Self-affine measures and vector-valued representations

Abstract

Abstract. Let A be a d × d integral expanding matrix and let Sj(x) = A −1 (x + dj) for some dj ∈ Z d , j = 1, . . . , m. The iterated function system (IFS) {Sj} m j=1 generates self-affine measures and scale functions. In general this IFS has overlaps, and it is well known that in many special cases the analysis of such measures or functions is facilitated by expressing them in vector-valued forms with respect to another IFS that satisfies the open set condition. In this paper we prove a general theorem on such representation. The proof is constructive; it depends on using a tiling IFS {ψj} l j=1 to obtain a graph directed system, together with the associated probability on the vertices to form some transition matrices. As applications, we study the dimension and Lebesgue measure of a self-affine set, the L q -spectrum of a self-similar measure, and the existence of a scaling function (i.e., an L 1 -solution of the refinement equation).

Multifractal formalism for self-affine measures with overlaps

“…Although false in general, the multifractal formalism has been verified for many natural measures, including self-similar and graph self-similar measures satisfying the open set condition (see [3,6,7,26] and the references therein), self-similar measures with the weak separation property (see [4,5,8,9,13,14,22,23,24,25]), and self-conformal measures (see [10,28,29,31]). However, little is known for selfaffine measures, except some special cases (see [1,2,20,27]), and this motivated our work.…”

Gibbs properties of self-conformal measures and the multifractal formalism

“…The result is based on the thermodynamic formalism for matrix-valued functions established in [19]. Some extensions were given recently in [10,55] for some specific non-equi-contractive and high-dimensional cases. We remark that even under the finite type condition, τ (ν, q) may be non-differentiable for some q < 0 and it may lead to intervals in which the multifractal formalism does not hold (see [16,20,21,28,33,[54][55][56]).…”

Laplace operators related to self-similar measures on Rd

“…Therefore, for 0 < r < 1 we have In order to study some important IFSs of contractive similitudes that do not satisfy the OSC, Lau and Ngai [21] generalized the OSC by introducing a weaker notion of separation on the IFSs called the weak separation condition (WSC). The properties of IFSs satisfying the WSC have been studied extensively in a series of papers [9,[19][20][21][22][23]25]. In particular, by making use of the renewal equation, they have given algorithms to calculate the L q -spectrum τ (q) for q = 2 as well as for integers q > 2 for self-similar measures defined by several important classes of IFSs satisfying the WSC [12,25].…”