Hausdorff dimension estimates by use of a tubular Carathéodory structure

Katrin Gelfert  Gennady A. Leonov
gelfert@math.tu-dresden.de leonov@math.lgu.spb.su

Volker Reitmann^{∗}

reitmann@rcs.urz.tu-dresden.de

Abstract. The paper is concerned with upper bounds for the Hausdorff dimension of flow invariant compact sets on Riemannian manifolds. The proof of the main theorem uses a special Carathéodory dimension structure in order to get contraction conditions for the considered Carathéodory measures which majorize the Hausdorff measures. The Hausdorff dimension bounds are formulated in terms of the eigenvalues of the symmetric part of the operator which generates the associated system in normal variations with respect to the direction of the vector field.

Keywords: Hausdorff dimension, Carathéodory dimension structure, outer measures via tube covers, system in normal variations, Riemannian manifolds AMS subject classification: Primary 58F12, 58F25, secondary 28A78

1 Introduction

Carathéodory structures were introduced by Pesin [15] (see also [16]) in order to give a general concept for dimension-like characteristics of sets and a given dynamics on the sets may be considered as a generalization of a well-known measure-theoretic construction of Carathéodory [3, 5]. The essential parts of such a structure are the following .

Let X be a set, F be a family of subsets of X, ℙ = [d^{∗}, +∞) for finite d^{∗} or ℙ = ℝ be a parameter set, and let ξ, η : F × ℙ → [0, ∞) and ψ : F → [0, ∞) be functions. A sub-family G ⊂ F is said to be an ε-cover of a set Y ⊂ X if Y ⊂ ⋃_{u∈G} U and ψ(G) := sup{ψ(U)|U ∈ G} ≤ ε hold. The following conditions are assumed to be satisfied:

(A1) ∅ ∈ F, ψ(∅) = 0 and ξ(∅, d) = 0 for all d ∈ ℙ.

(A2) ξ(U, d + d′) = η(U, d)ξ(U, d′) for all d, d′ ∈ ℙ and all U ∈ F .

(A3) For any d ∈ ℙ for any Δ > 0 there exists ε > 0 such that for all U ∈ F\{∅} with ψ(U) ≤ ε we have η(U, d) ≤ Δ if d > 0 and η(U, d) ≥ Δ^{−1} if d < 0.

(A4) For any subset Y ⊂ X and for arbitrary ε > 0 there exists a countable ε-cover of Y.

Typeset by L^{A}TEX 2ε

∗Supported by the DFG-Schwerpunktprogramm ”Dynamik: Analysis, effiziente Simulation und Ergodentheorie”

In analogy to [16] we call a collection (F, ℙ, ξ, η, ψ) which satisfies (A1)-(A4) a Carathéodory (dimension) structure on X. For a given Carathéodory structure (F, ℙ, ξ, η, ψ), an arbitrary set Y ⊂ X, d ∈ ℙ and ε > 0 we define the Carathéodory d-measure at level ε of Y with respect to (F, ℙ, ξ, η, ψ) by

μ_{C}(Y, d, ε) = inf_{G} ∑_{U∈G} ξ(U, d),

where the infimum is taken over all countable sub-collections G ⊂ F being ε-covers of the set Y. For fixed Y and d the function μ_{C}(Y, d, ε) is non-increasing with respect to ε. Therefore, there exists the limit

μ_{C}(Y, d) = lim_{ε→0+0} μ_{C}(Y, d, ε)

which is called the Carathéodory d-measure of Y with respect to (F, ℙ, ξ, η, ψ). For arbitrary d ∈ ℙ and arbitrary ε > 0 the functions μ_{C}(·, d, ε) and μ_{C}(·, d) are outer measures on X. It turns out that for any Y ⊂ X there exists a unique number d_{cr}(Y) ∈ \overline{ℙ} having the property that

μ_{C}(Y, d) = \begin{cases}0 & for d > d_{cr}(Y) \\ +∞ & for d < d_{cr}(Y) \end{cases}

holds for d ∈ ℙ. This critical value d_{cr}(Y) is called Carathéodory dimension dim_{C} Y of Y with respect to the structure (F, ℙ, ξ, η, ψ).

Note that our system of conditions (A1)-(A4) which leads to a Carathéodory structure is slightly different from the system in [15, 16]. In contrast to these papers we assume that our family of objects in the Carathéodory construction depends on parameters which come from a (possible proper) subset of ℝ.

As an example let X be a metric space, F the family of open balls B(u, r) in X with center u and radius r, ℙ = ℝ_{+}, ξ(B(u, r), d) = η(B(u, r), d) = r^{d} and ψ(B(u, r)) = r for each u ∈ X, r > 0 and each d ≥ 0. It is easy to see that such a system (F, ℙ, ξ, η, ψ) defines a Carathéodory structure on X. We denote by μ_{H}(., d, r), μ_{H}(., d) and dim_{H} the resulting Carathéodory measures and Carathéodory dimension which are in fact the Hausdorff d-measure at level r, the Hausdorff d-measure and the Hausdorff dimension, respectively. The concept of the Carathéodory dimension covers not only several dimension type characteristics of sets but also characteristics of dynamical systems such as topological pressure and topological entropy (see [15, 16]) or a dimension introduced for Poincaré recurrences [1].

In the sequel the paper is organized as follows. In Section 2 we give the definition of a special Carathéodory structure adapted for the dimension investigation of flow invariant sets. It is defined via covering elements which are tubular neighborhoods of arcs of smooth curves to approximate the fiber structure of the sets. In Section 3 we present a short review of basic facts on Riemannian geometry, and we introduce the variational system written in normal variations, transversal to the evolution direction of the flow. Similar variational systems written in such normal coordinates are also used in stability theory to show orbital stability of solutions of a differential equation [8, 12, 13]. The main results of the paper are contained in Section 4. For flow negatively invariant sets which do not contain singular points of the vector field an upper bound of the Hausdorff dimension is given. The estimates are derived by means of the Carathéodory measures which are contractive under the flow and majorize the Hausdorff measure. These results generalize those from [10, 11] on Riemannian manifolds. Hausdorff dimension estimates of flow invariant sets using the eigenvalues of the symmetric part of the operator part of the (standard) equation in variation are deduced in [17, 14]. In the case of attractors of differential equations it is natural to investigate the variation transversal to the flow lines. The estimates in this paper are formulated in terms of the eigenvalues of the symmetric part of the generated operator of the associated system in normal variation. Assuming special properties of the stable and unstable manifolds of equilibrium points the results of Section 4 are generalized in Section 5 for vector fields having a finite number of such equilibrium points in the considered invariant set. The used Carathéodory measures show in many cases a better contracting behavior under the positive semi-flow than the Hausdorff measures do. The paper ends with a discussion of various examples which show the effectivity of the obtained Hausdorff dimension estimates. In addition we obtain results about the asymptotic behavior of the dynamical system using the dimension bounds, which are closely related to results in [2, 7, 8].

2 Tubular Carathéodory structure

In this section we define a special Carathéodory structure on Riemannian manifolds. The Carathéodory measures which arise from this structure will majorize the Hausdorff measures and will be applied to obtain for Hausdorff dimension estimates of flow-invariant sets on the manifold.

