Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 3, 733–751

On Some Dimension Problems for Self-Affine Fractals

M. P. Bernardi and C. Bondioli

Abstract. We deal with self-affine fractals in IR^{2}. We examine the notion of affine dimension of a fractal proposed in [26]. To this end, we introduce a generalized affine Hausdorff dimension related to a family of Borel sets. Among other results, we prove that for a suitable class of self-affine fractals (which includes all the so-called general Sierpiński carpets), under the “open set condition”, the affine dimension of the fractal coincides – up to a constant – not only with its Hausdorff dimension arising from a non-isotropic distance D_{θ} in IR^{2}, but also with the generalized affine Hausdorff dimension related to the family of all balls in (IR^{2}, D_{θ}). We conclude the paper with a comparison between this assertion and results already known in the literature.

Keywords: Self-affine fractals, Hausdorff measures, dimensions, homogeneous spaces AMS subject classification: Primary 28 A 80, secondary 28 A 78, 43 A 85

Introduction

This paper deals with self-affine fractals in IR^{2}, that is, fractals arising from systems of contractive affinities. Even though not so intensively studied as the case of systems consisting of similitudes, the construction of fractals by affine maps has been widely investigated in recent years. A list of references on the subject can be found, for instance, in [17: Chapter 4, Remarks 4.15].

Our work is suggested by the recent book [26] by H. Triebel, where fractal geometry is studied in connection with Fourier analysis, function spaces on self-affine fractals and fractal differential operators.

Here we are interested in the geometrical background of the contents of this book. In the first chapter of [26] the author gives the definition of affine dimension of a selfaffine fractal. This notion is the “affine counterpart” of the definition of the similarity dimension introduced by Mandelbrot for a system of similitudes: in the exponential equation defining the similarity dimension, one has to replace every similarity ratio with the square root of the corresponding affinity ratio. Here square roots appear, since the underlying space is IR^{2}. So the affine dimension of a fractal Γ reduces to its similarity dimension if every affinity of the system defining Γ is a similitude.

It is clear that a priori the similarity dimension refers to a system Σ of similitudes, but – as is well known – under a suitable hypothesis of “minimal overlapping” it also coincides with the Hausdorff dimension of the fractal Γ arising from Σ. Consequently, it turns out to be an intrinsic property of Γ.

Both authors: Univ. of Pavia, Dept. Math. “F. Casorati”, via Ferrata 1, I-27100 Pavia, Italy ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

This paper is motivated by the analogous question on the affine dimension, that is, by the desire to understand to what extent the affine dimension is not only a property of a system Ψ of affinities, but a property of the fractal associated with Ψ. We shall prove that for a suitable class of self-affine fractals, still under conditions of “minimal overlapping”, the affine dimension coincides – up to a constant – with the Hausdorff dimension arising from a distance in IR^{2}, which differs from the Euclidean one, but accords with the non-isotropicity of the situation. Also, if we agree to work with quasidistances, we can choose a quasi-distance in IR^{2} so that these two dimensions are equal.

Now we give a brief description of the content of the paper. Let us underline that for simplicity we work in IR^{2}; still, all the ideas of Sections 2 - 4, at least, apply equally in IR^{n}.

In Section 1 we present the basic material that will be needed later on.

In Section 2 we report a simple example, which in part motivated these investigations: we define an infinite family of systems of contractive affinities satisfying even stronger conditions than the usual one of “minimal overlapping”. All these systems lead to the same fractal, but nevertheless any two of them have different affine dimensions.

In Sections 3 and 4 we provide an extension of the results proved by Hutchinson in [11] in the case of similitudes for the affine context. To this end, we need a new notion of dimension with which to compare the affine dimension. We therefore introduce in Section 3 the definitions of generalized affine Hausdorff measures and generalized affine Hausdorff dimension. These definitions depend on the choice of a family F of Borel sets: different families can lead to different dimensions for the same set. We obtain conditions which ensure that the affine dimension is an intrinsic property of the fractal, or at least of the fractal and the family F (see Proposition 4.1(ii) and (iii)).

In Section 5 we consider a subclass of the family of self-affine fractals that nevertheless is wide enough to contain all the so-called general Sierpiński carpets. For this class we can reinterpret the affinities of the system as similitudes with respect to a new nonisotropic distance. Therefore we are able to prove Theorem 5.1, which can be considered the main result of the paper: if a condition of “minimal overlapping” is satisfied, then the affine dimension of the fractal coincides – up to a constant – with the Hausdorff dimension arising from the new distance. Also, the affine dimension coincides with the generalized affine Hausdorff dimension arising from the family of all balls defined by the same non-isotropic distance.

Finally, in Section 6 we briefly compare the propositions of Section 5 with results already known about the calculation of the Hausdorff dimension – with respect to the usual distance – of self-affine fractals in IR^{2}.

We would like to thank: H. Triebel, who allowed us to read the manuscript of his latest book in advance and explained to us some of the leading ideas; U. Mosco, whose suggestions simplified some of our previous proofs; W. Farkas, M.V. Marchi and L. Skrzypczak for helpful discussions.

1. Notation and preliminaries

1.1 Carathéodory’s construction of measures. We follow the exposition of [17: Chapters 4 and 5]. Another classical reference is the book [21].

Let (X, D) be a metric space. For E ⊆ X we denote by diam E the diameter of E. Let F be a family of subsets of X and ζ a non-negative function on F, satisfying the following assumptions:

(F1) For every δ > 0 there are F_{1}, F_{2}, … ∈ F such that X = ⋃_{i=1}^{∞} F_{i} and diam F_{i} ≤ δ.

(F2) For every δ > 0 there is F ∈ F such that ζ(F) ≤ δ and diam F ≤ δ.

For E ⊆ X and δ > 0, we define

β_{δ}(E) = inf { ∑_{i=1}^{∞} ζ(F_{i})| E ⊆ ⋃_{i=1}^{∞} F_{i} , diam F_{i} ≤ δ , F_{i} ∈ F },

and letting δ → 0^{+} we define

β(E) = lim_{δ→0^{+}} β_{δ}(E) .

So β is a Borel (outer) measure in X and, if the members of F are Borel sets, β is Borel regular. If F consists of all subsets of X, s is a non-negative real number, and ζ(E) = (diam E)^{s}, then the resulting (outer) measure is called the s-dimensional Hausdorff measure H^{s}. If F consists of all balls in X and ζ is defined as before, then the resulting (outer) measure is called the s-dimensional spherical Hausdorff measure S^{s}. The two measures H^{s} and S^{s} are related by the inequalities H^{s}(E) ≤ S^{s}(E) ≤ 2^{s}H^{s}(E). The properties of the measures H^{s} allow us to define the Hausdorff dimension of a set E ⊆ X as

dim_{H}E = sup{s | H^{s}(E) = ∞}

and to prove that dim_{H}E = inf{t| H^{t}(E) = 0}.

