the limit fractal set, in a recursive construction. This sheds new light on the concept of Hausdorff dimensionality. We show that the dependence of the random order statistics is at the core of the apparent anomaly of consistently smaller Hausdorff dimensions of the random sets, when compared with the corresponding "expected" deterministic counterparts. We also recover Falconner's, Pesin's and Weiss' (among others) ideas on recursive geometric constructions as a straightforward approach to important issues in fractality and chaos.