Lower Hausdorff dimension estimates for k -1-endomorphisms

Astrid Franz

Department of Mathematics University of Technology Dresden Mommsenstr. 13 D-01062 Dresden

e-mail: mirle@math.tu-dresden.de

Abstract. For a special class of non-injective maps on Riemannian manifolds a lower bound for the Hausdorff dimension of invariant sets is given in terms of the singular values of the tangent map. This estimation can be reached using Frostman’s lemma based on an outer measure which is concentrated on the invariant set. But it can also be proved using a technique by Douady and Oesterlé and its generalization to Riemannian manifolds by Noack and Reitmann, which in general only gives upper dimension estimates. Here this technique is applied to modify a lower dimension estimate by Shereshevskij for geometric constructions on the real line (also described by Barreira). The lower dimension estimate for k -1-endomorphisms can for instance be applied to Julia sets of quadratic maps on the complex plane, but it can also be useful for injective maps like horseshoe maps. It can be combined with an upper estimate to get an interval for the Hausdorff dimension.

1 Introduction

Many chaotic systems are characterized by a very complicated dynamics on invariant sets and a complex geometric structure of these sets, given by a non-integer Hausdorff dimension. In a lot of examples the invariant sets are not known, only the dynamics can be described by a map. Therefore the Hausdorff dimension of an invariant set often can not be computed explicitly, but properties of the map can be used to find upper and lower estimates.

In [3] an upper bound for the Hausdorff dimension of compact negatively invariant sets K of smooth maps φ defined on open subsets of ℝ^{n} is given. Using properties of φ for a given ε > 0 a cover of K by balls of radii at most ε is constructed such that the sum of the radii to the power d is contracting under the linearized map. Then d is an upper bound for the Hausdorff dimension of K. This estimate is generalized in [9] to maps on Riemannian manifolds. In [8] a special class of non-injective maps, the so-called k -1-endomorphisms, are considered. They always assign at least k different points to the same image. For such maps the conditions of the theorems described above can be modified considering the k different inverse maps.

Lower dimension estimations often are based on Frostman’s lemma using an outer measure concentrated on the invariant set (see for instance [10]). For k -1-endomor-phisms a suitable outer measure exists which can be used to find a lower Hausdorff dimension estimate of the invariant set. In [12] Shereshevskij described a geometric construction on the real line (see also [1]) and gave a lower bound of the Hausdorff dimension of such a set without using a measure construction. There analogously to [3] for a given cover of the invariant set by balls the sum over the radii to the power d is estimated. In this paper ideas of this method are applied to invariant sets of k -1-endomorphisms on general Riemannian manifolds. In this way we can prove the lower dimension estimate without using a measure construction.

2 Hausdorff dimension and singular values

Let us first recall the basic notations used in this paper. For a metric space (X, ϱ), a compact set K ⊂ X and arbitrary real numbers ε > 0 and d ≥ 0 we consider an outer measure at level ε and of order d of K given by

μ(K, d, ε) = inf ∑_{i} r_{i}^{d},

where the infimum is taken over all finite covers of K by balls of radii r_{i} ≤ ε. For fixed d and K the limit of the monotone decreasing function μ(K, d, ·)

μ(K, d) = lim_{ε→0+0} μ(K, d, ε)

is called the Hausdorff outer measure of order d of K. For every compact set K there exists a critical number d^{∗} ≥ 0 with

μ(K, d) = \{ \begin{matrix}∞ & for & 0 ≤ d < d^{∗} \\ 0 & for & d > d^{∗}. \end{matrix}

This critical number is the Hausdorff dimension of K and denoted by dim_{H}(K).

To estimate the Hausdorff dimension we need the singular values of linear operators. Consider a linear operator L : E → E′ between two n-dimensional Euclidean spaces E and E′ with the scalar products ⟨·, ·⟩_{E} and ⟨·, ·⟩_{E′}, respectively, and let L^{∗} denote the adjoint operator of L, i.e. the unique linear operator L^{∗} : E′ → E satisfying ⟨Lx, y⟩_{E′} = ⟨x, L^{∗}y⟩_{E} for all x ∈ E and y ∈ E′. The singular values of L are defined as the eigenvalues of the positive semi-definite operator \sqrt{L^{∗}L}. They are denoted by α_{1}(L) ≥ … ≥ α_{n}(L). For an arbitrary j ∈ {0, 1, …, n} we introduce for L the notation

ω_{j}(L) = \{ \begin{matrix}α_{1}(L) · . . . · α_{j}(L) & for j > 0, \\ 1 & for j = 0. \end{matrix}

For an arbitrary number d ∈ (0, n], further on written in the form d = d_{0} + s with d_{0} ∈ {0, 1, …, n − 1} and s ∈ (0, 1], we define a singular value function

ω_{d}(L) = ω_{d_{0}}^{1−s}(L) · ω_{d_{0}+1}^{s}(L).

3 The class of k -1-endomorphisms

Now we want to define a special class of non-injective maps which always assign at least k different points to the same image.

Definition 1. Let (M, g) be an n-dimensional Riemannian manifold. A k -1-en-domorphism system (φ, K, K_{1}, …, K_{k}) is a (k + 2)-tuple (k ∈ ℕ) consisting of a map φ : K → M and (k + 1) compact sets K ⊂ M, K_{1} ⊂ M, …, K_{k} ⊂ M with K = ⋃_{i=1}^{k} K_{i}, φ(K_{i}) = K for all i = 1, …, k and K_{i} ∩ K_{j} = ∅ for any 1 ≤ i < j ≤ k.

