The Hausdorff dimension of general Sierpiński carpets

Abstract

In this note we determine the Hausdorff dimension of a family of planar sets which are generalizations of the classical Cantor set.

Self-Affine Sets and Graph-Directed Systems

“…McMullen [25] and Bedford [2] had considered the special case for A = n 0 0 m and the IFS has no overlap. Based on the directed graph version of Kenyon and Peres [17] we can use the above construction to include the case that the IFS has overlaps (Theorem 4.4).…”

“…If A is not a similarity, then the only way we can handle this is for A = n 0 0 m , 0 < m < n, the special case of McMullen [25] and Bedford [2]. Let D ⊆ Z 2 + be a digit set.…”

“…It is also known that for this case the Hausdorff dimension and the box dimension are not the same in general [25]. By using [17, Proposition 3.5], we can also calculate the box dimension of the set T .…”

A Class of Self-Affine Sets and Self-Affine Measures

“…Even under the open set condition we know how to compute the Hausdorff dimension of K(I) only for very special I's, and for which the solutions are quite nontrivial. McMullen [21] and Bedford [1] independently computed the Hausdorff and box dimensions of K(I) for I = {φ j } m j =1 in which all φ j have the form…”

“…Lalley and Gatzouras [17], in a highly technical article along the same spirit of [21], computed the Hausdorff and box dimensions for a broader class of IFS I = {φ j } m j =1 , in which φ j map the unit square (0, 1) 2 into disjoint rectangles having certain geometric arrangement inside the unit square. More precisely, in the Lalley-Gatzouras class all rectangles φ j ((0, 1) 2 ) are parallel to the axes and have longer sides parallel to the x-axis.…”