Measure of Self-Affine Sets and Associated Densities

Abstract

Let B be an n × n real expanding matrix and D be a finite subset of R n with 0 ∈ D. The self-affine set K = K(B, D) is the unique compact set satisfying the setvalued equation BK = d∈D (K + d). In the case where card(D) = |det B|, we relate the Lebesgue measure of K(B, D) to the upper Beurling density of the associated measure µ = lim s→∞ ℓ0,...,ℓs−1∈D δ ℓ0+Bℓ1+···+B s−1 ℓs−1 . If, on the other hand, card(D) < |det B| and B is a similarity matrix, we relate the Hausdorff measure H s (K), where s is the similarity dimension of K, to a corresponding notion of upper density for the measure µ.

Open set condition and pseudo Hausdorff measure of self-affine IFSs

“…It will be used to find a different expression for the pseudo Hausdorff measure of K(A, D). This is motivated by the connection between the upper s-density of µ in (1.1) which was first introduced in [6] and the Hausdorff measure of a self-similar set K(A, D). Definition 5.1.…”

“…For self-similar sets satisfying certain separating conditions (e.g. open set condition [17], weak separation condition [18,19], nite type condition [20]), there exist methods to calculate their Hausdorff dimensions [15,17,20,21] and the corresponding Hausdorff measures [6,22,[23][24][25][26][27][28]. However, no many results are available in that direction for self-af ne sets.…”

“…For the proof of Theorem 1.1, we utilize the connection between pseudo norm and Euclidean norm as well as the technique used by Schief [29], Bishop and Peres [3] for the self-similar case. We also would like to mention that there have been several equivalent characterizations for the OSC under special cases given by Lagarias and Wang (Theorem 1.1 in [18]), by He and Lau (Theorem 4.4 in [10]) and by Fu and Gabardo (Theorem 3.2 in [6]).…”

“…In this paper, we are interested in the computation of the pseudo Hausdorff measure of self-affine sets in the case that #D ≤ q. This is motivated by the results in [6], which gave an exact expression for the Lebesgue measure of K(A, D) with #D = q and the Hausdorff measure of the self-similar set K(A, D) associated with its similarity dimension in the case that #D ≤ q. One of the main results of this paper is to relate the pseudo Hausdorff measure of K(A, D), namely H s w (K(A, D)) where s = n ln(#D)/ ln(q) is the pseudo similarity dimension of K, to a notion of upper density with respect to (w.r.t.)…”