The map φ is then called a k -1-endomorphism. The definition implies φ(K) = K, i.e. K is invariant under φ. Furthermore every point of the set K has at least k different preimages under the map φ, one in every of the sets K_{1}, …, K_{k}. If the manifold M is smooth (for simplicity C^{∞}) then we can additionally define the following property.

(C1) Every partial map φ|_{K_{i}} can be extended to a C^{1}-map φ_{i} on an open neighbourhood U_{i} ⊂ M of K_{i} (i = 1, …, k).

The following property is sharper than (C1) and ensures that every point of K has exactly k preimages under φ.

(C2) Every partial map φ|_{K_{i}} can be extended to a C^{1}-diffeomorphism φ_{i} on an open neighbourhood U_{i} ⊂ M of K_{i} (i = 1, …, k).

Since K_{1}, …, K_{k} are compact and pairwise disjoint the neighbourhoods U_{i} (i = 1, …, k) defined in (C1) or (C2) can be chosen pairwise disjoint. Thus the map φ can be extended to a C^{1}-map defined on U = ⋃_{i=1}^{k} U_{i}, which coincides with φ_{i} on U_{i} (i = 1, …, k). This extension will be denoted by φ as well.

Example 1. We consider M = ℝ and the tent map φ : ℝ → ℝ defined as

φ(u) = \{ \begin{matrix}3u & for u < \frac{1}{2}, \\ 3 − 3u & for u ≥ \frac{1}{2}. \end{matrix}

For points u < 0 or u > 1 we have lim_{p→∞} φ^{p}(u) = −∞, thus a compact invariant set of this map must be contained in [0, 1]. Let K be the maximal compact invariant set of φ, i.e. K = ⋂_{p=0}^{∞} φ^{−p}([0, 1]) = ⋂_{p=0}^{∞} K^{p} with K^{p} = ⋂_{i=0}^{p} φ^{−i}([0, 1]). Then the sets K^{p} (p = 1, 2, …) have the form K^{0} = [0, 1], K^{1} = [0, \frac{1}{3}] ∪ [\frac{2}{3}, 1], K^{2} = [0, \frac{1}{9}] ∪ [\frac{2}{9}, \frac{1}{3}] ∪ [\frac{2}{3}, \frac{7}{9}] ∪ [\frac{8}{9}, 1] and so on, i.e. K is the middle-third Cantor set.

The sets K_{1} = K ∩ [0, \frac{1}{3}] and K_{2} = K ∩ [\frac{2}{3}, 1] are compact as well. They are mapped onto the whole set K and do not intersect each other. Thus (φ, K, K_{1}, K_{2}) is a 2-1-endomorphism system.

The map φ, restricted to each set K_{i} (i = 1, 2), is linear and therefore, φ_{i} := φ|_{K_{i}} can be extended to a C^{1}-map on a neighbourhood U_{i} of K_{i}. Thus the here considered 2-1-endomorphism system satisfies the property (C1). Furthermore the linear maps φ_{i} (i = 1, 2) are invertible, therefore also (C2) is satisfied. □

Example 2. A well-known example of maps, which in general assign k different points to the same image, are polynomials of degree k in the two-dimensional manifold ℂ. Let us consider on ℂ the quadratic map

φ(z) = z^{2} + c

where c ∈ ℂ is a parameter. The closure K of the set of all repelling periodic points yields the Julia set (see for instance [11]). This set is invariant under the map. In [4] in Theorem 14.5 Falconer considered parameters c ∈ ℂ with |c| > \frac{1}{4}(5+2\sqrt{6}). He showed that in this case the Julia set is totally disconnected and consists of two disjoint sets K_{1} and K_{2} with φ(K_{1}) = φ(K_{2}) = K. Thus (φ, K, K_{1}, K_{2}) is a 2-1-endomorphism system. Here also conditions (C1) and (C2) are satisfied. □

Now we analyse the iteration of k -1-endomorphisms. It will be shown that the p-th power φ^{p} of a k -1-endomorphism φ is a k^{p}-1-endomorphism for any natural number p.

Notation. For any natural number p we will denote the subsets of K related to the map φ^{p} and the corresponding partial maps by

\begin{matrix}K_{i_{1},... ,i_{p}} := φ^{−(p−1)}(K_{i_{1}}) ∩ … ∩ φ^{−1}(K_{i_{p−1}}) ∩ K_{i_{p}}, \\ φ_{i_{1},... ,i_{p}} := φ^{p}|_{K_{i_{1},... ,ip}}. \end{matrix}

Lemma 1. Let (M, g) be a smooth n-dimensional Riemannian manifold, and let (φ, K, K_{1}, …, K_{k}) be a k -1-endomorphism system satisfying property (C1). Then for any natural number p the (k^{p} + 2)-tuple (φ^{p}, K, {K_{i_{1},... ,i_{p}}}_{(i_{1},... ,i_{p})∈{1,... ,k}^{p}}) is a k^{p}-1-endomorphism system and satisfies (C1).