1.2 Similitudes and affinities. Let (X, D) be a metric space. For a map f : X → X the terms Lipschitz-continuous function, contraction and similitude have the usual meaning (see, e.g., [11: p. 716 - 717]). We write Lip f for the Lipschitz constant of f. If σ is a similitude, we denote Lip σ by ρ_{σ} and call ρ_{σ} the (similarity) ratio of σ.

Throughout the paper X is the real space IR^{2}. In IR^{2} we consider also distances D different from the Euclidean one, which we denote by D_{e}. Nevertheless, all the distances we use give rise to a complete metric space, induce the Euclidean topology and therefore define the same family of Borel sets. We denote by L the usual Lebesgue (outer) measure. It is well known that, if D = D_{e}, then the Hausdorff measure H^{2} coincides – up to a constant – with L.

If ψ : IR^{2} → IR^{2} is of the form ψ(x) = Ax + h, where A ∈ GL(2, IR) and h ∈ IR^{2}, then ψ is called an affinity. We briefly write ψ = (A, h). We denote | det A| by α_{ψ} (or simply α) and call α_{ψ} the (affinity) ratio of ψ. Recall that every affinity ψ has the following properties: for any subset E ⊆ IR^{2}, L(ψ(E)) = α_{ψ} L(E); ψ is a Lipschitz continuous function with α_{ψ} ≤ (Lip ψ)^{2}.

Finally, a one-parametric group {δ_{t} | t > 0} of non-isotropic dilations of IR^{2} is defined as follows: we fix two strictly positive real numbers θ_{1}, θ_{2} and we consider the family of maps

δ_{t}(x) = δ_{t}(x_{1}, x_{2}) = (t^{θ_{1}}x_{1}, t^{θ_{2}}x_{2})

(see, e.g., [24: Chapter I, Example 2.3]).

1.3 Systems of contractions and fractals. We follow [11: §3, §4, and §5] for the following subsections (i) - (ii) and [26: Chapter I, Section 4] for subsection (iii).

(i) Let Φ = {φ_{1}, …, φ_{N}} be a family of contractions of IR^{2} (endowed with a distance D).

There exists a unique (non-empty) compact set Γ in IR^{2} invariant with respect to Φ, that is, such that Γ = φ_{1}(Γ) ∪ … ∪ φ_{N}(Γ).

We call Γ the fractal associated with the system Φ. In addition, let r = {r_{1}, …, r_{N}} be a family of N real numbers in ]0, 1[ with ∑_{i=1}^{N} r_{i} = 1. Then:

There exists a unique Borel regular (outer) measure μ in IR^{2} compactly supported and of total mass 1, such that μ is invariant with respect to (Φ, r), that is,

μ(E) = ∑_{i=1}^{N} r_{i} μ(φ_{i}^{−1}(E)) (E ⊆ IR^{2}).

Moreover, the support of μ is the fractal Γ.

We say that the system Φ satisfies the open set condition (OSC) if there exists a bounded open set O in IR^{2} such that

\begin{matrix}φ_{i}(O) ⊆ O \\ φ_{i}(O) ∩ φ_{j}(O) = ∅ for i ≠ j (i = 1, 2, …, N) . \end{matrix} \}

Stronger statements than OSC have been considered. The system Φ is said to satisfy the strong open set condition (SOSC) if there exists a bounded open set O such that OSC is verified by O and in addition O ∩ Γ ≠ ∅, where Γ is the fractal associated with Φ (see, e.g., [22]). The system Φ is said to satisfy the separated open set condition if there exists a bounded open set O such that OSC is verified by O and in addition the closures of the sets φ_{i}(O) are disjoint (see, e.g., [25]).

(ii) Now we suppose furthermore that the contractions of Φ are similitudes. Then we write Φ = Σ = {σ_{1}, …, σ_{N}}. The number d_{Σ} > 0 uniquely defined by the relation ∑_{i=1}^{N} ρ_{σ_{i}}^{d_{Σ}} = 1 is called the similarity dimension of Σ. Let Γ again be the fractal associated with the system Σ. By abuse of language d_{Σ} is called the similarity dimension of Γ. Therefore we will write d_{Σ} = d_{Σ}Γ and, when no confusion arises, we simply write d_{Σ}Γ = d.

Let k be the Hausdorff dimension of Γ. The fractal Γ is called self-similar (with respect to Σ) if H^{k}(Γ) > 0 and H^{k}(σ_{i}(Γ) ∩ σ_{j}(Γ)) = 0 for i ≠ j.

The following results are proved in [11: §5] for D = D_{e}: (I) (i) H^{d}(Γ) < +∞ and so k ≤ d.

(ii) Let Γ be self-similar with H^{k}(Γ) < +∞. Then d = k.

(iii) If H^{d}(Γ) > 0, then d = k and Γ is self-similar.

(II) If the system Σ satisfies OSC, then H^{d}(Γ) > 0. Moreover, suppose r_{i} = ρ_{σ_{i}}^{d} for i = 1, …, N. Then the restriction of H^{d} to Γ is – up to a multiplicative constant – the invariant measure with respect to (Σ, r).

The first statement in (II) can be made more precise as follows (cf. [22, 23]):

SOSC ⟺ OSC ⟺ H^{d}(Γ) > 0.

Let us note that Hutchinson’s results are extended in [16] to quasi-metric spaces of homogeneous type in the sense used by Coifman and Weiss in [3]. For a description of the frame within which the results of [16] are included we refer to [19, 20].

(iii) Now let us suppose that the contractions of Φ are affinities. Then we write Φ = Ψ = {ψ_{1}, …, ψ_{N}}.

Let Γ be the fractal associated with Ψ. The real number d_{Ψ}Γ > 0 uniquely defined by the relation ∑_{i=1}^{N} α_{ψ_{i}}^{(d_{Ψ}Γ)/2} = 1 is called the affine dimension of Γ (see [26: Definition 4.12]). We will often write d_{Ψ}Γ as d_{Ψ}. It should be observed that in the case of diagonal affinities d_{Ψ}Γ was already considered in [15: Part I, Section 4] under the name of gap dimension. The number d_{Ψ}Γ plays a key role in describing the asymptotic behaviour of the eigenvalues of some fractal differential operator in [26: Theorem 30.7] and [9].

A particular case deserves attention. Let n_{1}, n_{2} be integers with n_{1} ≥ 2 and n_{2} ≥ 2, and let T be the subset of IR^{2} given by

T = {(t_{1}/n_{1}, t_{2}/n_{2}) | t_{1}, t_{2} integers, 0 ≤ t_{1} < n_{1}, 0 ≤ t_{2} < n_{2} } .

Suppose that every element ψ_{i} of the system Ψ is of the form (A, h_{i}), where the translation vectors h_{i} are in T and

A = ( \begin{matrix}1/n_{1} & 0 \\ 0 & 1/n_{2} \end{matrix} ) .

Thus each ψ_{i} maps the unit square Q onto a rectangle contained in Q. The fractal associated with Ψ is called general Sierpiński carpet in [18].

