Local time of self-affine sets of Brownian motion type and the jigsaw puzzle problem

摘要

Let Ω ? [0, 1] × [0, 1] be the solution of the set equation:where for an interval I = [a, b] ? [0,1] and τ ∈ {-1, 1},? I τ:[0,1] → I is the linear map such that Ф I, 1(0) = a, Ф I, (1) = b, Ф I, -1 (0) = b, Ф I, -1 (1) = a, and {I_2;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 o f_Ω~(-1). The density ρΩ = dμΩ/dλ, if it exists, is called the local time of Ω and it has been studied. It is known that dimH_Ω = 3/2 if ρΩ exists. In the present study, ρΩ is obtained by solving the so-called jigsaw puzzle on {J_i, Τ_i;i =1, …, k}, i.e., the problem of decomposing ρΩ into a sum of its self-similar images with the support J_i and the orientation Τ_i for i = 1, …,k.