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首页> 外文会议>International Conference on Chaotic Modeling, Simulation, and Applications >Beta(p,q)-Cantor sets: Determinism and randomness
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Beta(p,q)-Cantor sets: Determinism and randomness

机译：Beta（p，q） - 班塔尔集：决定论和随机性

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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.
机译：通常出现的随机性作为一个复杂的延伸确定性模型，即，然后提交一些类随机模型中的预期（这种做法非常好古典Barucha - 里德的专着管理上的随机函数和随机过程）。作品[1]，[2]，[3]和[5]总结以往由作者的研究，采用Cantor公司的分形的扩展，随机定义，提出了相应的确定性模型，在精确的意义上是一个结构化的期望类的模型，并研究了分叉，阿利”的效果，和豪斯多夫尺寸。 β（P，Q）的模型，与是p = 1和或q = 1，或经典Verhulsts模型（P = Q = 2），对于该计算两者的Hausdorff尺寸和概率可被明确地评价相称有趣可计算模型，无论是分析或使用蒙特卡洛方法。本发明的扩展，削减了对任意的符号动力系统，进一步发展几何构造的新的基础类，并利用确定性和随机的相互作用的极限分形集的丰富性，在递归结构。这对豪斯多夫维度的概念，揭示新的光。我们发现，在随机顺序统计的依赖是在随机集的一贯较小的维数的明显的异常的核心，当与相应的“预期”确定性的同行相比。我们还恢复Falconner的，Pesin的和Weiss'（等等）递归几何构造为一个简单的方法来在分形和混乱的重要问题的想法。

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