Let (M, g) be a smooth n-dimensional Riemannian manifold and ρ the metric induced by g. For a piecewise smooth curve γ : I → M (I ⊂ ℝ an interval) of finite length and arbitrary ε > 0 we define the ε-tubular neighborhood Ω(γ, ε) of γ by

Ω(γ, ε) = ⋃_{u∈γ(I)} B(u, ε),

where B(u, ε) = {p ∈ M|ρ(u, p) < ε} is a metric ε-ball on M centered in u. For simplicity we call the ε-tubular neighborhood Ω(γ, ε) around the curve γ of length l shortly tube of length l.

For a given compact set K ⊂ M and a given number l_{0} > 0 we denote by Γ = {γ} a family of piecewise smooth curves of a finite length l(γ) = l_{0} such that for any ε > 0 the following condition is satisfied :

(A) K is contained in the union of ε-tubular neighborhoods Ω(γ, ε) with γ ∈ Γ.

Condition (A) guarantees the existence of arbitrarily fine covers of the set K which are generated by the family Γ. For a family Γ satisfying (A) we define a family of subsets F, a parameter set ℙ and the functions ξ, η : F × ℙ → [0, ∞) and ψ : F → [0, ∞) by

\begin{matrix}F = \{Ω(γ, ε) ∩ K | γ ∈ Γ, ε > 0\} ∪ \{∅\},  ℙ = [1, +∞) \\ ξ(Ω(γ, ε) ∩ K, d) = ε^{d−1}, η(Ω(γ, ε) ∩ K, d) = ε^{d}, ψ(Ω(γ, ε) ∩ K) = ε \end{matrix}  (2.1)

for γ ∈ Γ, ε > 0, d ∈ ℙ and set ξ(∅, d) = η(∅, d) = ψ(∅) = 0 for all d ∈ ℙ.

Straight forward, one can verify that the collection (F, ℙ, ξ, η, ψ) defined via (2.1) with Γ satisfying (A) is a Carathéodory structure on K in the sense as considered above. In the sequel we will call such a structure simply a Carathéodory structure with tubes of length l_{0} or tubular Carathéodory structure, if the underlying set K and the family Γ are clear from the context. The next proposition shows the relations between the Carathéodory measure, generated by this structure, and the Hausdorff measure.

Proposition 2.1 Suppose that K is a compact set on the smooth n-dimensional Riemannian manifold (M, g). Suppose that (F, ℙ, ξ, η, ψ) is a tubular Carathéodory structure on K with tubes of length l_{0} defined by (2.1) and μ_{C}(., d, ε) is the Carathéodory d-measure at level ε with respect to this structure. Then there exist two numbers C > 0 and ε_{0} > 0 depending only on K such that for any set Y ⊂ K and any d ≥ 1

μ_{H}(Y, d, ϵ) ≤ 2l_{0}Cμ_{C}(Y, d, ϵ).  (2.2)

holds for all ε ∈ (0, ε_{0}].

Proof. Because of the compactness of the set K there exist two numbers ε′ > 0 and C > 0 such that for all p ∈ K and for r ∈ (0, ε′] every ball B(p, 2r) can be covered by at least C balls of radius r. Set ε_{0} = min{l_{0}, ε′}.

Fix an arbitrary ε ∈ (0, ε_{0}] and consider the tubular Carathéodory d-measure at level ε

μ_{C}(Y, d, ε) = inf { ∑_{Ω(γ_{i},r_{i})∩K∈F} r_{i}^{d−1} | Ω(γ_{i}, r_{i}) ∩ Y ≠ ∅, r_{i} ≤ ε} .

Therefore, for arbitrary small number Δ > 0 there exists an ε-cover G ⊂ F consisting of elements Ω(γ_{i}, r_{i}) ∩ K which satisfy r_{i} ≤ ε and

∑_{i≥1} r_{i}^{d−1} ≤ μ_{C}(Y, d, ε) + Δ.  (2.3)

Every Ω(γ_{i}, r_{i}) ∩ K ∈ G can be covered by ([l_{0}/r_{i}] + 1) balls centered in a point of γ_{i} and with radius 2r_{i}, where [a] denotes the integer part of a. Thus, we get

μ_{H}(Y, d, ε) ≤ inf_{Ω(γ_{i},r_{i})∩K∈G} ∑_{i≥1} ([\frac{l_{0}}{r_{i}} ] + 1) Cr_{i}^{d}
≤ inf_{Ω(γ_{i},r_{i})∩K∈G} ∑_{i≥1} 2l_{0}Cr_{i}^{d−1}
≤ 2l_{0}C(μ_{C}(Y, d, ε) + Δ),

where the last inequality follows from (2.3). Since the value Δ > 0 can be chosen arbitrarily small we obtain the inequality (2.2). ■

Corollary 2.1 Let on the manifold (M, g) the compact set K, the structure (F, ℙ, ξ, η, ψ) and l_{0} be defined as in Proposition 2.1 and let μ_{C}(., d) and dim_{C} be the Carathéodory dmeasure and the Carathéodory dimension with respect to this structure, respectively. Then there exists some constant C > 0 such that for any set Y ⊂ K and any d ≥ 1 there hold the inequality

μ_{H}(Y, d) ≤ 2l_{0}Cμ_{C}(Y, d)

and thus

dim_{H} Y ≤ dim_{C} Y.

Now we specify the family Γ of curves which will be used further for the considerations of sets being negatively invariant with respect to a flow. Let (M, g) be a n-dimensional Riemannian manifold (M, g) of class C^{∞}. Here and in the sequel let f : M → T M be a vector field of class C^{2} and let us consider the corresponding differential equation

u̇ = f(u).  (2.4)

In addition, we assume that the global flow φ : ℝ×M → M of (2.4) exists. This flow φ can be written as one-parameter family of C^{2}-diffeomorphisms {φ^{t}}_{t∈ℝ} with φ^{t}(.) = φ(t, .). Let K, K̃ ⊂ M be two compact sets satisfying

K ⊂ φ^{t}(K) ⊂ K̃ for all t ≥ 0.  (2.5)

(A set K satisfying such an inclusion is usually called negatively invariant with respect to the flow.) We suppose that the set K does not contain equilibrium points of (2.4).

To construct the family Γ we denote by Λ the set of all equilibrium points of (2.4) and set e_{1} = \frac{1}{2}dist(Λ, K), where dist(Λ, K) = inf_{u∈Λ,p∈K} ρ(u, p) is the usual metric distance between two sets in M and define

Φ = K̃ ∩ ⋃_{p∈K} B(p, e_{1}).  (2.6)

With respect to the vector field f, the compact set K̃ from (2.5), and the set Φ from (2.6) define the following coefficient, which will be important for the proofs in Section 4. Let

V (f, K̃, Φ) = \frac{max_{u∈K̃} ||f(u)||}{min_{u∈Φ} ||f(u)||}.  (2.7)

For any p ∈ K there exists a time b_{p} > 0 such that φ^{t}(p) ≠ ⋃_{p∈Λ} B(p, e_{1}) for all t ∈ [0, b_{p}]. Since K is compact there exists a number e_{2} > 0 such that for the length of the integral curve pieces l(φ(., p)|_{[0,b_{p}]}) ≥ e_{2} holds for all p ∈ K. We define

l_{0} = \frac{1}{2} min\{e_{1}, e_{2}\} and

Γ = { φ(., q)|_{[0,τ(q)]} | q ∈ K, τ(q) > 0 such that l(φ(., q)|_{[0,τ(q)]}) = l_{0}}.

