Local time of self-affine sets of Brownian motion type and the jigsaw puzzle problem
Local time of self-affine sets of Brownian motion type and the jigsaw puzzle problem
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DOI：
10.1016/j.jmaa.2014.04.018
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发表时间：
2014-11
影响因子：
1.3
通讯作者：
Yumei Xue;T. Kamae
中科院分区：
文献类型：
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作者：
Yumei Xue;T. Kamae
设Ω，k}是[0，1]的一个划分，|J I| =|我我|1/2。因此，Ω几乎必然是Borel函数f Ω的图，它被称为布朗运动型的自仿射集。设λ是[0，1]上的勒贝格测度，μ Ω= λ f Ω− 1。密度ρ Ω= d μ Ω d λ，如果存在的话，称为Ω的局部时，并且已经被研究过了。已知如果ρ Ω存在，则dim H Ω= 3/2。在本研究中，ρ Ω是通过解决所谓的拼图游戏{J i，τ i; i= 1，n，k}，即分解ρ Ω的自相似图像与支持集J i和方向τ i（i= 1，n，k）的和的问题。
Abstract Let Ω⊂[0, 1]×[0, 1] be the solution of the set equation: Ω=⋃ i= 1 k (φ I i, 1× φ J i, τ i)(Ω), where for an interval I=[a, b]⊂[0, 1] and τ∈{− 1, 1}, φ I, τ:[0, 1]→ I is the linear map such that φ I, 1 (0)= a, φ I, 1 (1)= b, φ I,− 1 (0)= b, φ I,− 1 (1)= a, and {I i; i= 1,⋯, k} is a partition of [0, 1] with| J i|=| I i| 1/2. Thus, Ω is a graph of a Borel function f Ω almost surely and it is called a self-affine set of Brownian motion type. Let λ be the Lebesgue measure on [0, 1] and let μ Ω= λ∘ f Ω− 1. The density ρ Ω= d μ Ω d λ, if it exists, is called the local time of Ω and it has been studied. It is known that dim H Ω= 3/2 if ρ Ω exists. In the present study, ρ Ω is obtained by solving the so-called jigsaw puzzle on {J i, τ i; i= 1,⋯, k}, ie, the problem of decomposing ρ Ω into a sum of its self-similar images with the support J i and the orientation τ i for i= 1,⋯, k.