Proof. This proof can be done by induction. It is clear that for p = 1 the assertion is true. Suppose that for an arbitrary number p ∈ ℕ the (k^{p} + 2)-tuple (φ^{p}, K, {K_{i_{1},... ,i_{p}}}_{(i_{1},... ,i_{p})∈{1,... ,k}^{p}}) is a k^{p}-1-endomorphism system. Now we are going to prove the assertion for p + 1. Let (i_{1}, …, i_{p+1}) ∈ {1, …, k}^{p+1} be an arbitrary (p + 1)-tuple. The set φ^{−1}(K_{i_{1},... ,i_{p}}) is the preimage of the compact set K_{i_{1},... ,i_{p}} under the continuous map φ with compact domain K and therefore, it is compact. Furthermore the relation

φ^{−1}(K_{i_{1},... ,i_{p}}) = φ^{−1}(φ^{−p+1}(K_{i_{1}}) ∩ … ∩ K_{i_{p}}) = φ^{−p}(K_{i_{1}}) ∩ … ∩ φ^{−1}(K_{i_{p}})

holds. Therefore any set K_{i_{1},... ,i_{p+1}} is compact as the intersection of the two compact sets K_{i_{p+1}} and φ^{−1}(K_{i_{1},... ,i_{p}}). Obviously two sets K_{i_{1},... ,i_{p+1}} and K_{j_{1},... ,j_{p+1}} with (i_{1}, …, i_{p+1}) ≠ (j_{1}, …, j_{p+1}) are disjoint. Furthermore we have

⋃_{(i_{1},... ,i_{p+1})∈{1,... ,k}^{p+1}} K_{i_{1},... ,i_{p+1}} = ⋃_{(i_{1},... ,i_{p+1})∈{1,... ,k}^{p+1}} (φ^{−1}(K_{i_{1},... ,i_{p}}) ∩ K_{i_{p+1}})

= ⋃_{(i_{1},... ,i_{p})∈{1,... ,k}^{p}} ⋃_{i_{p+1}=1}^{k} (φ^{−1}(K_{i_{1},... ,i_{p}}) ∩ K_{i_{p+1}})

= ⋃_{(i_{1},... ,i_{p})∈{1,... ,k}^{p}} φ^{−1}(K_{i_{1},... ,i_{p}})

= φ^{−1} (  ⋃_{(i_{1},... ,i_{p})∈{1,... ,k}^{p}} K_{i_{1},... ,i_{p}} )

= φ^{−1}(K) = K.

From the equality

\begin{aligned}φ(K_{i_{1},... ,i_{p+1}}) & = φ (φ^{−1}(K_{i_{1},... ,i_{p}}) ∩ K_{i_{p+1}}) = K_{i_{1},... ,i_{p}} ∩ φ(K_{i_{p+1}}) \\ & = K_{i_{1},... ,i_{p}} ∩ K = K_{i_{1},... ,i_{p}} \end{aligned}

together with the assumption φ^{p}(K_{i_{1},... ,i_{p}}) = K the relation φ^{p+1}(K_{i_{1},... ,i_{p+1}}) = K follows.

Finally (C1) has to be shown for φ^{p+1}. For every partial map φ_{i_{1},... ,i_{p+1}} we have the equality φ_{i_{1},... ,i_{p+1}} = φ_{i_{1},... ,i_{p}} ∘ φ_{i_{p+1}} on the set K_{i_{1},... ,i_{p+1}}. Because of (C1) the map φ_{i_{p+1}} can be extended to a C^{1}-map on a neighbourhood U_{i_{p+1}} of K_{i_{p}+1}. The same is true for φ_{i_{1},... ,i_{p}} and some neighbourhood U_{i_{1},... ,i_{p}} of K_{i_{1},... ,i_{p}}. Therefore the set U_{i_{1},... ,i_{p+1}} := U_{i_{p+1}} ∩ φ_{i_{p+1}}^{−1}(U_{i_{1},... ,i_{p}}) is a neighbourhood of K_{i_{1},... ,i_{p+1}}, on which the map φ_{i_{1},... ,i_{p},i_{p+1}} can be extended as C^{1}-map.

Consequently the validity of the result for p + 1 is shown and therefore, Lemma 1 is proved. ■

Analogously the following statement can be proved.

Lemma 2. Let (M, g) be a smooth n-dimensional Riemannian manifold, and let (φ, K, K_{1}, …, K_{k}) be a k -1-endomorphism system satisfying property (C2). Then for any natural number p the (k^{p} + 2)-tuple (φ^{p}, K, {K_{i_{1},... ,i_{p}}}_{(i_{1},... ,i_{p})∈{1,... ,k}^{p}}) is a k^{p}-1-endomorphism system and satisfies (C2).

Under the condition (C2) in [8] the following upper bound for the Hausdorff dimension of invariant sets of k -1-endomorphisms is given.

Theorem 1. Suppose that (M, g) is a smooth n-dimensional Riemannian manifold and (φ, K, K_{1}, …, K_{k}) is a k -1-endomorphism system satisfying (C2). If a number d ∈ (0, n] exists with

max_{i=1,... ,k} sup_{u∈K} ω_{d}(d_{u}φ_{i}^{−1}) < \frac{1}{k}  (1)

then the Hausdorff dimension of K can be estimated by

dim_{H}(K) ≤ d.