(2.8)

Obviously this family Γ satisfies condition (A) and (F, ℙ, ξ, η, ψ) defined by (2.1) on the base of this family is a Carathéodory structure on K - a Carathéodory structure with tubes of length l_{0} - which will be used in Section 4.

3 The system in normal variation

In this section we introduce a modified variational equation which will be used for modeling the variation of time translated pieces of hypersurfaces orthogonal to the considered orbit. This idea originates from investigations on stability behavior of solutions of a differential equation (see [8, 12, 13]) where instead of the movements of phase points along a trajectory one considers their movements in transversal direction only. Projecting the covariant derivative of the vector field along a reference orbit in the (n − 1)-dimensional tangent space lying orthogonal to the vector field we get a variational equation describing the normal variation. For the first time this type of variational equation has been applied to dimensional estimates in [10, 11].

Let us recall some notation of linear algebra and differential geometry used later. If V and W are m-dimensional Euclidean spaces with scalar products ⟨., .⟩_{V} and ⟨., .⟩_{W}, respectively, and L : V → W is the linear operator, then the adjoint operator L^{∗} : W → V is a linear operator uniquely determined by the relation ⟨Lξ, η⟩_{W} = ⟨ξ, L^{∗}η⟩_{V} for all ξ ∈ V, η ∈ W. The singular values of the operator L are the eigenvalues of the positive semidefinite operator (L^{∗}L)^{\frac{1}{2}} : V → V. We denote them by σ_{1}(L) ≥ … ≥ σ_{m}(L) ≥ 0 ordered with respect to size and algebraic multiplicity. Let an arbitrary number d ∈ [0, m] written in the form d = d_{0} + d_{1} for d > 0 with d_{0} ∈ {0, 1, …, m − 1} and d_{1} ∈ (0, 1]. We define by

ω_{d}(L) = \begin{cases}1 & for d = 0, \\ σ_{1}(L) · . . . · σ_{d_{0}}(L)σ_{d_{0}+1}^{d_{1}}(L) & for d ∈ (0, m], \end{cases}

the singular value function of order d in L. Let E be an ellipsoid in the m-dimensional Euclidean space V and let σ_{1}(E) ≥ … ≥ σ_{m}(E) ≥ 0 denote the length of its semi-axes. For an arbitrary number d ∈ [0, m], for d > 0 written in the form d = d_{0} + d_{1} with d_{0} ∈ {0, 1, …, m − 1} and d_{1} ∈ (0, 1], we introduce the d-dimensional ellipsoid volume by

ω_{d}(E) = \begin{cases}1 & for d = 0, \\ σ_{1}(E) · . . . · σ_{d_{0}}(E)σ_{d_{0}+1}^{d_{1}}(E) & for d ∈ (0, m]. \end{cases}

For the linear operator L : V → W and a ball B(0, r) of radius r around 0 the image LB(0, r) is an ellipsoid with length of semi-axes σ_{i}(L)r. It follows that for d as considered above

ω_{d}(LB(0, r)) = ω_{d}(L) r^{d}.  (3.1)

Consider now a Riemannian manifold (M, g) of dimension n and, for simplicity, of class C^{∞}, which we call smooth. Denote by T_{p}M the tangent space at p ∈ M. The Christoffel symbols of second kind on (M, g) with respect to a chart x : D(x) → R(x) are given by the n^{3} smooth functions Γ_{ij}^{k} = \frac{1}{2}g^{ks}(g_{js,i} + g_{si,j} − g_{ij,s}) (here and throughout this paper with summation on repeated indices), where g_{kl,r} = \frac{∂g_{kl}}{∂x^{r}}. Let f : M → T M be a C^{1}-vector field, {∂_{i}(p)} the canonical basis of T_{p}M in the chart x and f(p) = f^{i}∂_{i}(p) the representation of the vector field f. The covariant derivative of f in p is the linear operator ∇f(p) : T_{p}M → T_{p}M defined by ∇f(p)v = ∇_{i}f^{k}v^{i}∂_{k}(p) = (\frac{∂f^{k}}{∂x^{i}} v^{i} + Γ_{ij}^{k}f^{j}v^{i}) ∂_{k}(p) for all v = v^{i}∂_{i}(p) ∈ T_{p}M. For the linear operator ∇f(p) : T_{p}M → T_{p}M in the Euclidean space (T_{p}M, ⟨. , .⟩_{T_{p}M}) ∇f(p)^{∗} is the adjoint operator and S∇f(p) := \frac{1}{2}[∇f(p) + ∇f(p)^{∗}] the symmetric part of ∇f(p). For a C^{1}-curve c : [a, b] → M let x^{i}(t) be local coordinates of c(t) in the chart x. For a vector field F (t) along c, i.e., F (t) ∈ T_{c(t)}M for all t ∈ [a, b], the absolute derivative of F along c is \frac{DF (t)}{dt} ∈ T_{c(t)}M defined in the chart x by

\frac{DF (t)}{dt} ≡ ∇ċF (t) := (\frac{dF^{k}}{dt} + Γ_{ij}^{k}F^{j}ċ^{i}) ∂_{k}(c(t)).

For a given C^{1}-curve c : [a, b] → M, any v ∈ T_{c(a)}M and any s ∈ [a, b] there exists a unique vector field F_{v} along c such that F_{v} is parallel along c, i.e., ∇ċF ≡ 0 and F_{v}(a) = v. This defines for any s, t ∈ [a, b] with s ≤ t the parallel transport τ_{c(s)}^{c(t)} : T_{c(s)}M → T_{c(t)}M along c from c(s) to c(t) which relates to any v ∈ T_{c(s)}M the vector F_{v}(t) ∈ T_{c(t)}M. Recall that a geodesic on (M, g) is a smooth curve c : [a, b] → M satisfying Dċ/dt ≡ 0. For any p ∈ M and v ∈ T_{p}M we denote the maximal geodesic by c_{p,v}. Let Ω^{1} ⊂ T M be the set of pairs {(p, v)} with p ∈ M and v ∈ T_{p}M such that c_{p,v}(1) exists. Then the exponential map exp : Ω^{1} → M on (M, g) is given by exp((p, v)) = c_{p,v}(1) for all (p, v) ∈ Ω_{1} and exp_{p} is the restriction exp |_{T_{p}M∩Ω^{1}}. It is well-known (see [9]) that Ω^{1} is open in T M, that exp : Ω^{1} → M is smooth, and for any p ∈ M there exists an open set Ω_{p}^{1} ⊂ T_{p}M such that exp_{p} is a diffeomorphism on Ω_{p}^{1} and ||d_{O_{p}} exp_{p} || = 1.

As in the previous section we consider on the smooth n-dimensional Riemannian manifold (M, g) a vector field f : M → T M of class C^{2} and the corresponding differential equation (2.4). Since the vector field f is assumed to be of class C^{2} the same holds for φ^{t}(.) for every t ∈ ℝ.

The behavior of system (2.4) near a given solution φ^{(.)}(p) is described by the variational equation

\frac{Dy}{dt} = ∇f(φ^{t}(p))y  (3.2)

(see [9]). In local coordinates of a chart x around φ^{t}(p) system (3.2) takes the form

\frac{Dy^{k}}{dt} = \frac{∂f^{k}}{∂x^{i}} y^{i} + Γ_{ij}^{k}f^{j}y^{i} = ∇_{i}f^{k}y^{i}.

