Hausdorff dimension estimates for non-injective

maps using the cardinality of the preimage sets

Astrid Franz

Institute of Analysis, Department of Mathematics, University of Technology Dresden, Mommsenstr. 13, 01069 Dresden

Abstract. Upper bounds for the Hausdorff dimension of compact invariant sets of C^{1}-maps on smooth Riemannian manifolds are given in terms of the singular values of the tangent map and the multiplicity function of the map, describing the number of preimages of a certain point in a given set. For non-injective maps this improves previous results using only the singular values.

1. Introduction

The long-time behaviour of a dynamical system {φ^{t}}_{t∈Γ}is mainly determined by asymptotic limit sets which are invariant under the maps φ^{t}. A complex geometric structure of these sets, described by a non-integer Hausdorff or box dimension, is often connected with a chaotic dynamics. Therefore it is useful to determine the dimension of mapinvariant sets. Unfortunately asymptotic limit sets often are not known explicitely, but the maps describing the evolution are given. Therefore properties of the maps can be investigated to find dimension bounds for the invariant sets. This paper is concerned with Hausdorff dimension estimates, but similar results, based on [2], can be formulated for the box dimension, too.

First results in this direction are given in [3] for compact subsets of ℝ^{n} that are invariant under C^{1}-maps. These are generalized in [9] using Lyapunov functions in ℝ^{n}, in [14, 15] to maps on Hilbert and Banach spaces and in [8, 10] to maps on Riemannian manifolds. The main idea consists in showing that the map is contracting with respect to the Hausdorff outer measure on the invariant set. The contraction constant can be estimated by means of a singular value function, i. e. if the singular value function is less than 1, then the map is contracting.

In the present paper it is shown, how the multiplicity function of a map, describing the cardinality of the preimage sets, can be used to prove a contraction of the weighted Hausdorff outer measure although the singular value function in points with more than one preimage does not need to be less than 1. Weighted Hausdorff measures have the same critical value like the original Hausdorff measures and, therefore, can analogously be used to get dimension estimates.

This paper is organized as follows. Definitions and notations are given in Section 2. In Section 3 the multiplicity function is introduced, and basic properties of this function are stated. Section 4 in concerned with the definition of Hausdorff outer integrals. Transformation formulas for these integrals under smooth maps are derived in Section 5. These formulas are used in Section 6 to get upper dimension estimates for map-invariant sets. In Section 7 these estimates are illustrated for two simple examples.

2. Basic definitions and notations

Let us first recall the definition of the Hausdorff dimension. Let (X, ρ) be a metric space and K ⊂ X a compact subset. For arbitrary real numbers ε > 0 and d ≥ 0 we consider the Hausdorff outer (d, ε)-measure 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 B(u_{i}, r_{i}) of radii r_{i} ≤ ε centered in u_{i} ∈ X. For fixed d and K the function μ(K, d, ·) is monotone nonincreasing. Hence the limit

μ(K, d) := lim_{ε→0+0} μ(K, d, ε)

exists, although it may be infinite, and is called the Hausdorff outer d-measure 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 called the Hausdorff dimension of K and denoted by dim_{H}(K).

In order to use singular value functions let us briefly recall the definition of singular values. 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 α_{1}(L) ≥ α_{2}(L) ≥ … ≥ α_{n}(L) of the positive semi-definite operator \sqrt{L^{∗}L}, ordered with respect to their size and multiplicity.

Now let d ∈ [0, n] be an arbitrary number, and let ⌊d⌋ denote the largest integer smaller than d. Then the singular value function of order d of L is defined as

ω_{d}(L) := \{ \begin{matrix}1 & for d = 0, \\ α_{1}(L) · . . . · α_{⌊d⌋}(L)α_{⌊d⌋+1}(L)^{d−⌊d⌋} & for d ∈ (0, n], \end{matrix}

and the inverse singular value function of order d of L is

\overline{ω}_{d}(L) := \{ \begin{matrix}1 & for d = 0, \\ α_{n}(L) · . . . · α_{n−⌊d⌋+1}(L)α_{n−⌊d⌋}(L)^{d−⌊d⌋} & for d ∈ (0, n]. \end{matrix}

In the case d = n we have ω_{n}(L) = \overline{ω}_{n}(L) = | det L|.

A similar concept can be introduced for ellipsoids, too. Let E be an ellipsoid in an n-dimensional Euclidean space E and let α_{1}(E) ≥ … ≥ α_{n}(E) ≥ 0 denote the lengths of the half-axes. Then we define

ω_{d}(E) := \{ \begin{matrix}1 & for d = 0, \\ α_{1}(E) · . . . · α_{⌊d⌋}(E)α_{⌊d⌋}(E)^{d−⌊d⌋} & for d ∈ (0, n]. \end{matrix}

The function ω_{d} has a few important properties which we cite from [2, 3].

Lemma 1. Let E, E′ and E′′ be three n-dimensional Euclidean spaces and L : E → E′ and L′ : E′ → E′′ be two linear maps. Then for all d ∈ [0, n] the relation

ω_{d}(L′ ∘ L) ≤ ω_{d}(L) · ω_{d}(L′)

holds.

Lemma 2. Let E be an n-dimensional Euclidean space and k > 0, m > 0, η > 0 and d ∈ (0, n] be numbers satisfying k ≤ m^{d}. Furthermore let E ⊂ E be an ellipsoid with α_{1}(E) ≤ m and ω_{d}(E) ≤ k. Then E + B(O, η) is contained in an ellipsoid E′ ⊂ E with

ω_{d}(E′) ≤ [1 + (\frac{m^{⌊d⌋}}{k} )^{1/(d−⌊d⌋)} η]^{d} k.

Lemma 3. Let E be an n-dimensional Euclidean space, E ⊂ E an ellipsoid and r > 0 a number. Then E can be covered by ⌊\frac{2^{l}ω_{l}(E)}{r^{l}}⌋ balls of radius r\sqrt{l + 1}, where l is defined as

l := \{ \begin{matrix}0 & for r > α_{1}(E), \\ m & for α_{m+1}(E) ≤ r ≤ α_{m}(E), m ∈ \{1, …, n − 1\}, \\ n & for r ≤ α_{n}(E). \end{matrix}

3. Multiplicity function

The multiplicity function of a map describing the cardinality of the preimage sets is for instance introduced in [5].

