HAUSDORFF DIMENSION OF SELF-AFFINE LIMIT SETS WITH AN INVARIANT DIRECTION

. We determine the Hausdorﬀ dimension of self-aﬃne limit sets for some class of iterated function systems in the plane with an invariant direction. In particular, the method applies to some type of generalized non-self-similar Sierpi´nski triangles. This partially answers a question asked by Falconer and Lammering and extends a result by Lalley and Gatzouras.


1.
Introduction. In 1997, Falconer and Lammering [7] studied generalized Sierpiński triangles, which are self-affine limit sets of an iterated function system obtained by mapping a triangle ∆ in an affine way into three sub-triangles ∆ 1 , ∆ 2 , ∆ 3 contained in ∆, such that each ∆ i has one common vertex with ∆ and each pair of the sub-triangles intersect at one point, which is their common vertex and lies on a side of ∆. The authors considered the case, when one of the maps is a similarity. Then, changing the coordinates in an affine way, one can assume that ∆ has vertices at (0, 0), (1, 0), (0, 1) and the maps have the form f 1 (x, y) = a 0 0 b x y , where a, b ∈ (0, 1). The system is presented in Fig. 1.
The generalized Sierpiński triangle is defined as the limit set The case a = b = 1/2 corresponds to the standard self-similar Sierpiński triangle, where the Hausdorff and box dimension are equal to log 3/ log 2. In all other cases at least one map is not a similarity. Such affine iterated function systems are more difficult to handle than the self-similar ones, and their dimension theory is less developed. For some results in this area refer e.g. to [1,3,4,5,6,8,9,10,11]. In [7], the authors computed the box dimension of the generalized Sierpiński triangle in the case when b ≥ max(a, 1 − a) or b ≤ min(a, 1 − a) and asked several questions, e.g. on the Hausdorff dimension of the limit set. In this paper we partially answer this problem, computing the Hausdorff dimension of Λ in the case b ≥ max(a, 1−a). In fact, we determine the Hausdorff and box dimension for a wider class of self-affine limit sets of iterated function systems in the plane preserving a given direction and fulfilling some separation conditions. The theorem, which extends a result by Lalley and Gatzouras from [8], is based on the paper [1].

2.
Definitions and results. Consider an affine iterated function system on the plane R 2 (i.e. a finite number of affine planar automorphisms), which preserves a direction given by a straight line ℓ ⊂ R 2 (i.e. a straight line parallel to ℓ is always mapped to another line parallel to ℓ ). By a linear change of coordinates, we can assume that ℓ is horizontal, which means that the matrices of linear parts of the maps are upper triangular. The system can be written as {f i,j } (i,j)∈I , where I = {(i, j) : i = 1, . . . , k, j = 1, . . . , m i } for some positive integers k, m 1 , . . . , m k and Notation. The symbols dist and diam denote, respectively, the Euclidean distance and diameter, while U, int U and ∂U are, respectively, the closure, interior and boundary of a set U . For a set U ⊂ R 2 denote by U (y) the horizontal section of U at level y ∈ R, i.e.
Then the Hausdorff dimension of the limit set is equal to the maximum of the function (with the convention 0 log 0 = 0) over the simplex Moreover, the box dimension of the limit set is equal to the unique real number D, such that where t is the unique real number such that k i=1 |b i | t = 1. Remark 1. The condition (b) says that the horizontal strips Remark 2. Obviously, the condition (d) is satisfied, if all non-empty horizontal sections of V have 1-dimensional Lebesgue measure greater than δ (e.g. when V is a parallelogram with a pair of horizontal sides, see Fig. 2).
Remark 3. The limit set can be defined as Note that the maps f i,j need not be contractions. However, we have diam f i1,j1 • · · · • f in,jn (V ) → 0 as n → ∞ (see Lemma 3.3). This implies that Λ is the unique non-empty compact set, for which Λ = (i,j)∈I f i,j (Λ).   Some examples of fractal limit sets satisfying the assumptions of the above theorem are presented in Fig. 3.
It is easy to check that the system generating the generalized Sierpiński triangle described in Section 1 satisfies the conditions of the above theorem for V = int ∆ in the case b ≥ max(a, 1 − a). Hence, we have the following.
Note that in the case b = 1−a the corollary follows directly from [1] (Example 3.3 and Proposition 3.6). Essentially, it is also a consequence of a result from the earlier paper [8] (formally, in [8] the considered maps are not similarities, but the arguments remain the same). In [2], the Hausdorff and box dimension were computed for a nonlinear modification of the Sierpiński triangle, where the triangles ∆ and ∆ 1 , ∆ 2 , ∆ 3 are isosceles instead of equilateral.
To prove our theorem, we will use results from [1] (Theorems A and B and Remark 3.8), which give formulae for the Hausdorff and box dimension of the limit sets for so-called rectangle-like geometric constructions in the plane. For convenience, we present the definition of a rectangle-like construction suited to our case (note that the notation from [1] is changed).  j 1 ), . . . , (i n , j n ) ∈ I (where I is defined as in Section 2) be compact sets in the plane. Let a i,j and b i for (i, j) ∈ I be real numbers, such that 0 < |a i,j | ≤ |b i | < 1. Set A i1,j1,...,in,jn = a i1,j1 · · · a in,jn , B i1,...,in = b i1 · · · b in . We say that ∆ i1,j1,...,in,jn and ∆ i ′ 1 ,j ′ 1 ,...,i ′ n ,j ′ n have the same type, if i m = i ′ m for m = 1, . . . , n. Otherwise, we say that the sets have different types.