For any p ∈ M the differential Y (t, p) = d_{p}φ^{t} is the operator solution of (3.2) with initial condition Y (0, p) = id_{T_{p}M}.

All points p ∈ M with f(p) ≠ O_{p}, where O_{p} denotes the origin of the tangent space T_{p}M, we call regular points of f. If p is a regular point we may consider the system in normal variations with respect to the solution φ^{(.)}(p) of (2.4)

\frac{Dz}{dt} = A(φ^{t}(p))z,  (3.3)

where the linear operator A(p) : T_{p}M → T_{p}M is given by

A(p) = ∇f(p) − B(p), where
B(p)v = 2\frac{f(p)}{||f(p)||^{2}} ⟨f(p), S∇f(p)v⟩ for all v ∈ T_{p}M. (3.4)

The scalar product ⟨., .⟩ and the associated norm ||.|| are taken in the tangent space T_{p}M. In coordinates of an arbitrary chart x : D(x) → R(x) around the regular point p the linear operator A(p) is given by

A_{i}^{k} = ∇_{i}f^{k} − \frac{2}{g_{mn}f^{m}f^{n}} f^{k}g_{jl}f^{l}S_{i}^{j},

where f^{k} and g_{jl} are the coordinates of the vector field f and the Riemannian metric tensor g in the chart x and S_{i}^{j} = \frac{1}{2} [g^{jk}∇_{k}f^{p}g_{pi} + ∇_{i}f^{j}] is the representation in coordinates of the symmetric part S∇f(p) of the covariant derivative of the vector field f in this chart. Suppose that p ∈ M is regular point of f and y(.) is a solution of (3.2) along φ^{(.)}(p). This solution can be splitted for any t ≥ 0 into two orthogonal components as

y(t) = z(t) + μ(t)f(φ^{t}(p)),  (3.5)

where z(.) is the solution of (3.3) with respect to φ^{(.)}(p) with initial condition z(0) = y(0) and μ(.) is a scalar valued C^{1}-function for t ≥ 0 given by

μ(t) = \frac{⟨y(t), f(φ^{t}(p))⟩}{||f(φ^{t}(p))||^{2}}.  (3.6)

For every regular point p ∈ M of f we introduce the (n−1)-dimensional linear subspace

T^{⊥}(p) = {v ∈ T_{p}M : ⟨v, f(p)⟩ = 0}

of the tangent space T_{p}M. A straight forward calculation shows that for all v ∈ T^{⊥}(p) the following two relations

⟨f(p), [A(p) + A(p)^{∗}]v⟩ = 0  (3.7)

and

⟨v, A(p)v⟩ = ⟨v, ∇f(p)v⟩  (3.8)

are satisfied. Hence, we have \frac{1}{2}[A(p) + A(p)^{∗}] : T^{⊥}(p) → T^{⊥}(p). Because of (3.7) the subspace T^{⊥}(p) is an invariant space of the symmetric part of A(p), i.e., of the operator SA(p) := \frac{1}{2}[A(p) + A(p)^{∗}]. Using this fact one can easily prove the following lemma.

Lemma 3.1 For an arbitrary regular point p ∈ M of the vector field (2.4) the eigenvalues of the operator SA(p) : T_{p}M → T_{p}M are the eigenvalues of the operator SA(p) which is restricted to the linear subspace T^{⊥}(p) and the value −⟨∇f(p)f(p), f(p)⟩\||f(p)||^{2}.

Remark 3.1 For an arbitrary regular point p ∈ M of the vector field (2.4) we have

S∇f(p)z − \frac{f(p)}{||f(p)||^{2}} ⟨f(p), S∇f(p)z⟩ = SA(p)z for all z ∈ T^{⊥}(p).

In the following we denote at any regular point p of f the eigenvalues of the operator SA(p) restricted to the subspace T^{⊥}(p) by β_{1}(p) ≥ … ≥ β_{n−1}(p), which are ordered with respect to size and algebraic multiplicity. By Z(t, p) we denote the operator solution of (3.3) with initial condition Z(0, p) = id_{T^{⊥}(p)}. For every t ∈ ℝ the linear operator Z(t, p) : T^{⊥}(p) → T^{⊥}(φ^{t}(p)) maps between the subspaces T^{⊥}(p) and T^{⊥}(φ^{t}(p)) being orthogonal to the vector field in p and φ^{t}(p), respectively. The next lemma will be needed in the sequel and can be proved analogously to [14].

Lemma 3.2 Suppose that d ∈ (0, n − 1], written as d = d_{0} + d_{1} with d_{0} ∈ {0, …, n − 2} and d_{1} ∈ (0, 1], p ∈ M is a regular point of the vector field (2.4) and Z(., p) is the operator solution of (3.3). Then

ω_{d}(Z(t, p)) ≤ exp ∫_{0}^{t} [β_{1}(φ^{τ}(p)) + … + β_{d_{0}}(φ^{τ}(p)) + d_{1}β_{d_{0}+1}(φ^{τ}(p))] dτ.

holds for all t ≥ 0.

Let us consider again the vector field (2.4) and the sets K, K̃ satisfying (2.5) and Φ in (2.6). Choose a number δ > 0 in such a way that for every p ∈ Φ the exponential map exp_{p} is defined on B(O_{p}, δ) (the open ball in T_{p}M centered in the origin O_{p} with radius δ). For a regular point p ∈ M of f let B^{⊥}(O_{p}, r) = B(O_{p}, r) ∩ T^{⊥}(p) be the ball in the subspace T^{⊥}(p) centered in O_{p} with radius r < δ. Fix p and r and consider for any t ≥ 0 the ellipsoid E(t) = Z(t, p)B^{⊥}(O_{p}, r) in the subspace T^{⊥}(φ^{t}(p)). If σ_{1}(E(t)) ≥ … ≥ σ_{n−1}(E(t)) are the lengths of the semi-axes of E(t) and d is an arbitrary number in (0, n − 1] written as d = d_{0} + d_{1} with d_{0} ∈ {0, …, n − 2} and d_{1} ∈ (0, 1] we have by (3.1)

ω_{d}(E(t)) = ω_{d}(Z(t, p)) r^{d}.  (3.9)

Our aim is to describe the variation of time translated pieces of hypersurfaces orthogonal to a considered orbit of (2.4). For this purpose we will use methods from [12] extended there for stability investigations of flows on manifolds, in order to get information for the Hausdorff dimension of underlying flow invariant sets. Considering a non-equilibrium solution φ^{(.)}(p) of (2.4) with p ∈ K the local transformation of small pieces of a hypersurface can be described by a reparametrized flow ϕ^{t}(u) = φ^{s(t,u)}(u) in a neighborhood of the given semi-trajectory O_{+}(p) through p satisfying

⟨exp_{φ^{t}(p)}^{−1} (ϕ^{t}(u)) , f(φ^{t}(p))⟩ = 0

on a certain finite time interval [0, T]. We consider the (n − 1)-dimensional submanifold

B^{⊥}(p, δ) := exp_{p}(T^{⊥}(p) ∩ B(O_{p}, δ))

of M through p which is local transversal at the point p to the trajectory of the vector field passing through the point p. Every point u ∈ B^{⊥}(p, δ) can be uniquely written in the form u = exp_{p}(rv), where v ∈ T^{⊥}(p) is a vector of length ||v|| = 1 and r ∈ [0, δ) measures the arc length of the geodesic connecting p and u. This defines us a representation u = u(r, v) of a point u ∈ B^{⊥}(p, δ).