4 The lower Hausdorff dimension estimate

Theorem 2. Suppose that (M, g) is a smooth n-dimensional Riemannian manifold, (φ, K, K_{1}, …, K_{k}) is a k -1-endomorphism system with k ≥ 2 satisfying (C1) and

sup_{u∈K} α_{1}(d_{u}φ) ≥ k^{\frac{1}{n}} .  (2)

Then the Hausdorff dimension of K can be estimated by

dim_{H}(K) ≥ \frac{ln k}{ln sup_{u∈K} α_{1}(d_{u}φ)}.

Lower Hausdorff dimension estimates often are based on Frostman’s lemma (see for instance [10]). Here we use a modified version as in [5].

Lemma 3. Let μ be a finite outer measure defined on the subsets of a metric space (X, ϱ) which is concentrated on K ⊂ X and μ(K) > 0 holds. If there exist numbers d ≥ 0, c > 0 and ε_{0} > 0 such that for any r ∈ (0, ε_{0}] and any x ∈ K the relation μ(B(x, r)) ≤ cr^{d} is satisfied, then dim_{H}(K) ≥ d.

For a k -1-endomorphism-system (φ, K, K_{1}, …, K_{k}) on a smooth n-dimensional Riemannian manifold M we can construct an outer measure as follows. Let S be the set system containing all sets of the form K_{i_{1},... ,i_{p}} ((i_{1}, …, i_{p}) ∈ {1, …, k}^{p}, p ∈ ℕ). For all sets in S the outer measure μ is defined as

μ(K_{i_{1},... ,i_{p}}) = \frac{1}{k^{p}} .

For an arbitrary set A ⊂ M we define

μ(A) = inf ∑ μ(E_{i}), A ⊂ ⋃ E_{i}, E_{i} ∈ S.

In this way we get an outer measure defined for all subsets of M which is concentrated on K, and furthermore μ(K) = 1 holds. With this outer measure we can prove Theorem 2 using Lemma 3.

Proof 1 of Theorem 2. Let d ∈ (0, \frac{ln k}{ln sup_{u∈K} α_{1}(d_{u}φ)}) be an arbitrary number (since k ≥ 2 this interval is not empty). Because of sup_{u∈K} α_{1}(d_{u}φ) > 1 we find a number α > sup_{u∈K} α_{1}(d_{u}φ) with

k = α^{d}.

There exists an open set V ⊂ U containing K such that α_{1}(d_{u}φ) ≤ α for any u ∈ V holds. Let ϱ(·, ·) denote the geodesic distance on M and δ_{1} := inf{ϱ(u_{i}, u_{j}) | u_{i} ∈ K_{i}, u_{j} ∈ K_{j}, 1 ≤ i < j ≤ k} denote the least distance between the sets K_{1}, …, K_{k}. The sets K_{1}, …, K_{k} are compact and pairwise disjoint, therefore δ_{1} > 0. Furthermore there exists a number δ_{2} > 0 such that any ball of radius δ_{2} containing points of K is entirely contained in V. Let δ ∈ (0, min{δ_{1}, δ_{2}}) be an arbitrary number.

Let us consider an arbitrary number p ∈ ℕ, two different p-tuples (i_{1}, …, i_{p}), (j_{1}, …, j_{p}) ∈ {1, …, k}^{p} and two arbitrary points u ∈ K_{i_{1},... ,i_{p}}, v ∈ K_{j_{1},... ,j_{p}}. By induction we want to show ϱ(u, v) > δα^{1−p}. For p = 1 this is obviously satisfied. For p > 1 let us consider two cases. If i_{p} ≠ j_{p} then ϱ(u, v) ≥ δ_{1} > δ and, because of α > 1, the relation δ > δα^{1−p} holds. If i_{p} = j_{p} then (i_{1}, …, i_{p−1}) ≠ (j_{1}, …, j_{p−1}), therefore by assumption we have ϱ(φ(u), φ(v)) > δα^{2−p}. For ϱ(u, v) > δ the assertion is true. In the case ϱ(u, v) ≤ δ we have v ∈ B(u, δ_{2}) ⊂ V ⊂ U, i.e. the map φ is defined on the whole ball B(u, δ_{2}) containing v. Since φ is C^{1} on U, the Taylor formula yields ϱ(φ(u), φ(v)) ≤ αϱ(u, v). Therefore we get ϱ(u, v) > α^{−1}δα^{2−p} = δα^{1−p}.

Let 0 < r ≤ \frac{δ}{2} be an arbitrary number. Then there exists a number p ∈ ℕ with \frac{δ}{2}α^{−p} < r ≤ \frac{δ}{2}α^{1−p}. Every ball B(u, r) with arbitrary centre u ∈ M can intersect at most one of the sets K_{i_{1},... ,i_{p}} ((i_{1}, …, i_{p}) ∈ {1, …, k}^{p}). Therefore with the outer measure μ defined above we have

μ(B(u, r)) ≤ \frac{1}{k^{p}} = α^{−pd} ≤ (\frac{2r}{δ})^{d} = (\frac{2}{δ})^{d} r^{d}.

Lemma 3 yields dim_{H}(K) ≤ d. ■

Now we want to show a way of proving Theorem 2 similar to the techniques of Douady and Oesterlé where we do not need an outer measure. In [12] a lower Hausdorff dimension estimate (together with an upper dimension bound) of a geometric construction on the real line is given which can also be compared with results of [1].