The definition of regular anisotropic fractal in [26: Chapter I, Subsection 4.18] is slightly more general. In fact, every ψ_{i} is of the form (A_{i}, k_{i}), where

A_{i} = ( \begin{matrix}±1/n_{1} & 0 \\ 0 & ±1/n_{2} \end{matrix} )

(the signs depend on i and indicate a possible reflection); the vectors k_{i} still have as components integer multiples of 1/n_{1} and 1/n_{2}, and are chosen so that each ψ_{i}(Q) is still contained in Q and ψ_{i}(Q) is different from ψ_{j}(Q) if i ≠ j. In [26] the matrices A_{i} are written in the form

A_{i} = ( \begin{matrix}±(2^{−κ})^{a_{1}} & 0 \\ 0 & ±(2^{−κ})^{a_{2}} \end{matrix} )  (1.1)

with

a_{1} = 2 log n_{1} ( log(n_{1}n_{2}))^{−1} , a_{2} = 2 log n_{2} ( log(n_{1}n_{2}))^{−1}  (1.2)
κ = \frac{1}{2} log_{2}(n_{1}n_{2}) .  (1.3)

Observe that κ and a_{1}, a_{2} are chosen so that a_{1} + a_{2} = 2.

2. An example as motivation

In this section we refer to the pathological phenomenon sketched in [26: Chapter I, Remark 5.12/p. 31], which is also connected with [6: Example 9.10/ p. 127 - 128].

Let η be a real number in ]0, 1[. We consider the system Ψ = {ψ_{1}, ψ_{2}} of affine contractions

\begin{aligned}ψ_{1}(x_{1}, x_{2}) & = (\frac{1}{2}x_{1}, ηx_{2}) \\ ψ_{2}(x_{1}, x_{2}) & = (\frac{1}{2}x_{1} + \frac{1}{2}, ηx_{2}). \end{aligned}

The associated fractal set Γ is [0, 1] × {0}; if η^{−1} is an integer number, Γ is a regular anisotropic fractal. An easy calculation shows that d_{Ψ}Γ = 2 (1 − log_{2} η)^{−1}. So we reach the apparently surprising result that the fractal set Γ is independent of η, whereas d_{Ψ}Γ does depend on it and can even assume any value in ]0, 2[.

We will often return to this example and use it to illustrate certain situations. In Remark 5.9 we will try to reinterpret it in an appropriate context.

Remark 2.1. The system Ψ satisfies OSC (e.g., with O =]0, 1[×]0, 1[) and also SOSC (e.g., with O =]0, 1[×] − 1, 1[).

Remark 2.2. We can slightly modify the above construction and obtain as resulting fractal the set Γ_{q} = [0, 1] × {q}, for any q ∈ [0, 1]. In fact, if we start with the two affinities

\begin{aligned}ψ_{1}(x_{1}, x_{2}) & = (\frac{1}{2}x_{1}, ηx_{2} + q(1 − η)) \\ ψ_{2}(x_{1}, x_{2}) & = (\frac{1}{2}x_{1} + \frac{1}{2}, ηx_{2} + q(1 − η)) \end{aligned}

we reach the conclusion that Γ_{q} = ψ_{1}(Γ_{q}) ∪ ψ_{2}(Γ_{q}).

Remark 2.3. The previous example can be easily generalized. If η ∈]0, 1[ and n, N are integers such that 1 < N ≤ n, let us consider the system Ψ = {ψ_{1}, …, ψ_{N}}, where ψ_{i} is of the form (A, h_{i}) with A = ({1/n \atop 0} {0 \atop η} ), and h_{i} are N translation vectors chosen in {(0, 0), (1/n, 0), (2/n, 0), …, ((n − 1)/n, 0)}. The associated fractal Γ is of “Cantor set” type with dim_{H}Γ = log_{n} N and d_{Ψ}Γ = 2 (1 − log_{n} η)^{−1} · log_{n} N. So the affine dimension can vary only in ]0, 2 · dim_{H}Γ[. Note that the system Ψ satisfies SOSC and, under a suitable choice of N and h_{i}, also the separated open set condition.

3. Generalized affine Hausdorff measures

In this section we define a family of generalized Hausdorff measures which behaves in a natural way with respect to the affine maps of IR^{2}.

Let F denote a family of Borel sets in IR^{2} satisfying conditions (F1) and (F2). We will fix F from time to time, since F has to be adapted to the specific situation.

Definition 3.1. Let s ≥ 0. We denote by A_{F}^{s} (and call it the s-dimensional generalized affine Hausdorff measure) the outer measure ensued from Carathéodory’s construction, by setting ζ(F) = (LF)^{s/2} for every F ∈ F.

Remark 3.1. For s = 0 we interpret as usual 0^{0} = 1; therefore A_{F}^{0} coincides with the counting measure.

Remark 3.2. Let us observe that if F is the family of all balls in (IR^{2}, D_{e}), then – up to a constant – A_{F}^{s} = S^{s}. Hence A_{F}^{s} and H^{s} are both zero, positive and finite, or infinite. The same result holds if F is the family of all squares in IR^{2}, or of all squares with sides parallel to the axes. Note that Definition 3.1 and this remark already appeared in [13: Section 9.3: “Volume interpretation of the Hausdorff dimension”].

We list now some properties of A_{F}^{s}. Since the proofs are standard, we will omit them.

Proposition 3.1.

(i) A_{F}^{s} is a Borel regular measure.

(ii) Let E be a subset of IR^{2}. If 0 ≤ s < t, then A_{F}^{s}(E) < +∞ implies A_{F}^{t}(E) = 0.

As usual, this result leads to the following definition.

Definition 3.2. Let E be a subset of IR^{2}. We define

dim_{F}E = sup {s | A_{F}^{s}(E) = ∞} .

We call dim_{F}E the generalized affine Hausdorff dimension of the set E.

It follows that dim_{F}E = inf {t | A_{F}^{t}(E) = 0} and that dim_{F}E is invariant under the action of bi-Lipschitz transformations f, such that f and f^{−1} map F in F.

Remark 3.3. Suppose F = B, where B is the family of all Borel sets in IR^{2}, and D = D_{e}. Let B ∈ B with LB = 0. For every fixed δ > 0 we can cover B with a countable family {F_{i}} of Borel sets with diam F_{i} ≤ δ and LF_{i} = 0 (take, for instance, F_{i} = B ∩ R_{i}, where {R_{i}} is a suitable family of rectangles). Therefore, if s > 0, then A_{B}^{s}(B) = 0. Now let B ∈ B with LB > 0. Since LB = A_{B}^{2}(B), it follows that A_{B}^{s}(B) = ∞ for s ∈]0, 2[. Thus for every Borel set B in IR^{2} we have dim_{B}B = 0 if LB = 0 and dim_{B}B = 2 if LB > 0. This is the reason why F will usually be a proper subset of B.