The main properties of the described reparametrization are summarized in the following lemma which proof we omit since argumentation is similar to [8, 12] where a slightly different reparametrization is considered. A detailed proof is done in [6].

Lemma 3.3 Suppose that φ^{(.)}(p) is a non-equilibrium solution of the C^{2}-vector field (2.4). Then for any finite number T > 0 there exists a number ε_{1} > 0 such that for every u ∈ B^{⊥}(p, ε_{1}) there is a monotonously increasing differentiable function s(., u) : ℝ_{+} → ℝ_{+} satisfying

⟨exp_{φ^{t}(p)}^{−1} (φ^{s(t,u)}(u)) , f(φ^{t}(p))⟩ = 0  (3.10)

with the initial condition s(t, p) = t for all t ∈ [0, T].

p  f(φ^{t}(p)) exp_{φ^{t}(p)}(T^{⊥}(φ^{t}(p))) φ^{t}(p) f(p) exp_{p}(T^{⊥}(p))  φ^{t}(u) φ^{s(t,u)}(u) u(r, v) v

Figure 1: Reparametrization of the flow

The next lemma states that for any regular point p ∈ M of f for the locally defined reparametrized flow ϕ^{t}(.) = φ^{s(t,.)}(.) the linearization d_{p}ϕ^{t} of ϕ^{t} restricted to T^{⊥}(p) is exactly the operator solution of (3.3). This provides the desired description of variation of time translated pieces of hypersurfaces orthogonal to the considered orbit.

Lemma 3.4 Suppose that φ^{(.)}(p) is a non-equilibrium solution of (2.4) and s(., .) : [0, T]× B^{⊥}(p, ε_{1}) → ℝ_{+} as given in Lemma 3.3 defines a reparametrized flow ϕ^{t}(u) := φ^{s(t,u)}(u). Then for all t ∈ [0, T] there holds

d_{p}ϕ^{t}|_{T^{⊥}(p)} = Z(t, p),

where Z(t, p) denotes the operator solution of (3.3) with Z(0, p) = id_{T^{⊥}(p)}.

Proof. Using the representation u = u(r, v) = exp_{p}(rv) with v ∈ T^{⊥}(p), ||v|| = 1, and r ∈ [0, ε_{1}) of a point u ∈ B^{⊥}(p, ε_{1}) we write for brevity s(t, r, v) ≡ s(t, u(r, v)). Let the reparametrized flow be defined by the map ϕ : [0, T] × B^{⊥}(p, ε_{1}) → M,

ϕ(t, u(r, v)) := φ^{s(t,r,v)}(exp_{p}(rv)).

Write for brevity ϕ(t, r, v) ≡ ϕ(t, u(r, v)). Recall from the beginning of this section that τ_{p}^{u} is the isometry between T_{p}M and T_{u}M given by parallel transport along a geodesic connecting p and u. We fix some t ∈ [0, T] and v ∈ T^{⊥}(p) with ||v|| = 1. For r ∈ [0, ε_{1}) there holds

D_{2}ϕ(t, r, v) = τ_{φ^{s(t,r,v)}(u)}^{φ^{t}(p)}f(φ^{s(t,r,v)}(u))\frac{∂s}{∂r}(t, r, v) + τ_{φ^{s(t,r,v)}(u)}^{φ^{t}(p)}d_{u}φ^{s(t,r,v)}τ_{p}^{u}v,

where D_{2}ϕ stands for the derivative of ϕ with respect to the second argument. Thus, in particular for r = 0 we have

D_{2}ϕ(t, 0, v) = f(φ^{t}(p))\frac{∂s}{∂r}(t, 0, v) + d_{p}φ^{t}v.  (3.11)

Hence, the claimed property (3.10) of the reparametrization s can be formulated as

⟨D_{2}ϕ(t, 0, v), f(φ^{t}(p))⟩ = 0

for every r ∈ [0, ε_{1}) and every t ∈ [0, T]. Therefore, equation (3.11) transforms into

\frac{∂s}{∂r}(t, 0, v) = −\frac{⟨d_{p}φ^{t}v, f(φ^{t}(p))⟩}{||f(φ^{t}(p))||^{2}}  (3.12)

and we conclude

d_{p}ϕ^{t}v = D_{2}ϕ(t, 0, v) = d_{p}φ^{t}v − f(φ^{t}(p))\frac{⟨d_{p}φ^{t}v, f(φ^{t}(p))⟩}{||f(φ^{t}(p))||^{2}}.

From formulas (3.5) and (3.6) we obtain d_{p}ϕ^{t} = Z(t, p). ■

4 Negatively invariant sets without equilibria

In the present section we derive upper bounds for the Hausdorff dimension of compact sets being negatively invariant with respect to the flow of the differential equation (2.4). Investigating the deformation of such a set by shift maps generated by the flow the deformation transversal to the flow lines is of great importance.

Our main result is the following theorem which may be considered as a generalization of the results [10, 11].

Theorem 4.1 Let f : M → T M be the C^{2}-vector field (2.4) on the smooth n-dimensional Riemannian manifold (M, g) satisfying the following conditions :

a) The flow {φ^{t}}_{t∈ℝ} of (2.4) satisfies (2.5) with respect to compact sets K and K̃ and K contains no equilibrium points.

b) Let β_{1}(p) ≥ … ≥ β_{n−1}(p) be the eigenvalues of the symmetric part SA(p) = \frac{1}{2}[A(p)+ A(p)^{∗}] of the operator A(p) from (3.4). There exist a number d ∈ (0, n − 1], written as d = d_{0} + d_{1} with d_{0} ∈ {0, 1, …, n − 2} and d_{1} ∈ (0, 1], a number Θ > 0 and a time T > 0 such that

∫_{0}^{T} [β_{1}(φ^{τ}(p)) + … + β_{d_{0}}(φ^{τ}(p)) + d_{1}β_{d_{0}+1}(φ^{τ}(p))] dτ ≤ −Θ  (4.1)

is satisfied for all regular points p ∈ K̃.

Then we have

1) dim_{H} K < d + 1.

2) If d_{0} = 1 and d_{1} = 0 we have dim_{H} K ≤ 1.

Before proving Theorem 4.1 we formulate some lemmata. The special structure of sets which are negatively invariant with respect to the flow allows us to obtain this dimension estimate. In order to describe the deformation under the map φ^{t} of tubular neighborhoods around an arc of trajectory we investigate the evolution of time translated pieces of hypersurfaces lying transversal to the considered trajectory. In the next lemma we consider the influence of φ^{t} on arcs of the trajectory. Recall that for an arbitrary piecewise smooth curve c : [t_{1}, t_{2}] → M we denote its length by l(c).

Lemma 4.1 Suppose that {φ^{t}}_{t∈ℝ} is the flow of (2.4), Φ ⊂ K̃ ⊂ M are compact sets such that Φ does not contain any equilibrium points of (2.4) and V (f, K̃, Φ) is the coefficient from (2.7). Let c^{t} : [t_{1}, t_{2}] → M for some p ∈ Φ be a restriction of the integral curve of (2.4) through p given by c^{t}(.) = φ(t + ., p)|_{[t_{1},t_{2}]} and satisfying c^{0}([t_{1}, t_{2}]) ⊂ Φ and c^{t}([t_{1}, t_{2}]) ⊂ K̃ for all t > 0. Then the lengths l(c^{t}) of these restrictions satisfy

l(c^{t}) ≤ V (f, K̃, Φ) l(c^{0}).

for all t ≥ 0.

