Markov Property of Kusuoka-Zhou ’ s Dirichlet Forms on Self-Similar Sets
Markov Property of Kusuoka-Zhou ’ s Dirichlet Forms on Self-Similar Sets
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自相似集上Kusuoka-Zhou狄利克雷形式的马尔可夫性质
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发表时间：
2004
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Jun
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作者：
Jun
本文的主要目的是填补Kusuoka-Zhou在自相似集上构造自相似Dirichlet型的空白。不幸的是，在[KZ]的定理6.9的证明中得到的自相似封闭形式E是否满足马尔可夫性质并不十分清楚。利用锥上保序可加映射的不动点定理证明了具有马氏性的自相似闭型的存在性。不动点定理将在§ 1中介绍。它也适用于其它问题，例如p.c.f.上调和结构的存在性问题。自相似集在§ 2中，我们将应用不动点定理证明自相似集上自相似Dirichlet型的存在性。1.不动点定理在这一节中，我们将介绍一个有序拓扑锥上的不动点定理。定义1.1（拓扑锥）。一个豪斯多夫拓扑空间U称为拓扑锥，如果它满足以下条件。(1)U是具有单位元的交换半群。我们用u + v表示U中u与v的半群和。单位用0表示。(2)存在一个映射[0，∞）× U → U，（s，u）→ su，它满足关于半群结构的标量乘法的标准性质：（a）对任意s1，s2 ∈ [0，∞）和任意u ∈ U，s1（s2 u）=（s1 s2）u和（s1 + s2）u = s1 u + s2 u . (b)s（u + v）= su + sv，对任意s ∈ [0，∞）和任意u，v ∈ U . 2000年数学学科分类。60J45、31C25、28A80。
The main purpose of this note is to fill a gap in Kusuoka-Zhou’s construction of self-similar Dirichlet forms on selfsimilar sets. Unfortunately, it is not quite clear whether or not the self-similar closed form E obtained in the proof of Theorem 6.9 of [KZ] satisfies the Markov property. We will use a kind of fixed point theorem of order preserving additive maps on a cone to prove existence of a self-similar closed form with the Markov property. The fixed point theorem will be introduced in § 1. It is also applicable to other problems, for example, the existence problem of a harmonic structure on a p.c.f. self-similar set. In § 2, we will apply the fixed point theorem to show existence of self-similar Dirichlet forms on self-similar sets. 1. A Fixed Point Theorem In this section, we will introduce a fixed point theorem on an ordered topological cone. Definition 1.1 (Topological cone). A Hausdorff topological space U is called a topological cone if it satisfies the following conditions. (1) U is a commutative semigroup with a unity. We use u + v to denote the semigroup sum of u and v in U . The unity is denoted by 0. (2) There exists a map [0,∞) × U → U , (s, u) → su, that satisfies the standard properties of a scalar multiplication with respect to the semigroup structure: (a) s1(s2u) = (s1s2)u and (s1 + s2)u = s1u + s2u for any s1, s2 ∈ [0,∞) and any u ∈ U . (b) s(u + v) = su + sv for any s ∈ [0,∞) and any u, v ∈ U . 2000 Mathematics Subject Classification. 60J45, 31C25, 28A80.