Lemma 4. Let k > 1 be a natural number and for any p ∈ ℕ and any p-tuple (ω_{1}, …, ω_{p}) ∈ {1, …, k}^{p} there exists an interval D_{ω_{1},... ,ω_{p}} ⊂ ℝ with the following properties:

1) If (ω_{1}, …, ω_{p}) ≠ (ω′_{1}, …, ω′_{p}) then D_{ω_{1},... ,ω_{p}} ∩ D_{ω′_{1},... ,ω′_{p}} = ∅.
2) For any (ω_{1}, …, ω_{p}, ω_{p+1}) ∈ {1, …, k}^{p+1} there is D_{ω_{1},... ,ω_{p+1}} ⊂ D_{ω_{1},... ,ω_{p}}.

If there are numbers α > 0 and r ∈ (0, 1) such that

|D_{ω_{1},... ,ω_{p}}| ≥ αr^{p} for any p ∈ ℕ, (ω_{1}, …, ω_{p}) ∈ {1, …, k}^{p}

then the Hausdorff dimension of

D = ⋂_{p=1}^{∞} ⋃_{(ω_{1},... ,ω_{p})∈{1,... ,k}^{p}} D_{ω_{1},... ,ω_{p}}

can be estimated by dim_{H}(D) ≥ −\frac{ln k}{ln r}.

For k -1-endomorphisms the structure of the sets K_{i_{1},... ,i_{p}} is similar to the sets D_{i_{1},... ,i_{p}} of Lemma 4: The conditions 1) and 2) are satisfied if we reverse the order of the indices. The only problem is that for the sets K_{i_{1},... ,i_{p}} we do not have a quantity like the interval length |D_{i_{1},... ,i_{p}}|. We can not use the n-dimensional volume since in general the dimension of K is less than n and therefore the n-volume of the sets K_{i_{1},... ,i_{p}} is zero. In the proof of Lemma 4 in [12] the interval length is used to estimate the number of sets D_{i_{1},... ,i_{p}} which can at most be covered by a ball of a given radius. The same can be done in general metric spaces using the distance between the sets.

Lemma 5. Let (X, ϱ) be a metric space, k > 1 be a natural number and for any p ∈ ℕ and any p-tuple (ω_{1}, …, ω_{p}) ∈ {1, …, k}^{p} there exists a nonempty compact set D_{ω_{1},... ,ω_{p}} ⊂ X with the properties 1) and 2) of Lemma 4. If there are numbers α > 0 and r ∈ (0, 1) such that

\begin{aligned}ϱ(x, y) ≥ αr^{p} for any & x ∈ D_{ω_{1},... ,ω_{p}}, y ∈ D_{ω′_{1},... ,ω′_{p}}, p ∈ ℕ, \\ & (ω_{1}, …, ω_{p}) ≠ (ω′_{1}, …, ω′_{p}) ∈ \{1, …, k\}^{p} \end{aligned}  (3)

then the Hausdorff dimension of

D = ⋂_{p=1}^{∞} ⋃_{(ω_{1},... ,ω_{p})∈{1,... ,k}^{p}} D_{ω_{1},... ,ω_{p}}

can be estimated by dim_{H}(D) ≥ −\frac{ln k}{ln r}.

Proof. For any p ∈ ℕ the sets D_{i_{1},... ,i_{p}} ((i_{1}, …, i_{p}) ∈ {1, …, k}^{p}) further on are called p-sets. For d = −\frac{ln k}{ln r} we have

k = r^{−d}.  (4)

Let \{B(u_{i}, r_{i})\}_{i∈I} be an arbitrary finite cover of D by balls of radii r_{i} ≤ \frac{αr}{2} where I is a finite index set. Then for any i ∈ I there exists a number p_{i} ∈ ℕ with

\frac{α}{2}r^{p_{i}+1} < r_{i} ≤ \frac{α}{2}r^{p_{i}}

With P := max_{i∈I} p_{i} and N_{p} := card{i ∈ I | p_{i} = p} (p = 1, …, P) we get

∑_{i∈I} r_{i}^{d} > ∑_{i∈I} (\frac{α}{2}r^{p_{i}+1})^{d} = ∑_{p=1}^{P} N_{p} (\frac{α}{2}r^{p+1})^{d} = (\frac{αr}{2})^{d} ∑_{p=1}^{P} N_{p}r^{pd}.

Using (4) yields

∑_{i∈I} r_{i}^{d} > (\frac{αr}{2})^{d} ∑_{p=1}^{P} N_{p}k^{−p}.  (5)

Because of r_{i} ≤ \frac{α}{2}r^{p_{i}} a ball of radius r_{i} can cover points of just one of the p_{i}-sets. Every set D_{j_{1},... ,j_{p_{i}}} contains k^{P −p_{i}} P -sets. Since {B(u_{i}, r_{i})}_{i∈I} is a cover of D it covers points of all P -sets, because D ∩ V ≠ ∅ for any p-set V and any p ∈ ℕ. Therefore the relation

∑_{i∈I} k^{P −p_{i}} = ∑_{p=1}^{P} N_{p}k^{P −p} ≥ k^{P}

has to be satisfied. This is equivalent to ∑_{p=1}^{P} N_{p}k^{−p} ≥ 1. Together with (5) we get ∑_{i∈I} r_{i}^{d} > (\frac{αr}{2})^{d}, and this relation holds for any arbitrary cover of D by balls of radii r_{i} ≤ \frac{αr}{2}. Therefore μ (D, d, \frac{αr}{2}) ≥ (\frac{αr}{2})^{d} > 0 holds. Since μ(D, d, ·) is monotone decreasing we get μ(D, d) > 0 and thus dim_{H}(D) ≥ d = −\frac{ln k}{ln r}. ■

Now we can prove Theorem 2 based on Lemma 5.