Proof. The statement follows immediately from

l(c^{t}) = l(φ(t + ., p)|_{[t_{1},t_{2}]}) = ∫_{t_{1}}^{t_{2}} ∥φ̇(τ, φ^{t}(p))∥ dτ

= ∫_{t_{1}}^{t_{2}} \frac{∥φ̇(τ + t, p)∥}{∥φ̇(τ, p)∥} ∥φ̇(τ, p)∥ dτ ≤ V (f, K̃, Φ)l(c^{0}). (4.2)

■

Recall the definition of the family Γ of curves of length l_{0} in (2.8) and the chosen Carathéodory structure (F, ℙ, ξ, η, ψ) on the compact set K with tubes of length l_{0} in (2.1). The next lemma estimates the tubular measures μ_{C}(., d, ε), generated with respect to (F, ℙ, ξ, η, ψ), of the flow-changed set K. Its proof is based on the consideration of the deformation of tubular neighborhoods around trajectory pieces.

Lemma 4.2 Suppose {φ^{t}}_{t∈ℝ} is the flow of (2.4) satisfying (2.5) with respect to the compact sets K and K̃. Suppose also that Φ, V (f, K̃, Φ), and l_{0} are given by (2.6), (2.7), and (2.8), respectively, α_{1}(p) ≥ … ≥ α_{1}(p) are the eigenvalues of S∇f(p) and β_{1}(p) ≥ … ≥ β_{n−1}(p) are the eigenvalues from SA(p) of the operator A(p) from (3.4). Let Λ denote the set of equilibrium points in K̃. Define for a number d ∈ (0, n−1], written as d = d_{0} + d_{1} with d_{0} ∈ {0, …, n − 2} and d_{1} ∈ (0, 1], and a time T > 0 the values

k = sup_{p∈K̃\Λ} exp ∫_{0}^{T} [β_{1}(φ^{τ}(p)) + … + β_{d_{0}}(φ^{τ}(p)) + d_{1}β_{d_{0}+1}(φ^{τ}(p))] dτ

and

a = exp [3l_{0} max_{p∈K̃} α_{1}(p)\frac{max_{p∈K̃} ||f(p)||}{min_{p∈Φ} ||f(p)||^{2}} ] ,
λ = 2^{4}\sqrt{d_{0} + 1} a,
C = (3V (f, K̃, Φ) + 1) 2^{d_{0}+1}λ^{d}.

(4.3)

Then for any l > k there exists an ϵ_{0} > 0 such that for all ϵ ∈ (0, ϵ_{0}] the Carathéodory (d + 1)-measure at level ε μ_{C}(., d + 1, ε), generated with respect to the Carathéodory structure (2.1) with tubes of length l_{0}, satisfies the inequality

μ_{C}(φ^{T} (K) ∩ K, d + 1, λl^{\frac{1}{d}} ϵ) ≤ Clμ_{C}(K, d + 1, ϵ).

Proof. For arbitrary l > k we can choose an open set V ⊂ M containing K such that the closure \overline{V} is compact and the inequality

k′ := sup_{u∈\overline{V} \Λ} exp ∫_{0}^{T} [β_{1}(φ^{τ}(u)) + … + β_{d_{0}}(φ^{τ}(u)) + d_{1}β_{d_{0}+1}(φ^{τ}(u))] dτ < l  (4.4)

is satisfied. We set

σ := sup_{u∈\overline{V} \Λ} exp ∫_{0}^{T} β_{1}(φ^{τ}(u))dτ  (4.5)

and take a number m > 0 such that k′ < m^{d} and σ ≤ m are satisfied. Since l > k′ the equation

[1 + (\frac{m^{d_{0}}}{k′} )^{1/d_{1}} η]^{d} k′ = l

uniquely defines a number η > 0. We can choose δ > 0 such that for all p ∈ \overline{V} the exponential map exp_{p} is defined on B(O_{p}, δ), satisfying ||d_{v} exp_{p} || ≤ 2 for all v ∈ B(O_{p}, δ), and therefore

ρ(exp_{p} v_{1}, exp_{p} v_{2}) ≤ 2ρ(v_{1}, v_{2})  (4.6)

holds for all v_{1}, v_{2} ∈ B(O_{p}, δ). We take ε > 0 small enough such that such that the following hold

- ε < inf{δ, l_{0}/2, dist(Φ, M\V)/2}

- For all p ∈ K there holds

sup_{q∈B(p,8\sqrt{d_{0}+1} σ2ε), t∈[0,τ_{0}]} ||τ_{φ^{t}(q)}^{φ^{t}(p)}d_{q}φ^{t}τ_{p}^{q} − d_{p}φ^{t}|| ≤ a,  (4.7)

where a is defined in (4.3) and τ_{0} denotes the maximal time for which φ(., p)|_{[0,τ_{0}]} for any p ∈ K is of maximal length 3l_{0}.

- For arbitrary p ∈ K the function s : [0, max{T, τ_{0}}] × B^{⊥}(p, 2ε) → ℝ_{+} as given in Lemma 3.3 defines the locally reparametrization of the flow φ by ϕ : [0, T] × B^{⊥}(p, ε) → M with ϕ(t, .) = ϕ^{t}(.) = φ^{s(t,.)}(.) for t ∈ [0, T], and

sup_{q∈B^{⊥}(p,ε)} ||τ_{ϕ^{T} (q)}^{ϕ^{T} (p)}d_{q}ϕ^{T} τ_{p}^{q} − d_{p}ϕ^{T} || ≤ η

holds for all p ∈ K. Analyzing this reparametrization for fixed p ∈ K but for arbitrary t ∈ [0, T] the deviation between the map ϕ^{t} and φ^{t} for u = exp_{p}(rv) ∈ B^{⊥}(p, 2ε) is of the form ρ(φ^{t}(u), ϕ^{t}(u)) = o(r) as r → 0. Suppose that for any p ∈ K for all t ∈ [0, T]

ρ(φ^{t}(u), ϕ^{t}(u)) ≤ L r

holds for all u = exp_{p}(rv) ∈ B^{⊥}(p, 2ε), where L is a constant defined by

L = \sqrt{d_{0} + 1} exp (min_{u∈Φ} ∫_{0}^{T} β_{n−1}(ϕ^{τ}(u))dτ) .  (4.8)

- For any p ∈ K and for some t(p) > 0 let γ_{p}(.) = φ(., p)|_{[0,t(p)]} be a restriction of an integral curve starting in p such that γ_{p}(.) is of length 2l_{0}. For such curves γ_{p} and ε ∈ (0, δ] we define

S(γ_{p}, ε) := ⋃_{u∈φ([0,t(p)],p)} B^{⊥}(u, ε).