Definition 1. Let M_{1} and M_{2} be arbitrary sets and φ : M_{1} → M_{2} be a map. The multiplicity function N(φ, K, u) of the map φ with respect to a set K ⊂ M_{1} in a point u ∈ M_{2} is defined as the cardinality of the set {v ∈ K | φ(v) = u}.

Let us for instance consider a quadratic polynomial φ : ℂ → ℂ in the complex plane, given by φ(z) = z^{2} + c, where c ∈ ℂ is a parameter. Every point of the complex plane except c has in ℂ two preimages, i.e.

N(φ, ℂ, z) = \{ \begin{matrix}2 & for z ≠ c, \\ 1 & for z = c. \end{matrix}

In the case of a C^{1}-map φ on a Riemannian C^{3}-manifold M and a compact set K ⊂ M without critical points, i. e. without points with | det d_{u}φ| = 0, the multiplicity function is bounded.

Lemma 4. Let (M, g) be an n-dimensional Riemannian C^{3}-manifold, U ⊂ M an open set, φ : U → M a C^{1}-map and K ⊂ U a compact set with | det(d_{u}φ)| > 0 for all u ∈ K. Then there exists a number k ∈ ℕ with N(φ, K, v) ≤ k for all v ∈ M.

Proof. Since K is compact and does not contain critical points with respect to φ the implicite function theorem yields a number ε_{K} > 0 such that φ|_{B(u,ε_{K})} is a C^{1}- diffeomorphism for any u ∈ K. The set system {B(u, ε_{K})}_{u∈K} is an open cover of the compact set K and thus contains a finite cover {B(u_{1}, ε_{K}), …, B(u_{k}, ε_{K})} with a certain number k ∈ ℕ. Since φ|_{B(u_{i},ε_{K})} (i = 1, …, k) is a diffeomorphism any point v ∈ M has in every of the balls B(u_{i}, ε_{K}) (i = 1, …, k) at most one preimage. Thus we have N(φ, K, v) ≤ k for any v ∈ M. ■

If K is additionally φ-invariant, then N(φ, K, v) ≥ 1 for all v ∈ K. Therefore terms of the form \frac{1}{N(φ,K,v)} are well-defined. We can consider orbits of arbitrary length p ∈ ℕ and multiply one over the multiplicity function along such orbits.

Lemma 5. Let (M, g) be an n-dimensional Riemannian C^{3}-manifold, U ⊂ M an open set, φ : U → M a C^{1}-map and K ⊂ U a compact φ-invariant set with | det(d_{u}φ)| > 0 for all u ∈ K. Then for any v ∈ K and any p ∈ ℕ we have

∑_{u∈K:φ^{p}(u)=v} (∏_{i=1}^{p} \frac{1}{N(φ, K, φ^{i}(u))}) = 1.

Proof. This statement can be proven by induction. For p = 1 we have

∑_{u∈K:φ(u)=v} \frac{1}{N(φ, K, v)} = N(φ, K, v)\frac{1}{N(φ, K, v)} = 1.

Now we assume that the statement holds for a number p ∈ ℕ. Then for p + 1 we get

∑_{u∈K:φ^{p+1}(u)=v} (∏_{i=1}^{p+1} \frac{1}{N(φ, K, φ^{i}(u))})

= ∑_{u∈K:φ^{p+1}(u)=v} (\frac{1}{N(φ, K, v)} ∏_{i=1}^{p} \frac{1}{N(φ, K, φ^{i}(u))})

= \frac{1}{N(φ, K, v)} ∑_{w∈K:φ(w)=v} ∑_{u∈K:φ^{p}(u)=w} (∏_{i=1}^{p} \frac{1}{N(φ, K, φ^{i}(u))}) .

Since we assumed ∑_{u∈K:φ^{p}(u)=w} (∏_{i=1}^{p} \frac{1}{N(φ,K,φ^{i}(u))}) = 1 the relation

∑_{u∈K:φ^{p+1}(u)=v} (∏_{i=1}^{p+1} \frac{1}{N(φ, K, φ^{i}(u))}) = \frac{1}{N(φ, K, v)}N(φ, K, v) = 1

follows. Thus the statement is true for p + 1. ■

Under the assumption | det(d_{u}φ)| > 0 the map φ is locally invertible. If we want to use these inverse maps for estimating the Hausdorff dimension of an invariant set, then we have to consider an open set Ũ containing K, because on a compact set the multiplicity function needs not to be continuous. Since we furthermore also need properties of compact sets, we additionally assume, that the closure of Ũ is compact. Under the assumption φ^{−1}(Ũ) ⊂ Ũ every point of Ũ has at least one preimage in Ũ. Therefore terms of the form \frac{1}{N(φ,K,v)} are also well-defined for points v ∈ Ũ, and we get a property analogous to Lemma 5.

Lemma 6. Let (M, g) be an n-dimensional Riemannian C^{3}-manifold, U ⊂ M an open set, φ : U → M a C^{1}-map and Ũ an open set, such that cl(Ũ) ⊂ U is a compact set and | det(d_{u}φ)| > 0 for all u ∈ cl(Ũ) and φ^{−1}(Ũ) ⊂ Ũ hold. then for any p ∈ ℕ and any v ∈ Ũ we have

∑_{u∈Ũ,φ^{p}(u)=v} ∏_{i=1}^{p} \frac{1}{N(φ, Ũ, φ^{i}(u))} = 1.

4. Hausdorff outer integrals

Let us again look at the definition of the Hausdorff outer measures. Introducing weight factors for every ball in a considered cover we get Hausdorff outer integrals (see for instance [16]).

Definition 2. Let (X, ϱ) be a metric space and d ≥ 0 and ε > 0 numbers. For a compact subset K ⊂ X and a nonnegative function f : X → ℝ the Hausdorff outer (d, ε)-integral of f on K is defined as

I_{H}(K, f, d, ε) := inf {∑_{i} c_{i}r_{i}^{d} | 0 ≤ c_{i} < ∞, ∑_{i} c_{i}χ_{B(u_{i},r_{i})} ≥ fχ_{K}} ,

where the infimum is taken over all finite covers of K by balls B(u_{i}, r_{i}) of radii r_{i} ≤ ε centered in u_{i} ∈ X. Furthermore

I_{H}(K, f, d) := lim sup_{ε→0+0} I_{H}(K, f, d, ε)

is the Hausdorff outer d-integral of f on K.