Proof 2 of Theorem 2. The sets K_{i_{1},... ,i_{p}}, (i_{1}, …, i_{p}) ∈ {1, …, k}^{p}, p ∈ ℕ, satisfy the conditions 1) and 2) of Lemma 4, if we reverse the order of the indices. Furthermore, analogously to Proof 1 for any d ∈ (0, \frac{ln k}{ln sup_{u∈K} α_{1}(d_{u}φ)}) and α = k^{\frac{1}{d}} condition (3) is satisfied in the form ϱ(u, v) > δα^{1−p} = δα (\frac{1}{α})^{p} for arbitrary p ∈ ℕ, two different p-tuples (i_{1}, …, i_{p}), (j_{1}, …, j_{p}) ∈ {1, …, k}^{p} and two arbitrary points u ∈ K_{i_{1},... ,i_{p}}, v ∈ K_{j_{1},... ,j_{p}}. Thus Lemma 5 gives dim_{H}(K) ≥ \frac{ln k}{ln α} = d. In the limit this yields dim_{H}(K) ≥ \frac{ln k}{ln sup_{u∈K} α_{1}(d_{u}φ)}. ■

5 Application examples

Example 3. (Example 1 continued) Let us again consider the tent map in ℝ. For this map we have

α_{1}(d_{u}φ) = 3

independent of the point u ∈ K where K is the invariant set introduced above. So condition (2) is satisfied. Theorem 2 yields the lower bound dim_{H}(K) ≥ \frac{ln 2}{ln 3}.

Applying Theorem 1 we also get an upper bound for the Hausdorff dimension of K. Condition (1) here has the form (\frac{1}{3})^{d} < \frac{1}{2} which is satisfied for any d > \frac{ln 2}{ln 3}. Thus we get dim_{H}(K) ≤ d for any such d and in the limit dim_{H}(K) ≤ \frac{ln 2}{ln 3}. Hence in this example Theorems 1 and 2 combined give the exact value dim_{H}(K) = \frac{ln 2}{ln 3} for the Hausdorff dimension of the middle-third Cantor set K. □

Example 4. (Example 2 continued) Now let us again look at the Julia set K of a quadratic map φ : ℂ → ℂ of the form φ(z) = z^{2} + c with |c| > \frac{1}{4}(5 + 2\sqrt{6}). Here for any z ∈ ℂ we have two equal singular values

α_{1}(d_{z}φ) = α_{2}(d_{z}φ) = 2|z|

and therefore,

sup_{z∈K} α_{1}(d_{z}φ) = 2 sup_{z∈K} |z|.

Following the proof of Theorem 14.5 of [4] we note that the set V = {z ∈ ℂ | |z| ≤ \sqrt{|2c|}\} contains K and for any point z ∈ V the relations

\frac{1}{2\sqrt{|c| + \sqrt{|2c|}}} ≤ α_{1}(d_{z}φ_{i}^{−1}) ≤ \frac{1}{2\sqrt{|c| − \sqrt{|2c|}}} (i = 1, 2)  (6)

hold. Thus we have

2\sqrt{|c| − \sqrt{|2c|}} ≤ α_{1}(d_{z}φ) ≤ 2\sqrt{|c| + \sqrt{|2c|}}

for any z ∈ K. Therefore sup_{z∈K} α_{1}(d_{z}φ) can be estimated by

2\sqrt{|c| − \sqrt{|2c|}} ≤ sup_{z∈K} α_{1}(d_{z}φ) ≤ 2\sqrt{|c| + \sqrt{|2c|}}.

So condition (2) is satisfied, if 2\sqrt{|c| − \sqrt{|2c|}} ≥ \sqrt{2} holds, which is equivalent to

|c| ≥ \frac{3 + \sqrt{8}}{2}.  (7)

In this case Theorem 2 can be applied and gives

dim_{H}(K) ≥ \frac{ln 2}{sup_{z∈K} α_{1}(d_{z}φ)} ≥ \frac{ln 2}{ln (2\sqrt{|c| + \sqrt{|2c|}})}.

Using Theorem 1 we can also get an upper bound for the Hausdorff dimension of K.

Relation (6) yields ω_{d}(d_{z}φ_{i}^{−1}) ≤ (_{\frac{1}{2\sqrt{|c|−\sqrt{|2c|}}}})^{d}. Thus condition (1) has the form

(\frac{1}{2\sqrt{|c| − \sqrt{|2c|}}})^{d} < \frac{1}{2}.

Under condition (7) this inequality can be satisfied for numbers d ≤ 2, namely for d ∈ (_{\frac{ln 2}{ln(2\sqrt{|c|−\sqrt{|2c|}})}}, 2]. For any such d Theorem 1 gives dim_{H}(K) ≤ d, in the limit

this yields

dim_{H}(K) ≤ \frac{ln 2}{ln (2\sqrt{|c| − \sqrt{|2c|}})}.