For fixed p ∈ K the deviation of points ϕ^{t}(u) = φ^{s(t,u)}(u) on an integral curve starting in u = exp_{p}(rv) ∈ B^{⊥}(p, ε) for fixed t ∈ [0, T] is of the form

ρ(φ^{t}(p), ϕ^{t}(u)) ≤ ||d_{p}ϕ^{t}|| · r(1 + O(r))

as r → 0. Using an extensive standard technique we can show that we can make ε > 0 so small such that for any r ∈ (0, ε] the following holds : Let γ′ = ϕ([0, t(p)], u) be some arc of trajectory intersecting S(γ_{p}, r). Then γ′ is completely contained in S(γ_{p}, 2r) and the length of γ′ does not exceed the value 3l_{0}.

Now let ε_{0} = min\{\sqrt{d_{0} + 1}l^{1/d}2δ, ε/2\}. We fix some cover \{Ω(γ_{i}, r_{i}) ∩ K\}_{i≥1} ⊂ F of K by tubes Ω(γ_{i}, r_{i}) of length l_{0}, r_{i} ≤ ε_{0} and γ_{i} ∈ Γ, where Γ is the set of curves defined in (2.8).

To simplify the use of the reparametrized flow we cover each tube Ω(γ_{i}, r_{i}) by a set S(γ_{p_{i}}, r_{i}), where γ_{p_{i}}(.) = φ(., p_{i})|_{[0,t(p_{i})]} is a curve of length 2l_{0} such that γ_{i} ⊂ γ_{p_{i}} and Ω(γ_{i}, r_{i}) ⊂ S(γ_{p_{i}}, r_{i}) hold. This way, we obtain a cover {S(γ_{p_{i}}, r_{i})}_{i≥1}, r_{i} ≤ ε_{0}, of K where γ_{p_{i}} are pieces of integral curves of length 2l_{0} starting in some p_{i} ∈ K for all i ≥ 1.

Every set B^{⊥}(p_{i}, 2r_{i}) is contained in the open set V. Taylor’s formula for differentiable maps provides that for every u ∈ B^{⊥}(p_{i}, 2r_{i})

|| exp_{ϕ^{T} (p_{i})}^{−1} ϕ^{T} (u) − d_{p_{i}}ϕ^{T} (exp_{p_{i}}^{−1}(u))||

≤ sup_{q∈B^{⊥}(p_{i},2r_{i})} ||τ_{ϕ^{T} (q)}^{ϕ^{T} (p_{i})}d_{q}ϕ^{T} τ_{p_{i}}^{q} − d_{p_{i}}ϕ^{T} || · || exp_{p_{i}}^{−1}(q)|| (4.9)

holds. Considering the image of B^{⊥}(p_{i}, 2r_{i}) under ϕ^{T} we obtain the inclusion

exp_{φ^{T} (p_{i})}^{−1} (ϕ^{T} (B^{⊥}(p_{i}, 2r_{i}))) ⊂ d_{p_{i}}ϕ^{T} (B^{⊥}(O_{p_{i}}, 2r_{i})) + B^{⊥}(O_{φ^{T} (p_{i})}, η2r_{i}).  (4.10)

The set d_{p_{i}}ϕ^{T} (B^{⊥}(O_{p_{i}}, 2r_{i})) is an ellipsoid with half-axes of length 2r_{i}σ_{k}(p_{i}), where σ_{k}(p_{i}) (k = 1, …, n − 1) denote the singular values of the linear operator d_{p_{i}}ϕ^{T} : T^{⊥}(p_{i}) → T^{⊥}(φ^{T} (p_{i})). Using the definition of k′, Lemma 3.2 and (3.9) we can conclude

ω_{d} (d_{p_{i}}ϕ^{T} (B^{⊥}(O_{p_{i}}, 2r_{i}))) ≤ (2r_{i})^{d} k′.  (4.11)

By standard methods (see e.g. [14]) an ellipsoid E_{i} ⊂ T^{⊥}(φ^{T} (p_{i})) can be found containing exp_{φ^{T} (p_{i})}^{−1} (ϕ^{T} (B^{⊥}(p_{i}, 2r_{i}))) and satisfying ω_{d}(E_{i}) ≤ l(2r_{i})^{d}.

Any set E_{i} can be covered by at least N_{i} balls of radius R_{i} = \sqrt{d_{0} + 1}σ_{d_{0}+1}(E_{i}). This number can be estimated from above by

N_{i} ≤ \frac{2^{d_{0}}ω_{d}(E_{i})}{σ_{d_{0}+1}(E_{i})^{d}} .

Projecting the cover of E_{i} onto the manifold M and using (4.6) we obtain that the set ϕ^{T} (B^{⊥}(p_{i}, 2r_{i})) can already be covered by N_{i} balls of radius 2R_{i}. Considering now the deviation of the reparametrized flow we see that any set φ^{T} (B^{⊥}(p_{i}, 2r_{i})) can be covered by N_{i} balls of radius 2R_{i} + 2L2r_{i}, where L is defined in (4.8).

Following considerations of [14] we obtain

L ≤ \sqrt{d_{0} + 1}σ_{d_{0}+1}(d_{p_{i}}ϕ^{T})

and therefore

2L2r_{i} ≤ 2\sqrt{d_{0} + 1}σ_{d_{0}+1}(d_{p_{i}}ϕ^{T})2r_{i} = 2R_{i}.

Thus, any set φ^{T} (B^{⊥}(p_{i}, 2r_{i})) can be covered by N_{i} balls of radius 4R_{i}. Fixing such a cover {B(ũ_{ij}, 4R_{i})}_{j≥1} we choose in every set K ∩B(ũ_{ij}, 4R_{i})∩φ^{T} (B^{⊥}(p_{i}, 2r_{i})) a point u_{ij} and obtain the cover {B_{ij}}_{j≥1} = {B(u_{ij}, 8R_{i})}_{j≥1} of the set φ^{T} (B^{⊥}(p_{i}, 2r_{i})) ∩ K. Shifting these balls B_{ij} along the flow lines we construct a cover of φ^{T} (S(γ_{p_{i}}, r_{i})) ∩ K.

Now fix t_{ij} > 0 such that γ_{ij}([0, t_{ij}]) = φ([0, t_{ij}], u_{ij}) lies in φ^{T} (S(γ_{p_{i}}, 2r_{i})) and γ_{ij}(t_{ij} + τ) ∉ φ^{T} (S(γ_{p_{i}}, 2r_{i})) for all τ > 0. Observing (4.7) we obtain ⋃_{τ∈[0,t_{ij}]} φ^{τ}(B_{ij}) ∩ K ⊂ Ω(γ_{ij}, a8R_{i}). Each curve γ_{ij}(.) can also be written as γ_{ij}(.) = φ(T + . , u)|_{[0,t_{ij}]} with φ(., u)|_{[0,t_{ij}]} = ϕ(., u)|_{[0,t(p_{i})]} for some u ∈ B^{⊥}(p_{i}, 2r_{i}). The value ε_{0} has been chosen in such a way that the length of any ϕ(., u)|_{[0,t(p_{i})]} does not exceed 3l_{0}. Because of Lemma 4.1 the length of each γ_{ij} does not exceed the value 3l_{0}V (f, K̃, Φ).