These outer integrals correspond to the upper integrals with respect to the Hausdorff outer (d, ε)-measure or d-measure, as introduced in [5]. For the special function f ≡ 1 on X the Hausdorff outer integrals are weighted Hausdorff measures as considered in [7] using general Hausdorff functions.

For fixed numbers d and ε the functions I_{H}(·, ·, d, ε) and I_{H}(·, ·, d) are nonnegative functionals on P(X)×F, where F denotes the family of all nonnegative real functions on X. These functionals have the following properties which can be proven straightforward.

Lemma 7. Let (X, ϱ) be a metric space, J, K ⊂ X compact subsets, f, g : X → ℝ nonnegative functions and d, α ≥ 0 and ε > 0 real numbers. Then

I_{H}(K, αf, d, ε) = αI_{H}(K, f, d, ε),
I_{H}(K, f + g, d, ε) ≤ I_{H}(K, f, d, ε) + I_{H}(K, g, d, ε),
I_{H}(J ∪ K, f, d, ε) ≤ I_{H}(J, f, d, ε) + I_{H}(K, f, d, ε),
I_{H}(K, 1, d, 3ε) ≤ μ_{H}(K, d, 3ε) ≤ 3^{d}I_{H}(K, 1, d, ε),

and, therefore,

I_{H}(K, αf, d) = αI_{H}(K, f, d),
I_{H}(K, f + g, d) ≤ I_{H}(K, f, d) + I_{H}(K, g, d),
I_{H}(J ∪ K, f, d) ≤ I_{H}(J, f, d) + I_{H}(K, f, d),
I_{H}(K, 1, d) ≤ μ_{H}(K, d) ≤ 3^{d}I_{H}(K, 1, d).

The last inequality means, that I_{H}(K, 1, d) as a function of d for a fixed compact set K has a critical value d^{∗∗} with

I_{H}(K, 1, d) = \{ \begin{matrix}∞ & for & 0 ≤ d < d^{∗∗}, \\ 0 & for & d > d^{∗∗}, \end{matrix}

and d^{∗∗} coincides with the critical value d^{∗} = dim_{H}(K) for the Hausdorff outer dmeasure of K. Therefore the Hausdorff outer integral of the function f ≡ 1 can be used in the same way like the Hausdorff outer measure to get dimension estimates.

5. Outer integrals under transformations

Now we want to analyse the behaviour of the Hausdorff outer integrals under smooth transformations.

Theorem 1. Let (M, g) be an n-dimensional Riemannian C^{3}-manifold, U ⊂ M an open set, φ : U → M a C^{1}-map and K ⊂ U a compact set with | det(d_{u}φ)| > 0 for all u ∈ K. Furthermore let f : K → (0, 1] be a nonnegative function with the property

∑_{v∈K:φ(v)=u} f(v) ≥ 1 for all u ∈ φ(K).  (1)

Then for all d ∈ (0, n] and sufficiently small ε > 0 the inequality

I_{H}(φ(K), 1, d, ε) ≤ 16^{n}n^{n}I_{H} (K, max \{fω_{d}(dφ)^{\frac{n}{d}} , f^{\frac{d}{n}} ω_{d}(dφ)\} , d, ε)

holds.

Proof. Let d ∈ (0, n] be an arbitrary fixed number. Since K is compact and φ is C^{1}, there is a sufficiently large number m > 0 with

sup_{u∈K} ω_{d}(d_{u}φ) < m^{d} and sup_{u∈K} ∥d_{u}φ∥ ≤ m,  (2)

where ∥d_{u}φ∥ denotes the operator norm of the linear operator d_{u}φ : T_{u}M → T_{φ(u)}M. Let η > 0 be the number satisfying the equation

[1 + (\frac{m^{⌊d⌋}}{sup_{u∈K} ω_{d}(d_{u}φ)})^{1/(d−⌊d⌋)} η]^{n} = 2^{n−⌊d⌋}.  (3)

Note that | det(d_{u}φ)| > 0 for all u ∈ K implies sup_{u∈K} ω_{d}(d_{u}φ) > 0, therefore all terms in equation (3) are well-defined. For any u ∈ K we define a number l_{u} > 0 by

l_{u} := 2^{n−⌊d⌋}ω_{d}(d_{u}φ)^{\frac{n}{d}} = ω_{d}(d_{u}φ)^{\frac{n}{d}} [1 + (\frac{m^{⌊d⌋}}{sup_{v∈K} ω_{d}(d_{v}φ)})^{1/(d−⌊d⌋)} η]^{n} .  (4)

Then for any u ∈ K there exists a number η_{u} ∈ (0, η] with

l_{u} = ω_{d}(d_{u}φ)^{\frac{n}{d}} [1 + (\frac{m^{⌊d⌋}}{ω_{d}(d_{u}φ)})^{1/(d−⌊d⌋)} η_{u}]^{n} .  (5)

These numbers satisfy

η_{u} ≥ η̃:= (\frac{inf_{v∈K} ω_{d}(d_{v}φ)}{sup_{v∈K} ω_{d}(d_{v}φ)})^{1/(d−⌊d⌋)} η > 0.

Now let r_{1} > 0 be sufficiently small that there exists an open set V with K ⊂ V ⊂ U and

∥τ_{φ(v)}^{φ(u)}d_{v}φτ_{u}^{v} − d_{u}φ∥ ≤ η̃  (6)

for all u, v ∈ V with ϱ(u, v) ≤ r_{1}. Here τ denotes the parallel transport along the geodesics on the manifold M. Furthermore let r_{2} > 0 be so small, that a ball B(u, r_{2}) with centre u ∈ V containing points of K is entirely contained in V and that φ|_{B(u,r_{2})} is a C^{1}-diffeomorphism for any u ∈ K. Resulting from the properties of the exponential map exp : T M → M there exists a number r_{3} > 0 with ∥d_{v} exp_{u} ∥ ≤ 2 for all u ∈ K and all v ∈ B(O_{u}, r_{3}). Thus we have