Therefore, for |c| ≥ \frac{3+\sqrt{8}}{2} we get an interval for the Hausdorff dimension of K as

\frac{ln 2}{ln (2\sqrt{|c| + \sqrt{|2c|}})} ≤ dim_{H}(K) ≤ \frac{ln 2}{ln (2\sqrt{|c| − \sqrt{|2c|}})}.

These are the same lower and upper bounds as reached in [4]. For large |c| this yields the asyptotic behaviour dim_{H}(K) ∼ \frac{2 ln 2}{ln |c|} (see also [11]). □

Remark. Following [6] the Hausdorff dimension of a Julia set K of the quadratic map φ(z) = z^{2} + c for |c| > \frac{1}{4}(5 + 2\sqrt{6}) can also be estimated from below using the thermodynamic formalism. The Hausdorff dimension dim_{H}(K) is equal to the unique root d of Bowen’s equation

P (−d ln ∥dφ∥) = 0

where P is the pressure defined as

P (−d ln ∥dφ∥) = sup_{μ} {h_{μ} − d ∫_{K} ln ∥dφ∥dμ}

and the supremum is taken over all φ-invariant probability measures μ on K. The outer measure μ for general k -1-endomorphisms used in Proof 1 of Theorem 2 in this special case of Julia sets is an invariant probability measure. Therefore dim_{H}(K) satisfies the inequality

h_{μ} − dim_{H}(K) ∫_{K} ln ∥dφ∥dμ ≤ 0

which is equivalent to dim_{H}(K) ≥ \frac{h_{μ}}{∫_{K} ln ∥dφ∥dμ}. The entropy h_{μ} of φ with respect to the measure μ is in this case ln 2 and the norm of dφ is its largest singular value, therefore we get

dim_{H}(K) ≥ \frac{ln 2}{∫_{K} ln α_{1}(dφ)dμ}.

If we estimate ∫_{K} ln α_{1}(dφ)dμ ≤ sup_{u∈K} α_{1}(d_{u}φ) this inequality results in the statement of Theorem 2.

This means if for an invariant set of a k -1-endomorphism an invariant probability measure is known and the thermodynamic formalism can be applied then the estimate of Theorem 2 may be improved. The advantage of Theorem 2 is that it is also applicable if no invariant probability measure is known.

Example 5. Let us consider a class of horseshoe maps (see [2]) defined in the following way. Let I ⊂ ℝ denote the unit interval [0, 1] and I^{m} = I × … × I ⊂ ℝ^{m} the m-dimensional unit cube for any m ∈ ℕ. A horseshoe map in ℝ^{m+1} (m ∈ ℕ) is an injective map f : I^{m+1} → ℝ^{m+1} with the following property:

(H1) There exists a set I^{∗} ⊂ I with f^{−1}(I^{m+1}) = I^{∗} × I^{m} that is a finite union of disjoint closed intervals I^{∗} = ⋃_{i=1}^{k} I_{i} (k ≥ 2), and f in the points u = (x, y) ⊂ I^{∗} × I^{m} can be written in the form

f ((x, y)) = (φ(x), ψ(x) + λy)

with a parameter λ ∈ (0, 1) and C^{1}-maps φ : I^{∗} → I and ψ : I^{∗} → [0, 1 − λ]^{m}.

Let us furthermore assume that the following properties hold.

(H2) The map φ|_{I_{i}} can be extended to a C^{1}-diffeomorphism φ_{i} defined on a neighbourhood U_{i} of I_{i} with φ_{i}(I_{i}) = I.

(H3) The map ψ is constant on every interval I_{i}, i = 1, …, k.

Then there exist functions ψ_{i} : I^{m} → I^{m} (i = 1, …, k) with ψ_{i}(y) = ψ(x) + λy for any x ∈ I_{i} and any y ∈ I^{m}. Every map ψ_{i} is linear and invertible and therefore can be extended to a C^{1}-diffeomorphism on an open neighbourhood of I_{m}. Since f is injective we have ψ_{i}(I^{m}) ∩ ψ_{j}(I^{m}) = ∅ for i ≠ j. Therefore there exists a kind of inverse map ψ̃ : ⋃_{i=1}^{k} ψ_{i}(I^{m}) → I^{m} with ψ̃_{i} = ψ̃|_{ψ_{i}(I^{m})} = ψ_{i}^{−1}, and the partial maps ψ̃_{i} can be extended to C^{1}-diffeomorphisms on an open neighbourhood Ũ_{i} of ψ_{i}(I^{m}).

Thus the maximal invariant set K = ⋂_{p=−∞}^{∞} f^{p}(I^{∗} × I^{m}) of the horseshoe map can be written as product

K = ⋂_{p=−∞}^{∞} φ^{p}(I^{∗}) × ⋂_{p=−∞}^{∞} ψ̃^{p}(I^{m}) = ⋂_{p=−∞}^{0} φ^{p}(I^{∗}) × ⋂_{p=−∞}^{0} ψ̃^{p}(I^{m}).

Let us denote K_{φ} = ⋂_{p=−∞}^{0} φ^{p}(I^{∗}) and Kψ̃ = ⋂_{p=−∞}^{0} ψ̃^{p}(I^{m}). Then K_{φ} ⊂ ℝ is a φinvariant set, and Kψ̃ ⊂ ℝ^{m} is ψ̃-invariant. Therefore we can estimate the Hausdorff dimension of these two sets seperately.

