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

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 } , i.e., 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 .