ϱ(exp_{u} v, exp_{u} w) ≤ 2ϱ(v, w)  (7)

for arbitrary u ∈ K and v, w ∈ B(O_{u}, r_{3}). We set

r_{0} := min \{r_{1}, r_{2}, (\frac{1}{2} + m [1 + (\frac{m^{⌊d⌋}}{sup_{u∈K} ω_{d}(d_{u}φ)})^{1/(d−⌊d⌋)} η])^{−1} r_{3}\}

and consider a fixed number ε ∈ (0, \frac{r_{0}}{2} ). Let \{B(ũ_{i}, r_{i})\}_{i∈I} be an arbitrary finite cover of K by balls of radii r_{i} ≤ ε, such that every ball contains at least one point of K. Furthermore let c_{i} ≥ 0 (i ∈ I) be arbitrary numbers with the property

∑_{i∈I} c_{i}χ_{B(ũ_{i},r_{i})} ≥ max \{fω_{d}(dφ)^{\frac{n}{d}} , f^{\frac{d}{n}} ω_{d}(dφ)\} χ_{K}.  (8)

Since ω_{d}(dφ) is continuous on K, for any i ∈ I there is a point u_{i} ∈ cl(B(ũ_{i}, r_{i})) ∩ K with

ω_{d}(d_{u_{i}}φ) = inf_{u∈B(ũ_{i},r_{i})∩K} ω_{d}(d_{u}φ).  (9)

Then the set system {φ(B(u_{i}, 2r_{i}))}_{i∈I} is a cover of K, too. Because of 2r_{i} ≤ r_{2} every of the balls of this new cover is entirely contained in V. The set sytem {φ(B(u_{i}, 2r_{i}))}_{i∈I} is a cover of φ(K). In the following we consider an arbitrary fixed number i ∈ I. Taylor’s formula for the C^{1}-map φ (see [10]) yields

∥ exp_{φ(u_{i})}^{−1} φ(v) − d_{u_{i}}φ(exp_{u_{i}}^{−1}(v))∥

≤ sup_{w∈B(u_{i},2r_{i})} ∥τ_{φ(w)}^{φ(u_{i})}d_{w}φτ_{u_{i}}^{w} − d_{u_{i}}φ∥ · ∥ exp_{u_{i}}^{−1}(v)∥.

Using (6) this results in

\begin{aligned}exp_{φ(u_{i})}^{−1}(φ(B(u_{i}, 2r_{i}))) & ⊂ d_{u_{i}}φ(B(O_{u_{i}}, 2r_{i})) + B(O_{φ(u_{i})}, 2η̃r_{i}) \\ & ⊂ d_{u_{i}}φ(B(O_{u_{i}}, 2r_{i})) + B(O_{φ(u_{i})}, 2η_{u_{i}}r_{i}). \end{aligned}

The set d_{u_{i}}φ(B(O_{u_{i}}, 2r_{i})) is an ellipsoid in T_{φ(u_{i})}M. Because of Lemma 2 and the relations (2) and (5) there exists an ellipsoid E_{i} containing exp_{φ(u_{i})}^{−1}(φ(B(u_{i}, 2r_{i}))) with the property

ω_{d}(E_{i}) ≤ (2r_{i})^{d}lu_{i}^{\frac{d}{n}}.  (10)

Since E_{i} contains the ball B(O_{φ(u_{i})}, 2η_{u_{i}}r_{i}) all half-axes have positive lengths. Using Lemma 3 we get that the ellipsoid E_{i} can be covered by

N_{i} := ⌊\frac{2^{⌊d⌋}ω_{d}(E_{i})}{α_{⌊d⌋+1}(E_{i})^{d}} ⌋

balls of radius α_{⌊d⌋+1}(E_{i})\sqrt{⌊d⌋ + 1}. If this radius is greater than \frac{r_{i}}{2} then every ball can be covered by

Ñ_{i} := ⌊2^{2n}n^{\frac{n}{2}} (\frac{α_{⌊d⌋+1}(E_{i})\sqrt{⌊d⌋ + 1}}{r_{i}})^{n}⌋

balls of radius \frac{r_{i}}{2} . All of these balls containing points of the set exp_{φ(u_{i})}^{−1}(φ(B(u_{i}, 2r_{i}))) lie inside a ball of radius

r_{i} + 2r_{i}m [1 + (\frac{m^{⌊d⌋}}{sup_{u∈K} ω_{d}(d_{u}φ)})^{1/(d−⌊d⌋)} η] < r_{3}

and centre O_{u_{i}}. Because of (7) the set φ(B(ũ_{i}, r_{i})) is then covered by balls of radius at most r_{i}.

Applying this technique for all indices i ∈ I we get a cover {B_{j}}_{j∈J} of φ(K) by balls of radii at most ε. The function i(j) gives for every j ∈ J the index i ∈ I of the ball in the cover of K which the ball B_{j} comes from. Since r_{i} ≤ r_{2} (i ∈ I) holds the map φ restricted to the ball B(ũ_{i}, r_{i}) is a C^{1}-diffeomorphism. Hence points with the same image are in different balls. We assumed | det(d_{u}φ)| > 0 for all u ∈ K, thus we can define numbers

c̃_{j} := min \{\frac{c_{i(j)}}{ω_{d}(d_{u_{i(j)}}φ)^{\frac{n}{d}}} , \frac{c_{i(j)}[sup_{u∈K} f(u)]^{\frac{n−d}{n}}}{ω_{d}(d_{u_{i(j)}}φ)} \} (j ∈ J).

Resulting from (9) we get the relations

∑_{j∈J:u∈B_{j}} c̃_{j} ≥ ∑_{v∈K:φ(v)=u} ∑_{i∈I:v∈B(ũ_{i},r_{i})} min \{\frac{c_{i}}{ω_{d}(d_{u_{i}}φ)^{\frac{n}{d}}} , \frac{c_{i}[sup_{u∈K} f(u)]^{\frac{n−d}{n}}}{ω_{d}(d_{u_{i}}φ)} \}

≥ ∑_{v∈K:φ(v)=u} ∑_{i∈I:v∈B(ũ_{i},r_{i})} min \{\frac{c_{i}}{ω_{d}(d_{v}φ)^{\frac{n}{d}}} , \frac{c_{i}[sup_{u∈K} f(u)]^{\frac{n−d}{n}}}{ω_{d}(d_{v}φ)} \}

= ∑_{v∈K:φ(v)=u} (min \{\frac{1}{ω_{d}(d_{v}φ)^{\frac{n}{d}}} , \frac{[sup_{u∈K} f(u)]^{\frac{n−d}{n}}}{ω_{d}(d_{v}φ)} \} ∑_{i∈I:v∈B(ũ_{i},r_{i})} c_{i})

for any u ∈ φ(K). From (8)

