月曜解析セミナー: Weighted partition of a compact metrizable space, its hyperbolicity and Ahlfors regular conformal dimension

開催日時
2018年   3月 27日 15時 00分 ~ 2018年   3月 27日 16時 30分
場所
理学部3号館210
講演者
木上 淳(京都大学)
 
Successive divisions of compact metric spaces appear in many different areas of mathematics such as the construction of self-similar sets, Markov partition associated with hyperbolic dynamical systems, dyadic cubes associated with a doubling metric space. The common feature in these is to divide a space into a finite number of subsets, then divide each subset into pieces and repeat this process again and again. In this paper we generalize such successive divisons and call them partitions. Given a partition, we consider the notion of a weight function  assigning  a ``size'' to each piece of the partition. Intuitively we believe that a partition and a weight function should provide a ``geometry'' and an ``analysis'' on the space of our interest. In this paper, we are going to pursue this idea in three parts. In the first part, the metrizability of a weight function is shown to be equivalent to the Gromov hyperbolicity of the graph associated with the weight function. In the second part, the notions like bi-Lipschitz equivalence, Ahlfors regularity, the volume doubling property and quasisymmetry will be shown to be equivalent to certain properties of weight functions. In particular, we find that quasisymmetry and the volume doubling property are the same notion in the world of weight functions. In the third part, a characterization of the Ahlfors regular conformal dimension of a compact metric space is given as the critical index p of p-energies associated with the partition and the weight function corresponding to the metric.

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参照 http://www.math.sci.hokudai.ac.jp/~aik/ma/MonAna.pdf

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研究集会・セミナー・集中講義の一覧へ