Remark 3.4. The generalized affine Hausdorff dimension dim_{F}E of a subset E in IR^{2} depends on the family F. In Remark 3.3 we considered the case F = B. Now let us suppose that F consists of all balls with respect to D_{e}. By Remark 3.2, A_{F}^{s} coincides – up to a constant – with S^{s} for every s. So, for E ⊆ IR^{2}, dim_{F}E = dim_{H}E.

The following property of A_{F}^{s}, with respect to the affinities leaving the family F invariant, is the exact counterpart of the behaviour of H^{s} with respect to the similitudes.

Proposition 3.2. Let ψ be an affinity of IR^{2} with ratio α. Suppose that ψ and ψ^{−1} map the family F in F. Then A_{F}^{s}(ψ(E)) = α^{s/2}A_{F}^{s}(E).

Remark 3.5. From Proposition 3.2 it is clear that we are interested only in the subgroup of all the affinities ψ such that ψ and ψ^{−1} leave F invariant. We will denote it by G_{F}. It is natural to suppose, in addition to conditions (F1) and (F2) of Subsection 1.1, that F verifies the following condition:

(F3) There are some contractive affinities in G_{F}.

Note that (F3) is not implied by (F1) and (F2).

4. Strongly self-affine fractals and invariant measures

Now we extend some of the results proved in [11: §5] for similitudes and Hausdorff measures H^{s} to affinities and affine measures A_{F}^{s}.

Let F be a family of Borel sets satisfying conditions (F1), (F2), (F3), and G_{F} be the corresponding group. Suppose that Ψ = {ψ_{1}, …, ψ_{N}} is a system of contractive affinities of G_{F} and let Γ be the associated fractal. We shorten dim_{F}Γ to h.

Definition 4.1. We say that Γ is strongly self-affine (with respect to Ψ and F) if

(i) Γ = ⋃_{i=1}^{N} ψ_{i}(Γ).

(ii) A_{F}^{h}(Γ) > 0 and A_{F}^{h}(ψ_{i}(Γ) ∩ ψ_{j}(Γ)) = 0 for i ≠ j.

Remark 4.1. Definition 4.1 follows the pattern of the definition of self-similar fractal in [11: §5]. The term “self-affine” is used in the literature in a wider sense, namely to mean a fractal arising from a system of contractive affinities in IR^{2} with respect to the distance D_{e}. Therefore we must point out the difference.

Remark 4.2. Our definition of strong self-affinity is strictly related to the family F. For instance, consider (IR^{2}, D_{e}). Let F be the family of all balls. Then G_{F} is the group of all similitudes. By Remarks 3.2 and 3.4, “strongly self-affine with respect to F” has the same meaning as “self-similar”.

On the contrary, let F = B be the family of all Borel sets; therefore G_{B} is the whole affine group. Suppose that the fractal Γ generated by Ψ verifies LΓ = 0. Since dim_{B}Γ = 0 and A_{B}^{0} is the counting measure (recall Remarks 3.1 and 3.3), Γ is strongly self-affine with respect to B if and only if ψ_{i}(Γ) ∩ ψ_{j}(Γ) = ∅ for i ≠ j.

So it is even possible that the same system Ψ gives rise to a fractal Γ, which is strongly self-affine with respect to a family F_{1} but not with respect to a different family F_{2}. For instance, recalling the first example given in Section 2, let us consider the system Ψ with η = 1/2; consequently Γ = [0, 1] × {0}. If F_{1} is the family of all balls, then Γ is strongly self-affine; on the contrary if F_{2} = B, then Γ is not strongly self-affine, since ψ_{1}(Γ) ∩ ψ_{2}(Γ) ≠ ∅. The latter result is still true if F_{3} = R is the family of all rectangles with sides parallel to the axes. Indeed, it is easy to prove that for every s > 0, A_{R}^{s}(Γ) = 0; so dim_{R}Γ = 0.

Our first goal is to compare dim_{F}Γ with d_{Ψ}Γ. For the sake of simplicity we denote dim_{F}Γ = h and d_{Ψ}Γ = d_{Ψ}. From now on let us suppose that the family F satisfies the following additional condition:

(F4) Any compact set in IR^{2} can be covered by a finite number of elements of F.

Observe that (F4) is not implied by (F1), (F2) and (F3). Then we can prove

Proposition 4.1.

(i) A_{F}^{d_{Ψ}}(Γ) < +∞ and so h ≤ d_{Ψ}.

(ii) Let Γ be strongly self-affine with A_{F}^{h}(Γ) < +∞. Then d_{Ψ} = h.

(iii) If A_{F}^{d_{Ψ}}(Γ) > 0, then d_{Ψ} = h and Γ is strongly self-affine.

Proof. (i): By (F4), we can suppose Γ ⊆ ∪i=1^{ℓ}F_{i} with F_{i} ∈ F. Then, for every p ∈ IN,

Γ = ∪_{j_{1},...,j_{p}}ψ_{j_{1}} ∘ … ∘ ψ_{j_{p}}(Γ) ⊆ ∪_{j_{1},...,j_{p}} ∪i=1^{ℓ} ψ_{j_{1}} ∘ … ∘ ψ_{j_{p}}(F_{i})

and diam ψ_{j_{1}} ∘ … ∘ ψ_{j_{p}}(F_{i}) → 0 as p → ∞. It follows that, for any δ > 0, Γ can be covered by a finite number of elements of the family F with diameter less than δ. Therefore

A_{F,δ}^{d_{Ψ}} ≤ ∑_{j_{1},...,j_{p}} ∑_{i=1}^{ℓ} L(ψ_{j_{1}} ∘ … ∘ ψ_{j_{p}}(F_{i}))^{d_{Ψ}/2}

= ∑_{j_{1},...,j_{p}} ∑_{i=1}^{ℓ} (α_{j_{1}} … α_{j_{p}})^{d_{Ψ}/2} (LF_{i})^{d_{Ψ}/2}

= ∑_{i=1}^{ℓ} (LF_{i})^{d_{Ψ}/2}

< ∞

and A_{F}^{d_{Ψ}}(Γ) < +∞ follows immediately.

(ii) and (iii): By Proposition 3.2 the proof is the same as in [11: Proposition 5.1(4)]

Remark 4.3. The inequality h ≤ d_{Ψ} (which appears in (i) of Proposition 4.1) shows that the affine dimension d_{Ψ} is a natural upper bound for the generalized affine Hausdorff dimension h = dim_{F}Γ for every family F such that Ψ ⊆ G_{F}.

Remark 4.4. Let us investigate the role played by the “open set condition”. We recall that for a system Σ of similitudes in (IR^{2}, D_{e}) with similarity dimension d_{Σ} the following implications are true:

\begin{aligned}OSC & ⟺ H^{d_{Σ}}(Γ) > 0 \\ OSC & ⟹ d_{Σ} = dim_{H}Γ \end{aligned}

(see, e.g., [11: Theorem 5.3.(1)] and, for a more detailed discussion, [22]). The analogous assertions are not in general true in the affinity context, at least if the family F is not properly chosen. So it may happen that

OSC is verified and A_{F}^{d_{Ψ}}(Γ) = 0 OSC is verified and d_{Ψ}Γ ≠ dim_{F}Γ.