∑_{i∈I:v∈B(ũ_{i},r_{i})} c_{i} ≥ max\{f(v)ω_{d}(d_{v}φ)^{\frac{n}{d}} , f^{\frac{d}{n}} (v)ω_{d}(d_{v}φ)\},

follows, and since

min{a_{1}, a_{2}} max{b_{1}, b_{2}} ≥ min{a_{1}b_{1}, a_{2}b_{2}}

holds for arbitrary positive numbers a_{1}, a_{2}, b_{1}, b_{2}, with (1) we get

∑_{j∈J:u∈B_{j}} c̃_{j} ≥ ∑_{v∈K:φ(v)=u}min \{\frac{f(v)ω_{d}(d_{v}φ)^{\frac{n}{d}}}{ω_{d}(d_{v}φ)^{\frac{n}{d}}} , \frac{f^{\frac{d}{n}} (v)ω_{d}(d_{v}φ)[sup_{u∈K} f(u)]^{\frac{n−d}{n}}}{ω_{d}(d_{v}φ)} \}

≥ ∑_{v∈K:φ(v)=u} f(v)

≥ 1.

This is equivalent to ∑_{j∈J} c̃_{j}χ_{B_{j}} ≥ χ_{φ(K)}. Therefore we have

I_{H}(φ(K), 1, d, ε)

≤ ∑_{i∈I:α_{⌊d⌋+1}(E_{i})\sqrt{⌊d⌋+1}≤ \frac{r_{i}}{2}} N_{i} min \{_{\frac{c_{i}}{ω_{d}(d_{u_{i}}φ)^{\frac{n}{d}}}} , \frac{c_{i}[sup_{u∈K} f(u)]^{\frac{n−d}{n}}}{ω_{d}(d_{u_{i}}φ)} \} (2α_{⌊d⌋+1}(E_{i})\sqrt{⌊d⌋ + 1})^{d}

+ ∑_{i∈I:α_{⌊d⌋+1}(E_{i})\sqrt{⌊d⌋+1}> \frac{r_{i}}{2}} N_{i}Ñ_{i} min \{_{\frac{c_{i}}{ω_{d}(d_{u_{i}}φ)^{\frac{n}{d}}}} , \frac{c_{i}[sup_{u∈K} f(u)]^{\frac{n−d}{n}}}{ω_{d}(d_{u_{i}}φ)} \} r_{i}^{d}

≤ ∑_{i∈I:α_{⌊d⌋+1}(E_{i})\sqrt{⌊d⌋+1}≤ \frac{r_{i}}{2}} \frac{c_{i}[sup_{u∈K} f(u)]^{\frac{n−d}{n}}}{ω_{d}(d_{u_{i}}φ)} \frac{2^{⌊d⌋}ω_{d}(E_{i})}{α_{⌊d⌋+1}(E_{i})^{d}} (2α_{⌊d⌋+1}(E_{i})\sqrt{⌊d⌋ + 1})^{d}

+ ∑_{i∈I:α_{⌊d⌋+1}(E_{i})\sqrt{⌊d⌋+1}> \frac{r_{i}}{2}} \frac{c_{i}}{ω_{d}(d_{u_{i}}φ)^{\frac{n}{d}}} \frac{2^{⌊d⌋}ω_{d}(E_{i})}{α_{⌊d⌋+1}(E_{i})^{d}} 2^{2n}n^{\frac{n}{2}} (\frac{α_{⌊d⌋+1}(E_{i})\sqrt{⌊d⌋ + 1}}{r_{i}})^{n}r_{i}^{d}

= ∑_{i∈I:α_{⌊d⌋+1}(E_{i})\sqrt{⌊d⌋+1}≤ \frac{r_{i}}{2}} \frac{c_{i}[sup_{u∈K} f(u)]^{\frac{n−d}{n}}}{ω_{d}(d_{u_{i}}φ)} 2^{d+⌊d⌋}(⌊d⌋ + 1)^{\frac{d}{2}} ω_{d}(E_{i})

+ ∑_{i∈I:α_{⌊d⌋+1}(E_{i})\sqrt{⌊d⌋+1}> \frac{r_{i}}{2}} \frac{c_{i}}{ω_{d}(d_{u_{i}}φ)^{\frac{n}{d}}} \frac{2^{2n+⌊d⌋}n^{\frac{n}{2}} (⌊d⌋ + 1)^{\frac{n}{2}} ω_{d}(E_{i})α_{⌊d⌋+1}(E_{i})^{n−d}}{r_{i}^{n−d}} .

With (10) and the relation α_{⌊d⌋+1}(E_{i}) ≤ ω_{d}(E_{i})^{\frac{1}{d}} for all i ∈ I this results in

I_{H}(φ(K), 1, d, ε) ; \begin{aligned}≤ & ∑_{i∈I:α_{⌊d⌋+1}(E_{i})\sqrt{⌊d⌋+1}≤ \frac{r_{i}}{2}} \frac{c_{i}[sup_{u∈K} f(u)]^{\frac{n−d}{n}}}{ω_{d}(d_{u_{i}}φ)} 2^{2d+⌊d⌋}(⌊d⌋ + 1)^{\frac{d}{2}} lu_{i}^{\frac{d}{n}}r_{i}^{d} \\ & + ∑_{i∈I:α_{⌊d⌋+1}(E_{i})\sqrt{⌊d⌋+1}> \frac{r_{i}}{2}} \frac{c_{i}}{ω_{d}(d_{u_{i}}φ)^{\frac{n}{d}}} 2^{3n+⌊d⌋}n^{\frac{n}{2}} (⌊d⌋ + 1)^{\frac{n}{2}} l_{u_{i}}r_{i}^{d}. \end{aligned}

Because of sup_{u∈K} f(u) ≤ 1 and (4) finally

I_{H}(φ(K), 1, d, ε) ≤ ∑_{i∈I:α_{⌊d⌋+1}(E_{i})\sqrt{⌊d⌋+1}≤ \frac{r_{i}}{2}} c_{i}16^{n}(⌊d⌋ + 1)^{\frac{d}{2}} r_{i}^{d}

+ ∑_{i∈I:α_{⌊d⌋+1}(E_{i})\sqrt{⌊d⌋+1}> \frac{r_{i}}{2}} c_{i}16^{n}n^{\frac{n}{2}} (⌊d⌋ + 1)^{\frac{n}{2}} r_{i}^{d}

≤ 16^{n}n^{n} ∑_{i∈I} c_{i}r_{i}^{d}

follows. Since this inequality holds for any arbitrary cover {B(ũ_{i}, r_{i})}_{i∈I} of K by balls of radii r_{i} ≤ ε and any numbers {c_{i}}_{i∈I} satisfying (8) the theorem is proved. ■

A special function f : K → (0, 1] satisfying the assumption (1) is \frac{1}{N(φ,K,φ(·))}.