Let us first consider the map φ. Since this map acts on a one-dimensional manifold, its derivative has only one singular value α_{1}(d_{x}φ) = |d_{x}φ|. Resulting from (H2) we have d_{x}φ ≠ 0 for all x ∈ I^{∗}, therefore max_{i=1,... ,k} sup_{x∈K_{φ}} ω_{d}(d_{x}φ_{i}^{−1}) = sup_{x∈K_{φ}} \frac{1}{|d_{x}φ|^{d}}. Thus condition (1) can be satisfied for a number d ∈ (0, 1] if sup_{x∈K_{φ}} \frac{1}{|d_{x}φ|} < \frac{1}{k}, i.e.

inf_{x∈K_{φ}} |d_{x}φ| > k.  (8)

In this case Theorem 1 yields dim_{H}(K_{φ}) ≤ \frac{ln k}{ln inf_{x∈Kφ} |d_{x}φ|}. Since condition (2) for φ fol-

lows from condition (8), Theorem 2 gives the lower bound dim_{H}(K_{φ}) ≥ \frac{ln k}{ln sup_{x∈Kφ} |d_{x}φ|}. Thus under condition (8) we get

\frac{ln k}{ln sup_{x∈K_{φ}} |d_{x}φ|} ≤ dim_{H}(K_{φ}) ≤ \frac{ln k}{ln inf_{x∈K_{φ}} |d_{x}φ|}.

Now let us consider the set Kψ̃. This set is a uniform Cantor set, since it consists of k linear copies of itself with contracting factor λ (see [4]). Therefore we have

dim_{H}(Kψ̃) = dim_{F} (Kψ̃).  (9)

The singular values of d_{y}ψ̃_{i}^{−1} are α_{1}(d_{y}ψ̃_{i}^{−1}) = … = α_{m}(d_{y}ψ̃_{i}^{−1}) = λ (y ∈ Kψ̃, i = 1, …, k). Thus condition (1) can be satisfied for a number d ∈ (0, m] if

λ^{m} < \frac{1}{k}.  (10)

In this case Theorem 1 yields dim_{H}(Kψ̃) ≤ \frac{ln k}{− ln λ}. Condition (2) for ψ̃ again follows from (10), therefore Theorem 2 is applicable as well and gives dim_{H}(Kψ̃) ≥ \frac{ln k}{− ln λ}. Hence we get

dim_{H}(Kψ̃) = \frac{ln k}{− ln λ}.

For products of two arbitrary sets A ⊂ ℝ, B ⊂ ℝ^{m} the relation

dim_{H}(A) + dim_{H}(B) ≤ dim_{H}(A × B) ≤ dim_{H}(A) + dim_{F} (B)

holds (see [4]). In our case with (9) this means dim_{H}(K) = dim_{H}(K_{φ}) + dim_{H}(Kψ̃). Thus we get

\frac{ln k}{ln sup_{x∈K_{φ}} |d_{x}φ|} + \frac{ln k}{− ln λ} ≤ dim_{H}(K) ≤ \frac{ln k}{ln inf_{x∈K_{φ}} |d_{x}φ|} + \frac{ln k}{− ln λ}  (11)

under the assumptions (8) and (10).

Let us consider an example of a horseshoe map in ℝ^{2} which contracts the unit square [0, 1] × [0, 1] horizontally with factor \frac{1}{4} and stretches it vertically with factor 3 and finally forms it to a horseshoe (see Figure 1).

[figure: ⟶; ⟶; 1; 1; \frac{1}{4}; 3]

Figure 1: Horseshoe map in ℝ^{2}

This map satisfies (H1) - (H3). The set I^{∗} here consists of the two intervals [0, \frac{1}{3}] and [\frac{2}{3}, 1], and the maps φ and ψ̃ are tent maps as described in Example 1:

φ(x) = \{ \begin{matrix}3x & for x ∈ [0, \frac{1}{3}], \\ 3 − 3x & for x ∈ [\frac{2}{3}, 1] \end{matrix} and ψ̃(y) = \{ \begin{matrix}4y & for y ∈ [0, \frac{1}{4}], \\ 4 − 4y & for y ∈ [\frac{3}{4}, 1]. \end{matrix}

With k = 2, sup_{x∈K_{φ}} |d_{x}φ| = inf_{x∈K_{φ}} |d_{x}φ| = 3 and λ = \frac{1}{4} relation (11) gives

\frac{ln 2}{ln 3} + \frac{ln 2}{ln 4} ≤ dim_{H}(K) ≤ \frac{ln 2}{ln 3} + \frac{ln 2}{ln 4},

thus we get the exact Hausdorff dimension dim_{H}(K) = \frac{1}{2} + \frac{ln 2}{ln 3} ≈ 1.131. □

Remark. This example shows that results for k -1-endomorphisms can also be applied to special injective maps to get a lower dimension estimate. Furthermore we get an upper bound for the Hausdorff dimension which is better than the estimate dim_{H}(K) ≤ 1 + \frac{ln 3}{ln 4} ≈ 1.792 reached with a theorem of [3] and the estimate dim_{H}(K) ≤ 2 − \frac{ln 3}{ln 4} ≈ 1.208 which follows from a theorem of [7] considering the hyperbolic structure of the invariant set.

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