To show this, we return again to the first example of Section 2 and choose F = R, that is the family of all rectangles with sides parallel to the axes. For every η ∈]0, 1[, the system Ψ satisfies OSC. Nevertheless, A_{R}^{d_{Ψ}}(Γ) = 0 and 0 = dim_{R}Γ ≠ d_{Ψ}Γ.

We conclude this section with some remarks on invariant measures. We refer again to [11] and we extend some of the results given there to contractive affinities. If X is any subset of IR^{2}, we denote by A_{F}^{s}|X the restriction of A_{F}^{s} to X, that is (A_{F}^{s}|X)(E) = A_{F}^{s}(X ∩ E), for E ⊆ IR^{2}.

The following propositions are the “affine counterpart” of two theorems proved in [11] for similitudes.

Proposition 4.2. Let ψ be an affinity of G_{F} with ratio α and let X be a subset of IR^{2}. Then, for every s ≥ 0 and for every E in IR^{2},

(A_{F}^{s}|ψ(X)) (E) = α^{s/2} (A_{F}^{s}|X)(ψ^{−1}(E)) .

Proof. See [11: Subsection 2.6(2)]

Proposition 4.3. Let Ψ = {ψ_{1}, …, ψ_{N}} be a family of N contractive affinities of G_{F}. Let α_{i} be the ratio of ψ_{i} and Γ the fractal associated to Ψ. We denote d_{Ψ}Γ = d_{Ψ} and consider the n-ple r = {α_{1}^{d_{Ψ}/2}, …, α_{N}^{d_{Ψ}/2}}. If A_{F}^{d_{Ψ}}(Γ) > 0, then the unique (Ψ, r)- invariant measure of total mass 1 is (A_{F}^{d_{Ψ}}(Γ))^{−1} · A_{F}^{d_{Ψ}}|Γ.

Proof. See [11: Theorem 5.3(1)]

Remark 4.5. Let Ψ and Γ be as in Proposition 4.3. If A_{F}^{d_{Ψ}}(Γ) > 0, then the normalized invariant measure μ = (A_{F}^{d_{Ψ}}(Γ))^{−1}·A_{F}^{d_{Ψ}}|Γ obeys the scaling law μ(ψ_{k}(Γ)) = α_{k}^{d_{Ψ}/2} for every k = 1, …, N. This is the natural extension of the scaling property for the invariant measure in the case of self-similar fractals. Let us also note that μ coincides with the Radon measure considered in [26: Theorem 4.15].

Remark 4.6. Proposition 4.1(iii), Proposition 4.3, and Remark 4.5 show that the condition A_{F}^{d_{Ψ}}(Γ) > 0 seems to be significant. However, it also seems rather hard to test. So it is natural to look for conditions which are simpler to check and are sufficient to imply it. In the next section we will come back to this problem and show that for a suitable, sufficiently large class of systems of contractive affinities, OSC still implies that the resulting fractal Γ satisfies A_{F}^{d_{Ψ}}(Γ) > 0.

5. Regular anisotropic fractals and homogeneous spaces

In this section we limit ourselves to some specific systems of contractive affinities in IR^{2}. Firstly, we use only diagonal affinities that is those affinities (A, h) such that A is diagonal (with respect to the usual orthonormal base of IR^{2}). Next, our main assumption on the system Ψ = {ψ_{1}, …, ψ_{N}} is the following: there exist two positive numbers θ_{1} and θ_{2} such that every matrix A_{i} (corresponding to the affinity ψ_{i} of Ψ) can be written in the form

A_{i} = ( \begin{matrix}±τ_{i}^{θ_{1}} & 0 \\ 0 & ±τ_{i}^{θ_{2}} \end{matrix} )  with τ_{i} ∈]0, 1[ .  (5.1)

As we will show, these two conditions together with OSC guarantee that the affine dimension d_{Ψ}Γ can be interpreted as an “intrinsic” number related to the fractal Γ associated with Ψ, in the sense that d_{Ψ}Γ depends only on Γ, θ_{1}, and θ_{2}. More precisely, this assertion has to be understood as follows: if the fractal Γ arises from two different systems Ψ and Ψ̃ of affinities, satisfying the previous two conditions and having the same exponents θ_{1} and θ_{2}, then d_{Ψ}Γ = dΨ̃Γ.

Remark 5.1. Systems of diagonal affinities give rise to fractals which in [26: Chapter I, Subsection 4.17] are called PXT-fractals. Our second assumption leads to a smaller class of fractals. Nevertheless, this class strictly contains the class of the regular anisotropic fractals. To see this it is sufficient to recall formulae (1.1) - (1.3) in Subsection 1.3./(iii).

Remark 5.2. In (5.1) we can suppose 0 < θ_{1} ≤ θ_{2}; otherwise it is sufficient to change the order of the orthonormal base in IR^{2} (this is always possible, since we are not interested in graphs of functions).

Remark 5.3. Let us observe that a matrix ({±τ^{θ_{1}} \atop 0} {0 \atop ±τ^{θ_{2}}} ) with τ ∈]0, 1[ can be

written as ({±σ^{λ_{1}} \atop 0} {0 \atop ±σ^{λ_{2}}} ) with σ ∈]0, 1[, if and only if the pair (λ_{1}, λ_{2}) is a multiple of (θ_{1}, θ_{2}), that is, if and only if there exists c > 0 such that λ_{i} = cθ_{i} (i = 1, 2).

So, for any matrix of the form (5.1) one can assume that θ_{1} and θ_{2} satisfy the property θ_{1} + θ_{2} = 2. This choice of θ_{1} and θ_{2} is usual in the theory of anisotropic function spaces (see [26: Chapter I, Subsection 4.17], [8: Subsection 2.2] and the references therein) and in the case of regular anisotropic fractals leads to formulae (1.1) and (1.2).

Our perspective draws us to conclude that, since the underlying space is IR^{2}, it is more convenient to assume θ_{1} = 1 and θ_{2} = θ ≥ 1. This choice leads to simpler notation and, as will be seen, to the advantage that we can define in IR^{2} a new distance related to θ, and not only a quasi-distance related to θ_{1} and θ_{2}. For these reasons we work with systems Ψ = {ψ_{1}, …, ψ_{N}} of affinities such that the matrix A_{i} of ψ_{i} has the form

( \begin{matrix}±t_{i} & 0 \\ 0 & ±t_{i}^{θ} \end{matrix} ) ,  (5.2)

where t_{i} ∈]0, 1[ and θ ≥ 1 is a fixed real number. Nevertheless, we often return to the case θ_{1} + θ_{2} = 2.

