非均质分形的马丁边界理论

IF 0.5 4区数学 Q3 MATHEMATICS

Asian Journal of Mathematics Pub Date : 2024-07-12 DOI:10.4310/ajm.2023.v27.n5.a2

Uta Freiberg, Stefan Kohl

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引用次数: 0

摘要

我们考虑的分形是由具有开放集条件的概率迭代函数方案生成的，我们将概率解释为分形每个部分的权重。在不考虑权重的同质情况下，Denker 和 Sato 于 2001 年引入了词空间上的马尔可夫链，并证明了马丁边界与分形集同构。我们的目的是重新定义与权重相关的过渡概率，并计算出马丁边界。我们将看到，非均质马丁边界与均质情况相吻合。

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Martin boundary theory on inhomogeneous fractals

We consider fractals generated by a probabilistic iterated function scheme with open set condition and we interpret the probabilities as weights for every part of the fractal. In the homogeneous case, where the weights are not taken into account, Denker and Sato introduced in 2001 a Markov chain on the word space and proved that the Martin boundary is homeomorphic to the fractal set. Our aim is to redefine the transition probability with respect to the weights and to calculate the Martin boundary. As we will see, the inhomogeneous Martin boundary coincides with the homogeneous case.

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