Corollary 1. Let (M, g) be an n-dimensional Riemannian C^{3}-manifold, U ⊂ M an open set, φ : U → M a C^{1}-map and K ⊂ U a compact set with | det(d_{u}φ)| > 0 for all u ∈ K. Then for all d ∈ (0, n] and sufficiently small ε > 0 we have

I_{H}(φ(K), 1, d, ε) ≤ 16^{n}n^{n}I_{H} (K, max \{\frac{ω_{d}(dφ)^{\frac{n}{d}}}{N(φ, K, φ(·))}, \frac{ω_{d}(dφ)}{N(φ, K, φ(·))^{\frac{d}{n}}} \} , d, ε)

and thus

I_{H}(φ(K), 1, d) ≤ 16^{n}n^{n}I_{H} (K, max \{\frac{ω_{d}(dφ)^{\frac{n}{d}}}{N(φ, K, φ(·))}, \frac{ω_{d}(dφ)}{N(φ, K, φ(·))^{\frac{d}{n}}} \} , d) .

Remark. The Hausdorff outer n-measure on smooth Riemannian manifolds differs from the Lebesgue measure or n-dimensional volume V only by a constant factor. Corollary 1 for d = n gives

I_{H}(φ(K), 1, n) ≤ 16^{n}n^{n}I_{H} (K, \frac{ω_{n}(dφ)}{N(φ, K, φ(·))}, n).  (11)

This inequality can be compared with the transformation formula for Lebesgue integrals under C^{1}-maps

∫_{φ(K)} |dM| = ∫_{K} \frac{|det(dφ)|}{N(φ, K, φ(·))}|dM|,  (12)

where |dM| denotes the canonical absolute volume form on M ([13]). In both formulas (11) and (12) on the left-hand side the constant 1 is integrated on the set φ(K), and on the right-hand side the function \frac{ω_{n}(dφ)}{N(φ,K,φ(·))} = \frac{|det(dφ)|}{N(φ,K,φ(·))} is integrated on the set K. Since in contrast to the Lebesgue integral the Hausdorff outer integral is only subadditive and furthermore the relation between the Hausdorff outer integral of a characteristic function and the Hausdorff outer measure of the corresponding set is only an inequality with correcting factors (Lemma 7), the transformation of the Hausdorff outer intagral under a C^{1}-map can only be described by an inequality with a correcting factor and not by an equality.

Using the local inverse maps and the inverse singular value function we can prove a statement analogous to Theorem 1.

Theorem 2. Let (M, g) be an n-dimensional Riemannian C^{3}-manifold, U ⊂ M an open set, φ : U → M a C^{1}-map and K ⊂ U a compact set and Ũ ⊃ K an open set with cl(Ũ) ⊂ U compact, | det(d_{u}φ)| > 0 for all u ∈ cl(Ũ) and φ^{−1}(Ũ) ⊂ Ũ. Furthermore let f : Ũ → (0, 1] be a nonnegative function such that there exists a number a > 0 with

f(u) = f(v) for all u, v ∈ Ũ with φ(u), φ(v) ∈ φ(K) and ϱ(u, v) < a  (13)

and the relation

∑_{v∈Ũ:φ(v)=u} f(v) ≤ 1 for all u ∈ φ(K)  (14)

is satisfied. Then for all d ∈ (0, n] and sufficiently small ε > 0 the inequality

I_{H}(K, f min\{\overline{ω}_{d}(dφ)^{\frac{n}{d}} , \overline{ω}_{d}(dφ)\}, d, ε) ≤ 16^{n}n^{n}I_{H} (φ(K), 1, d, ε)

holds.

6. Upper dimension estimates

Using the transformation formulas of the previous section for map-invariant sets we can derive upper estimates of the Hausdorff dimension.

Theorem 3. Let (M, g) be an n-dimensional Riemannian C^{3}-manifold, U ⊂ M an open set, φ : U → M a C^{1}-map and K ⊂ U a compact φ-invariant set with | det(d_{u}φ)| > 0 for all u ∈ K. If there exist numbers d ∈ (0, n] and ν ∈ (0, 1) with

ω_{d}(d_{u}φ)^{\frac{1}{d}} ≤ νN(φ, K, φ(u))^{\frac{1}{n}} for all u ∈ K,  (15)

then dim_{H}(K) ≤ d.

Proof. Because of ν < 1 there exists a sufficiently large number p ∈ ℕ with

ν^{p} < (\frac{1}{16n})^{\frac{n}{d}} .  (16)

Let this p be fixed further on. Then Lemma 1 and relation (15) leads to

ω_{d}(d_{u}φ^{p}) ≤ ∏_{j=1}^{p} ω_{d}(d_{φ^{j−1}(u)}φ)

≤ ∏_{j=1}^{p} ν^{d}N(φ, K, φ^{j}(u))^{\frac{d}{n}}

= ν^{pd} [∏_{j=1}^{p} N(φ, K, φ^{j}(u))]^{\frac{d}{n}}

and, therefore,

\frac{ω_{d}(d_{u}φ^{p})^{\frac{n}{d}}}{∏_{j=1}^{p} N(φ, K, φ^{j}(u))} ≤ ν^{pn}.  (17)

Resulting from Lemma 5 the function \frac{1}{∏_{j=1}^{p} N(φ,K,φ^{j}(·))} : K → (0, 1] satisfies the condition (1) with respect to the map φ^{p}. Thus we can apply Theorem 1 for the map φ^{p} and get for sufficiently small numbers ε > 0 the inequality

\begin{matrix}I_{H}(φ^{p}(K), 1, d, ε) \\ ≤ 16^{n}n^{n}I_{H} (K, max \{\frac{ω_{d}(dφ^{p})^{\frac{n}{d}}}{∏_{j=1}^{p} N(φ,K,φ^{j}(u))}, _{\frac{ω_{d}(dφ^{p})}{[∏_{j=1}^{p} N(φ,K,φ^{j}(u))]^{\frac{d}{n}}}} \} d, ε) . \end{matrix}

Since K is φ-invariant φ^{p}(K) = K holds as well. Hence we get

I_{H}(K, 1, d, ε) ≤ 16^{n}n^{n}I_{H} (K, max \{\frac{ω_{d}(dφ^{p})^{\frac{n}{d}}}{∏_{j=1}^{p} N(φ,K,φ^{j}(u))}, _{\frac{ω_{d}(dφ^{p})}{[∏_{j=1}^{p} N(φ,K,φ^{j}(u))]^{\frac{d}{n}}}} \} d, ε) .