744 M. P. Bernardi and C. Bondioli

Corresponding to any system of affinities, whose matrices are as in (5.2), let us consider the group G_{θ} of all the affinities (A, h) such that h ∈ IR^{2} and A varies between the matrices of the form ({±t \atop 0} {0 \atop ±t^{θ}} ) (now clearly with t ∈]0, +∞[). Any affinity in G_{θ} can be regarded as the composition of the non-isotropic dilation

δ_{t}(x) = δ_{t}(x_{1}, x_{2}) = (tx_{1}, t^{θ}x_{2}) (t ∈]0, +∞[) , (5.3)

the translation given by the vector h, and in some cases a reflection.

Definition 5.1. For θ ≥ 1, let N_{θ} : IR^{2} → IR be defined by

N_{θ}(x) = N_{θ}(x_{1}, x_{2}) = max {|x_{1}|, |x_{2}|^{1/θ}}  (5.4)

and let D_{θ} be defined by

D_{θ}(x, y) = N_{θ}(x − y) .  (5.5)

Proposition 5.1. N_{θ} satisfies the following conditions: N_{θ}(x) ≥ 0 N_{θ}(x) = 0 if and only if x = 0 N_{θ}(−x) = N_{θ}(x) N_{θ}(x + y) ≤ N_{θ}(x) + N_{θ}(y) N_{θ}(δ_{t}(x)) = t N_{θ}(x).

Therefore, D_{θ} is a distance in IR^{2}, and δ_{t} is a similitude in (IR^{2}, D_{θ}) with ratio t, that is D_{θ}(δ_{t}(x), δ_{t}(y)) = t D_{θ}(x, y). Moreover, (IR^{2}, D_{θ}) is a complete metric space.

Proof. The triangle inequality follows from the formula (a + b)^{p} ≤ a^{p} + b^{p} for a, b ≥ 0 and p ∈]0, 1]. All the rest is straightforward

Remark 5.4. An element of G_{θ} is a contraction in IR^{2} with respect to D_{e} if and only if it is a contraction with respect to D_{θ}, that is, if and only if t ∈]0, 1[.

Remark 5.5. N_{θ} satisfies all the conditions of homogeneous norm except for the smoothness condition in IR^{2}\{0} (see [10: Chapter 1, Section A] and [12]).

Remark 5.6. If we substitute the pair (1, θ) with one of its multiple (θ_{1}, θ_{2}) = (c, cθ) (where c > 0), then we can define N_{θ_{1}θ_{2}}(x) = max {|x_{1}|^{1/θ_{1}}, |x_{2}|^{1/θ_{2}}}. N_{θ_{1}θ_{2}} gives rise to the quasi-distance D_{θ_{1}θ_{2}}(x, y) = N_{θ_{1}θ_{2}}(x − y). This quasi-distance becomes a distance if and only if θ_{1} ≥ 1 and θ_{2} ≥ 1.

We recall now that a system Ψ = {ψ_{1}, …, ψ_{N}} ⊆ G_{θ} can be seen as a system of affinities in IR^{2} with α_{ψ_{i}} = t_{i}^{1+θ}, and as a system of similitudes in (IR^{2}, D_{θ}) with ρ_{ψ_{i}} = t_{i}. This remark suggests a comparison between the affine dimension and the similarity dimension in (IR^{2}, D_{θ}). To this aim, let us first introduce further notation.

In (IR^{2}, D_{θ}) we denote by diam_{θ} E the diameter of a subset E ⊆ IR^{2} by H_{θ}^{l} the l-dimensional Hausdorff measure by S_{θ}^{l} the l-dimensional spherical Hausdorff measure

by dim_{H,θ} the Hausdorff dimension.

We also denote

by R_{θ} the family of all balls in (IR^{2}, D_{θ}).

Let us remark that if F ∈ R_{θ} is a ball in (IR^{2}, D_{θ}) with radius r, then from the Euclidean point of view F is a rectangle with sides parallel to the axes and of length 2r and 2r^{θ}. It follows that the group G_{θ} introduced in this section coincides with the group G_{R_{θ}} (see Remark 3.5). Finally, if Ψ = {ψ_{1}, …, ψ_{N}} is a system of similitudes with respect to D_{θ}, we denote by d_{θ} the similarity dimension of Ψ.

Using the notation just introduced, let us compare the affine dimension d_{Ψ}Γ with the similarity dimension d_{θ}.

Proposition 5.2. Let Ψ = {ψ_{1}, …, ψ_{N}} be a family of contractive affinities in G_{θ} and let Γ be the associated fractal. Then d_{Ψ}Γ = 2 (1 + θ)^{−1} d_{θ}.

Proof. The affine dimension d_{Ψ} = d_{Ψ}Γ and the similarity dimension d_{θ} are uniquely determined by

\begin{aligned}1 & = ∑_{i=1}^{N} (t_{i}^{1+θ})^{d_{Ψ}/2} = ∑_{i=1}^{N} t_{i}^{(1+θ)d_{Ψ}/2} \\ 1 & = ∑_{i=1}^{N} t_{i}^{d_{θ}} . \end{aligned}

By the uniqueness of γ such that ∑_{i=1}^{N} t_{i}^{γ} = 1 we have d_{θ} = (1 + θ)2^{−1}d_{Ψ}

Corollary 5.1. Let Γ be a regular anisotropic fractal in the sense of Subsection 1.3.(iii). Let a_{1}, a_{2} be as in (1.2) with a_{1} ≤ a_{2}. Then d_{Ψ}Γ = a_{1} d_{θ}.

Remark 5.7. The constant 2 (1+θ)^{−1}, which appears in Proposition 5.2, depends on the choice of the exponents (1, θ) in (5.2). A different normalization of the exponents would lead to a different constant. In fact, let us substitute (1, θ) with (θ_{1}, θ_{2}) = (c, cθ) (where c > 0) and let us define consequently the quasi-distance D_{θ_{1}θ_{2}} in IR^{2} (see Remark 5.6). Then we obtain d_{θ_{1}θ_{2}} = cd_{θ}, where d_{θ_{1}θ_{2}} is the new similarity dimension of the system Ψ, and therefore d_{Ψ}Γ = 2 (1 + θ)^{−1} c^{−1} d_{θ_{1}θ_{2}}.

The choice of θ_{1} and θ_{2} in the framework of anisotropic function spaces leads to the following result.

Corollary 5.2. Let Ψ = {ψ_{1}, …, ψ_{N}} be a system of contractive affinities ψ_{i} = (A_{i}, h_{i}), where A_{i} is of the form (5.1) and θ_{1} + θ_{2} = 2. Then d_{Ψ}Γ = d_{θ_{1}θ_{2}}.

Now we want to state the main assertions of this section. To this end, let us first prove a lemma, which compares A_{R_{θ}}^{s} with S_{θ}^{l}.

Lemma 5.1. Let s > 0. Then A_{R_{θ}}^{s} = 2^{\frac{1−θ}{2} ·s} S_{θ}^{\frac{1+θ}{2} ·s}.

Proof. Let F ∈ R_{θ} be a ball centered at a point (x_{1}, x_{2}) of IR^{2} with radius r and therefore diam_{θ} F = 2r. ¿From the Euclidean point of view we have LF = 4r^{1+θ} and diam F = 2(r^{2} + r^{2θ})^{1/2}, where diam F stands for the Euclidean diameter of F. Suppose 0 < δ ≤ 1; if diam F ≤ δ, then diam_{θ} F ≤ δ, and if diam_{θ} F ≤ δ, then diam F ≤ \sqrt{2}δ. By standard arguments the lemma follows

Now we can prove the following theorem. We use the notation introduced after Remark 5.3 and simply write dim_{H,θ}Γ as h_{θ}.

Theorem 5.1. Let Ψ = {ψ_{1}, …, ψ_{N}} be a system of contractive affinities in G_{θ} satisfying OSC. Then d_{Ψ}Γ = dim_{R_{θ}} Γ = 2 (1 + θ)^{−1} h_{θ} and Γ is strongly self-affine (with respect to Ψ and R_{θ}).

Moreover, let r = (α_{1}^{d_{Ψ}/2}, …, α_{N}^{d_{Ψ}/2}), where α_{i} = t_{i}^{1+θ} is the affinity ratio of ψ_{i}. Then the (Ψ, r)-invariant measure is

(A_{R_{θ}}^{d_{Ψ}}(Γ))^{−1} · A_{R_{θ}}^{d_{Ψ}}|Γ = (H_{θ}^{h_{θ}}(Γ))^{−1} · H_{θ}^{h_{θ}}|Γ.

Proof. (IR^{2}, D_{θ}) is a homogeneous space in the sense used by Coifman and Weiss in [3]. Therefore we can apply the results of [16] recalled in Subsection 1.3/(ii). Since Ψ is a system of similitudes in (IR^{2}, D_{θ}) with similarity dimension d_{θ} and Ψ satisfies OSC, it follows that d_{θ} = h_{θ}. By Proposition 5.2 we have d_{Ψ} = 2(1 + θ)^{−1}h_{θ}. ¿From

OSC, it follows that H_{θ}^{d_{θ}}(Γ) > 0 and therefore S_{θ}^{d_{θ}}(Γ) = S_{θ}^{\frac{1+θ}{2} ·d_{Ψ}}(Γ) > 0 (as usually d_{Ψ} stands for d_{Ψ}Γ). By Lemma 5.1 we have A_{R_{θ}}^{d_{Ψ}}(Γ) > 0. Proposition 4.1/(iii) implies that d_{Ψ} = dim_{R_{θ}} Γ and Γ is strongly self-affine, so the first part of the proposition is proved. Proposition 4.3 shows that (A_{R_{θ}}^{d_{Ψ}}(Γ))^{−1} · A_{R_{θ}}^{d_{Ψ}}|Γ is a (Ψ, r)-invariant measure of total mass 1. By the uniqueness result (A_{R_{θ}}^{d_{Ψ}}(Γ))^{−1} · A_{R_{θ}}^{d_{Ψ}}|Γ has to coincide with (H_{θ}^{d_{θ}}(Γ))^{−1} · H_{θ}^{d_{θ}}|Γ. Since d_{θ} = h_{θ}, the proposition is proved

Corollary 5.3. Let Ψ be as in Corollary 5.2. Suppose that Ψ satisfies OSC. Then d_{Ψ}Γ = h_{θ_{1}θ_{2}}, where h_{θ_{1}θ_{2}} stands for the Hausdorff dimension of Γ in (IR^{2}, D_{θ_{1}θ_{2}}).

Remark 5.8. Also the other assertions of Theorem 5.1 can be easily extended to the quasi-metric space (IR^{2}, D_{θ_{1}θ_{2}}). One has only to observe that the family of all balls in (IR^{2}, D_{θ}) coincides with the family of all balls in (IR^{2}, D_{θ_{1}θ_{2}}) if (θ_{1}, θ_{2}) = (c, cθ).

Remark 5.9. We return once more to the first example of Section 2 and try to explain it in this new context. Now we have to fix θ. Therefore, since t = 1/2, only one η is possible, namely η = 2^{−θ}. It follows that the affine dimension is d_{Ψ}Γ = 2 (1 − log_{2} η)^{−1} = 2 (1 + θ)^{−1}. This result concurs with Proposition 5.2, since one can easily check that the similarity dimension d_{θ} of Γ with respect to D_{θ} is 1.

6. Final remarks

In this section we compare our previous considerations with results already known about the Hausdorff dimension of some self-affine fractals.

Firstly, let us look at the relationship between the usual Hausdorff dimension dim_{H} in (IR^{2}, D_{e}) and the Hausdorff dimension dim_{H,θ} in (IR^{2}, D_{θ}). To this end it is convenient to consider in IR^{2} the norm N_{1}(x) = max{|x_{1}|, |x_{2}|} and the induced distance D_{1} (notice that N_{1} and D_{1} are a particular case of N_{θ} and D_{θ}). The identity map from IR^{2} with the Euclidean distance to IR^{2} with the distance D_{1} is a bi-Lipschitz continuous transformation; so we have dim_{H}E = dim_{H,1}E, for every E ⊆ IR^{2}.

Now we can state the following proposition.

Proposition 6.1. For any E ⊆ IR^{2} the inequalities

\frac{1}{θ} dim_{H,θ} E ≤ dim_{H} E ≤ dim_{H,θ} E  (6.1)

hold.

Proof. It is easy to prove that if N_{1}(x) ≤ 1, then (N_{θ}(x))^{θ} ≤ N_{1}(x) ≤ N_{θ}(x), and that for E ⊆ IR^{2} the assertion diam_{1} E ≤ 1 is equivalent to diam_{θ} E ≤ 1. If diam_{1} E ≤ 1, the above inequalities lead to (diam_{θ} E)^{θ} ≤ diam_{1} E ≤ diam_{θ} E. This result allows us to prove by standard methods that H_{θ}^{sθ}(E) ≤ H_{1}^{s}(E) ≤ H_{θ}^{s}(E), for any s > 0. Now (6.1) is an easy consequence

From Theorem 5.1 and Proposition 6.1 we immediately obtain a relation between the affine dimension of a fractal and its usual Hausdorff dimension.

Corollary 6.1. Let Ψ be a system of contractive affinities in G_{θ} satisfying OSC and let Γ be the associated fractal. Then

\frac{1}{θ} d_{θ} ≤ dim_{H} Γ ≤ d_{θ}  (6.2)

and therefore

\frac{1 + θ}{2θ} d_{Ψ}Γ ≤ dim_{H} Γ ≤ \frac{1 + θ}{2} d_{Ψ}Γ .  (6.3)

In particular, let Γ be a regular anisotropic fractal in the sense of Subsection 1.3./(iii). Let a_{1}, a_{2} be as in (1.2), with a_{1} ≤ a_{2}. Then

\frac{1}{a_{2}} d_{Ψ}Γ ≤ dim_{H}Γ ≤ \frac{1}{a_{1}} d_{Ψ}Γ .  (6.4)

It remains to be shown that the inequalities in (6.1) - (6.4) are sharp. It would be sufficient to consider line segments on the x_{1}-axis and the x_{2}-axis, respectively. However, we prefer to achieve the result as a by-product of the next comparisons between our propositions and the ones proved in the papers [14, 18].