With (17) and Lemma 7 this results in

I_{H}(K, 1, d, ε) ≤ 16^{n}n^{n}I_{H}(K, ν^{pd}, d, ε) ≤ 16^{n}n^{n}ν^{pd}I_{H}(K, 1, d, ε).

Because of (16) we have 16^{n}n^{n}ν^{pd} < 1, and since I_{H}(K, 1, d, ε) is finite, I_{H}(K, 1, d, ε) = 0 follows for all sufficiently small ε. This means I_{H}(K, 1, d) = 0 and, therefore, dim_{H}(K) ≤ d. ■

Remark. Condition (15) is weaker than the assumption ω_{d}(d_{u}φ) < 1 in the generalized Theorem of Douady and Oesterlé ([10]), because the multiplicity function is always at least one. In case of an injective map both conditions coincide. But since we used properties of the multiplicity function resulting from | det(d_{u}φ)| > 0, we have additionally to assume, that the considered set does not contain critical points. Note that condition (15) contains the singular value function ω_{d} with an exponent \frac{1}{d}, since

in contrast to ω_{d} the function ωd^{\frac{1}{d}}, as a function of d, is always monotone decreasing in the whole interval (0, n].

Using the local inverse maps we get the following dimension estimate which can be proved with Theorem 2.

Theorem 4. Let (M, g) be an n-dimensional Riemannian C^{3}-manifold, U ⊂ M an open set, φ : U → M a C^{1}-map and K ⊂ U a compact φ-invariant set and Ũ ⊃ K an open set with cl(Ũ) ⊂ U compact, | det(d_{u}φ)| > 0 for all u ∈ cl(Ũ) and φ^{−1}(Ũ) ⊂ Ũ. If there exist numbers d ∈ (0, n] and ν ∈ (0, 1) with

ν\overline{ω}_{d}(d_{u}φ) ≥ N(φ, Ũ, φ(u)) for all u ∈ K,  (18)

then dim_{H}(K) ≤ d.

Remark. Theorems 3 and 4 can not be applied to the same map, since the conditions (15) and (18) exclude each other: If (15) is satisfied with a number d ∈ (0, n], then ω_{n}(d_{u}φ) < N(φ, K, φ(u)) for all u ∈ K follows, whereas (18) implies ω_{n}(d_{u}φ) > N(φ, K, φ(u)) for all u ∈ K.

7. Examples

Both Theorems 3 and 4 can be applied to a wide class of non-invertible maps and to iterated function systems. Some classes of injective maps like horseshoe maps can be splitted into partial non-invertible maps, or the state space for instance for Belykh maps ([1, 12]) can be factorized such that the resulting non-injectivity of the map can be used to get dimension bounds for an invariant set.

Let us illustrate the dimension estimate of Theorem 3 for a simple example of a piecewise linear map in ℝ^{2}. Analogously to the Theorem of Douady and Oesterlé there are no non-trivial examples in ℝ since condition (15) is for d ∈ (0, 1] independent of d. In ℝ^{2} we can consider a modified baker’s transformation defined on the unit square [0, 1] × [0, 1] in the form

φ(x, y) = \{ \begin{matrix}(6x, λy) & for 0 ≤ x < \frac{1}{6}, \\ (6x − 1, λy + \frac{1}{2}) & for \frac{1}{6} ≤ x < \frac{1}{3}, \\ (6x − 2, λy) & for \frac{1}{3} ≤ x < \frac{1}{2}, \\ (6x − 3, λy + \frac{1}{2}) & for \frac{1}{2} ≤ x < \frac{2}{3}, \\ (6x − 4, λy) & for \frac{2}{3} ≤ x < \frac{5}{6}, \\ (6x − 5, λy + \frac{1}{2}) & for \frac{5}{6} ≤ x ≤ 1, \end{matrix}

where λ ∈ (0, \frac{1}{2}) is a parameter. The maximal invariant set of this map is K = ⋂_{p∈ℕ_{0}} φ^{p}([0, 1] × [0, 1]). Points of K have three preimages in K, hence N(φ, K, u) = 3 holds for all u ∈ K. The set K is the product of the unit interval and a uniform Cantor set (see [4]), as shown in Figure 1. Therefore from [4] the relation dim_{H}(K) = 1 + \frac{ln 2}{− ln λ} follows.

0 1  1

Figure 1: Structure of the invariant set K for λ = \frac{1}{3} In ℝ^{2} the map φ is not continuous. But we can consider it on the torus T, defined by the equivalence relation

(x_{1}, y_{1}) ∼ (x_{2}, y_{2}) ⇔ x_{2} = x_{1} + b, y_{2} = y_{1} + a − \frac{b}{2} (a, b ∈ ℤ),

with the flat metric

ϱ((x_{1}, y_{1}), (x_{2}, y_{2})) = min_{(u,v)∼(x_{2},y_{2})} \sqrt{(u − x_{1})^{2} + (v − x_{2})^{2}}.

Let π : ℝ^{2} → T denote the canonical projection, then φ is a C^{1}-map on K̃ := π(K) and can be extended to a C^{1}-map on an open neighbourhood U of K̃. This extension we will denote by φ̃. Because of dim_{H}(K) = dim_{H}(K̃) we can consider the set K̃ further on.