In [18] the author studies general Sierpiński carpets (see Subsection 1.3/(iii)) and proves the following formula for the Hausdorff dimension of the fractal Γ:

dim_{H} Γ = log_{m} ( ∑_{j=1}^{m} r_{j}^{log_{n} m}) .  (6.5)

Here 1 < m < n and r_{j} (with 1 ≤ j ≤ m) are positive integers with the following meaning: m and n appear in the matrix A, that is, A = ({m^{−1} \atop 0} {0 \atop n^{−1}} ), and r_{j} denotes the number of rectangles located at the j-th column. The names of the axes are interchanged from those in [18], in accordance with our previous notation. Let us remark that a result similar to the one of [18] is obtained in [2] by different methods.

If we reinterpret the above system of affinities as in Section 5, that is as a system of similitudes in (IR^{2}, D_{θ}), we easily obtain dim_{H,θ}Γ = d_{θ} = log_{m} N, where N = ∑_{j=1}^{m} r_{j}.

Hence from Theorem 5.1 we have d_{Ψ}Γ = 2(θ + 1)^{−1} log_{m} N, where θ = log_{m} n. So formula (6.1) immediately leads to the estimate

\frac{1}{θ} log_{m} N ≤ dim_{H} Γ ≤ log_{m} N ,  (6.5)

which is equivalent to

log_{m} ( ∑_{j=1}^{m} rj^{\frac{1}{θ}} ) ≤ log_{m} N ≤ θ log_{m} ( ∑_{j=1}^{m} rj^{\frac{1}{θ}} ) .  (6.6)

By an elementary calculation we can conclude:

(i) dim_{H} Γ = log_{m} N if and only if r_{j} ≤ 1 for every j, that is at most one rectangle is chosen in every column.

(ii) log_{m} N = θ dim_{H} Γ if and only if there exists j_{0} such that r_{j_{0}} = N (hence r_{j} = 0 for j ≠ j_{0}), that is all the chosen rectangles belong to the same column.

In the proof of the “only if” part in (ii) we used the inequality (a + b)^{p} > a^{p} + b^{p} for a, b > 0 and p > 1. So it is additionaly shown that the inequalities in (6.1) and (6.2) are sharp.

These remarks can be extended to some of the fractals considered in the paper [14]. Let us underline the fact that not every fractal of [14] can be obtained via the construction explained in Section 5, and viceversa. Therefore we describe only those systems of affinities which both satisfy the hypotheses of [14] and are contained in G_{θ} for a suitable θ. They are precisely the systems with which [14: Remark 2/p.549] is concerned; as observed in [14], all Sierpiński carpets are of this type. Again, with respect to [14] we interchange the names of the axes.

Thus we consider systems Ψ = {ψ_{jℓ}}, where 1 ≤ j ≤ m and for j fixed, 1 ≤ ℓ ≤ r_{j}. Here ψ_{jℓ} = (A_{j}, h_{jℓ}), with A_{j} = ({t_{j} \atop 0} {0 \atop t_{j}^{θ}} ) and h_{jℓ} = ({c_{j} \atop d_{jℓ}} ); in addition we suppose

N = ∑_{j=1}^{m} r_{j} > 1. If we denote by Q the unit square of IR^{2}, the further hypotheses stated in [14: at p. 534] guarantee that the N open rectangles ψ_{jℓ}(Int Q) are pairwise disjoint subsets of Q. Note that the rectangles are arranged in m columns; moreover, all the r_{j} rectangles of the j-th column are congruent: they have width t_{j} and, since Ψ ⊆ G_{θ}, height t_{j}^{θ}.

In [14: Remark 2] the authors show that dim_{H} Γ is the unique positive real number δ defined by

∑_{j=1}^{m} r_{j}^{1/θ} t_{j}^{δ} = 1 .

If we reinterpret the system Ψ as in Section 5, we obtain that d_{θ} is the unique positive number such that

∑_{j=1}^{m} r_{j} t_{j}^{d_{θ}} = 1 .

Self-Affine Fractals 749

So, in order to compare dim_{H} Γ and d_{θ}, it is natural to consider the functions f_{1} and f_{2} defined as

f_{1}(ξ) = ∑_{j=1}^{m} r_{j}^{1/θ} t_{j}^{ξ}  and  f_{2}(ξ) = ∑_{j=1}^{m} r_{j} t_{j}^{ξ} .

Let us suppose θ > 1. It is easy to prove the following equivalences:

(i)′ The three conditions dim_{H} Γ = d_{θ} f_{1} = f_{2} r_{j} = 1 for all j = 1, …, m, that is, again, each column contains only one rectangle

are equivalent.

(ii)′ The three conditions

d_{θ} = θ dim_{H} Γ

f_{2}(θξ) = (f_{1}(ξ))^{θ} for all ξ

m = 1, that is, again, all the rectangles are located on the same column

are equivalent.

Remark 6.1. To conclude this section, let us come back to the normalization used in the theory of anisotropic function spaces (see Remark 5.3). So θ_{1} and θ_{2} in (5.1) satisfy the condition θ_{1} + θ_{2} = 2; as before we suppose θ_{1} ≤ θ_{2}, therefore we have θ_{1} ≤ 1 ≤ θ_{2}. If we consider in IR^{2} the quasi-distance D_{θ_{1}θ_{2}} induced by N_{θ_{1}θ_{2}} (see Remark 5.6), we can easily prove that Proposition 6.1 now assumes the following form:

For any E ⊆ IR^{2}, the inequalities

\frac{1}{θ_{2}} dim_{H,θ_{1},θ_{2}} E ≤ dim_{H} E ≤ \frac{1}{θ_{1}} dim_{H,θ_{1},θ_{2}} E  (6.1)′

hold where dim_{H,θ_{1},θ_{2}} stands for the Hausdorff dimension in (IR^{2}, D_{θ_{1}θ_{2}}).

To prove this assertion it is sufficient to observe that N_{1}(x) ≤ 1 implies

(N_{θ_{1}θ_{2}}(x))^{θ_{2}} ≤ N_{1}(x) ≤ (N_{θ_{1}θ_{2}}(x))^{θ_{1}} ,

and then to proceed as in the final part of Proposition 6.1.

Also Corollary 6.1 can be rewritten in this new framework. In fact, under the same hypotheses on the system Ψ, formula (6.3) takes the form

\frac{1}{θ_{2}} d_{Ψ}Γ ≤ dim_{H} Γ ≤ \frac{1}{θ_{1}} d_{Ψ}Γ ,  (6.3)′

which for regular anisotropic fractals coincides with (6.4).

Acknowledgement. This research was supported by M.U.R.S.T. and by the University of Pavia.

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Received 11.11.1998