The singular values of d_{u}φ̃ are 6 and λ, independent of the point u ∈ K̃. Therefore condition (15) can not be satisfied for numbers d ∈ (0, 1]. For d ∈ (1, 2] this condition has the form

(6λ^{d−1})^{\frac{1}{d}} < \sqrt{3},

this is equivalent to d > \frac{ln 6−ln λ}{ln \sqrt{3}−ln λ} for arbitrary parameters λ ∈ (0, \frac{1}{2}). Thus Theorem 3 gives

dim_{H}(K̃) = dim_{H}(K) ≤ \frac{ln 6 − ln λ}{ln \sqrt{3} − ln λ} for any λ ∈ (0, \frac{1}{2}) .

Compared with the exact value this is only an upper bound. But this upper bound is better than the estimate dim_{H}(K) ≤ 1+\frac{ln 6}{− ln λ} which can be reached with the generalized Theorem of Douady and Oesterlé only for parameters λ ∈ (0, \frac{1}{6}).

A well-known class of non-injective maps are polynomials in the complex plane. We want to illustrate the dimension estimate by Theorem 4 for Julia sets of quadratic polynomials φ : ℂ → ℂ in the form

φ(z) = z^{2} + c,

where c ∈ ℂ is a parameter, as considered in Section 3. The Julia set K is the closure of the set of repelling periodic points (see for instance [4, 11]). This set is compact, non-empty and invariant under φ. For different parameters c this set has different shapes. Therefore it is interesting to estimate the Hausdorff dimension of this set.

Let us consider ℂ as two-dimensional Riemannian C^{∞}-manifold with a constant metric tensor giving the metric

ϱ(z_{1}, z_{2}) = |z_{1} − z_{2}| = \sqrt{(Re(z_{1}) − Re(z_{2}))^{2} + (Im(z_{1}) − Im(z_{2}))^{2}},

where Re and Im denote the real and imaginary part of the corresponding complex number. Under this metric the singular values of d_{z}φ are

α_{1}(d_{z}φ) = α_{2}(d_{z}φ) = 2\sqrt{Re(z)^{2} + Im(z)^{2}} = 2|z|.

Thus for d ∈ (0, 2] and any z ∈ ℂ we have

ω_{d}(d_{z}φ) = \overline{ω}_{d}(d_{z}φ) = (2|z|)^{d}.

It can be shown (see [4]) that the absolute value of the points of K can be estimated by

\begin{matrix}\frac{1}{2} + \sqrt{\frac{1}{4} − |c|} < |z| < \frac{1}{2} + \sqrt{\frac{1}{4} + |c|} & for |c| < \frac{1}{4}, \\ \sqrt{|c| − \sqrt{|2c|}} ≤ |z| ≤ \sqrt{|c| + \sqrt{|2c|}} & for |c| > 2. \end{matrix}

The multiplicity function of φ with respect to K and any open neighbourhood Ũ of K satisfies

N(φ, K, φ(z)) = N(φ, Ũ, φ(z)) = 2 for all z ∈ K

under the condition, that the origin is not contained in K. Thus condition (18) has the form

(2|z|)^{d} > 2 for all z ∈ K

which is satisfied if

\begin{matrix}(1 + \sqrt{1 − 4|c|})^{d} > 2 & for |c| < \frac{1}{4}, \\ 2\sqrt{|c| − \sqrt{|2c|}} > 2 & for |c| > 2. \end{matrix}

Thus Theorem 4 gives the dimension estimate

dim_{H}(K) ≤ \{ \begin{matrix}\frac{ln 2}{ln(1 + \sqrt{1 − 4|c|})} & for |c| < \frac{\sqrt{2} − 1}{2}, \\ \frac{ln 2}{ln (2\sqrt{|c| − \sqrt{|2c|}})} & for |c| > \frac{3}{2} + \sqrt{2}, \end{matrix}

similar to [4]. In the case |c| > \frac{3}{2} + \sqrt{2} the Julia set of the quadratic polynomial φ(z) = z^{2} + c can also be considered as invariant set with an equivariant tangent bundle splitting (see [6]) which yields the same dimension bound.

References

[1] V. N. Belykh and I. L. Shelesnyak. Dimension estimates for a strange attractor of a two-dimensional discontinuous map. In: Methods of qualitative theory and theory of bifurcations, pages 12–17, University of Nishny Novgorod, 1992. (in Russian).

[2] V. A. Boichenko, A. Franz, G. A. Leonov, and V. Reitmann. Hausdorff and fractal dimension estimates for invariant sets of non-injective maps. ZAA, 17(1):207–223, 1998.

[3] A. Douady and J. Oesterlé. Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris Ser. A, 290:1135–1138, 1980.

[4] K. J. Falconer. Fractal Geometry. Mathematical Foundations and Applications. John Wiley & Sons, Chichester, 1990.

[5] H. Federer. Geometric Measure Theory. Springer, New York, 1969.

[6] A. Franz. Hausdorff dimension estimates for invariant sets with an equivariant tangent bundle splitting. Nonlinearity, 11:1063–1074, 1998.

[7] J. D. Howroyd. On dimension and on the existence of sets of finite positive Hausdorff measure. Proc. London Math. Soc., 70(3):581–604, 1995.

[8] F. Ledrappier. Some relations between dimension and Lyapunov exponents. Commun. Math. Phys., 81:229–238, 1981.

[9] G. A. Leonov and V. A. Boichenko. Lyapunov’s direct method in the estimation of the Hausdorff dimension of attractors. Acta Appl. Math., 26:1–60, 1992.

[10] A. Noack and V. Reitmann. Hausdorff dimension of invariant sets of timedependent vector fields. ZAA, 15(2):457–473, 1996.

[11] D. Ruelle. Repellers for real analytic maps. Ergodic Theory & Dynamical Systems, 2:99–108, 1982.

[12] J. Schmeling. A dimension formula for endomorphisms - the Belykh family. Preprint 29/96, DFG-Schwerpunktprogramm ”‘Dynamik: Analysis, effiziente Simulation und Ergodentheorie”’, 1996.

[13] R. Sulanke and P. Wintgen. Differentialgeometrie und Faserbündel. Deutscher Verlag der Wissenschaften, Berlin, 1972.

[14] R. Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York, 1988.

[15] P. Thieullen. Entropy and the Hausdorff dimension for infinite-dimensional dynamical systems. Journal of Dynamics and Differential Equations, 4(1):127–159, 1992.

[16] H. Wegmann. Die Hausdorff-Dimension von kartesischen Produktmengen in metrischen Räumen. J. Reine und Angew. Math., 234:163–171, 1969.