We resume that every finite cover of the compact set K with tubes of length l_{0} {Ω(γ_{i}, r_{i})}_{i≥1} generates a cover {Ω(γ_{ij}, a8R_{i})}_{i≥1,j≥1} of the set φ^{T} (K) ∩ K. Covering each curve arc γ_{ij}([0, t_{ij}]) by arcs of curves of length l_{0} we conclude

μ_{C}(φ^{T} (K) ∩ K, d + 1, a8\sqrt{d_{0} + 1}l^{1/d}2ε)

≤ 2 ∑_{i≥1} (3V (f, K̃, Φ) + 1) N_{i} (a8R_{i})^{d}

≤ 2 (3V (f, K̃, Φ) + 1) 2^{d_{0}}(a8\sqrt{d_{0} + 1}2)^{d}l ∑_{i≥1} r_{i}^{d}.

Since we covered the set K by an arbitrary finite family of tubes of radius less or equal than ε_{0} we can pass in the last expression to the infimum and obtain

μ_{C}(φ^{T} (K) ∩ K, d + 1, λl^{1/d}ε) ≤ Clμ_{C}(K, d + 1, ε)

where λ and C are defined by (4.3). The proof is complete. ■

Although we are mainly interested in upper estimates of the Hausdorff dimension of flow negatively invariant sets we can deduce upper bounds of its Carathéodory dimension with respect to the chosen tubular Carathéodory structure.

Proposition 4.1 Let the differential equation (2.4) satisfy the conditions of Theorem 4.1 with the number d = d_{0} +d_{1} ∈ (0, n−1] in (4.1) and the negatively invariant set K. Then the Carathéodory dimension of K, determined with respect to the Carathéodory structure (2.1) on K consisting of tubes with length l_{0} determined in (2.8), satisfies

dim_{C} K < d + 1.

Proof. It follows from (4.1) that for arbitrarily small number κ ∈ (0, 1) there exists some number m = m(κ) > 0 such that

k := sup_{p∈K} exp ∫_{0}^{mT} [β_{1}(φ^{τ}(p)) + … + β_{d_{0}}(φ^{τ}(p)) + d_{1}β_{d_{0}+1}(φ^{τ}(p))] dτ

≤ exp(−mΘ) < κ. (4.12)

Without loss of generality we can assume that this number k satisfies

λk^{1/d} < 1 and Ck < 1,  (4.13)

where λ and C are the constants given in (4.3). We choose l > k with

λl^{1/d} < 1 and Cl < 1.  (4.14)

Lemma 4.2, applied to the map φ^{mT}, guarantees that for the chosen number l there exists a number ε_{0} > 0 such that for all ε ∈ (0, ε_{0}] the inequality

μ_{C}(φ^{mT} (K) ∩ K, d + 1, λl^{1/d}ε) ≤ Clμ_{C}(K, d + 1, ε).  (4.15)

holds. Let ε ∈ (0, ε_{0}] be arbitrarily small and suppose that μ_{C}(K, d + 1, ε) is finite and μ_{C}(K, d + 1, ε) ≠ 0. Since K is negatively invariant with respect to φ^{mT} we have K = φ^{mT} (K) ∩ K. Using inequality (4.14) we conclude

μ_{C}(K, d + 1, λl^{1/d}ε) < μ_{C}(K, d + 1, ε).

By λl^{1/d} < 1 and the fact that μ_{C}(K, d + 1, ε) is monotonously increasing as ε → 0 + 0 we get μ_{C}(K, d + 1, ε) = 0, which contradicts our assumption.

Thus, the equality μ_{C}(K, d+1, ε) = 0 holds for every ε ∈ (0, ε_{0}]. We see that μ_{C}(K, d+ 1) = 0. This implies dim_{C} K ≤ d + 1. Since (4.12) holds true if we slightly reduce d = d_{0} + d_{1} we conclude dim_{C} K < d + 1. ■

Proof of Theorem 4.1. Applying Proposition 4.1 and Corollary 2.1 we obtain dim_{H} K < d + 1. If condition (4.1) is also satisfied for d = 1 it is satisfied for all d ∈ (0, n − 1]. Thus, dim_{H} K < d + 1 for all d ∈ (0, n − 1] and we obtain dim_{H} K ≤ 1. ■

5 Negatively invariant sets with a finite number of equilibrium points

In this section we consider compact sets K and K̃ satisfying (2.5) with respect to the flow of (2.4). Unilke in Section 4 now we assume that the set K may possess a finite number of equilibrium points and satisfies the following condition :

(S) The set K contains a finite number of equilibrium points of (2.4). Every such equilibrium point possesses a local stable manifold with dimension at least n − 1. Trajectories starting in local unstable manifolds or local center manifolds of an equilibrium point in K converge for t → +∞ to an asymptotically stable equilibrium point of (2.4) in K̃.

The special structure of equilibrium points satisfying (S) allows us to obtain the following theorem.

Theorem 5.1 Let f : M → T M be a C^{2}-vector field (2.4) on the smooth n-dimensional Riemannian manifold (M, g). Suppose that the flow {φ^{t}}_{t∈ℝ} of (2.4) satisfies (2.5) and condition (S), with respect to compact sets K and K̃, and condition b) of Theorem 4.1. Then the conclusion of Theorem 4.1 holds.

The basic idea for proving Theorem 5.1 is to exclude an open flow positively invariant set K_{1} containing all equilibrium points of K̃ and to apply Theorem 4.1 to the remaining set. First we formulate the following lemma.

Lemma 5.1 Let ψ : M → M be a C^{1}-map on the smooth n-dimensional Riemannian manifold (M, g). Suppose that the compact sets K̃, K, K_{1} ⊂ M exist such that

K ⊂ ψ^{k}(K) ⊂ K̃  and  ψ^{k}(K_{1}) ⊂ K_{1} ⊂ K̃  (5.1)

for all integers k ≥ 0. If d, d′ > 0 are numbers satisfying dim_{H}(K\K_{1}) < d′ < d and lim_{k→∞} μ_{H}(ψ^{k}(K_{1}), d′, ϵ) = 0 for all ϵ > 0 then dim_{H} K < d.

Proof. Since dim_{H}(K\K_{1}) < d′, we have μ_{H}(K\K_{1}, d′, ε) = 0 for all ε > 0. The C^{1}-property of ψ implies that for arbitrary integer k > 0 one has that

0 ≤ μ_{H}(ψ^{k}(K\K_{1}), d′, ε) ≤ C^{kd}μ_{H}(K\K_{1}, d′, ε) = 0,

where C is a global Lipschitz constant of ψ on K̃. We fix now an arbitrary number ε > 0, use (5.1) and the outer measure property of μ_{H}(., d′, ε) to obtain that

0 ≤ μ_{H}(K, d′, ε) ≤ μ_{H}(ψ^{k}(K), d′, ε)

≤ μ_{H}(ψ^{k}(K\K_{1}), d′, ε) + μ_{H}(ψ^{k}(K_{1}), d′, ε) = μ_{H}(ψ^{k}(K_{1}), d′, ε)

for every k > 0. Because of lim_{k→∞} μ_{H}(ψ^{k}(K_{1}), d′, ϵ) = 0 for arbitrary ε′ > 0 there exists a number N(ε′) > 0 such that μ_{H}(ψ^{N(ε′)}(K_{1}), d′, ε) < ε′ and thus μ_{H}(K, d′, ε) = 0. The number ε > 0 has been chosen arbitrarily. Thus, we conclude lim_{ε→0+0} μ_{H}(K, d′, ε) = μ_{H}(K, d′) = 0 and dim_{H} K ≤ d′ < d follows. ■

Proof of Theorem 5.1. We restrict ourself to the case that K̃ contains three equilibrium points p_{1}, p_{2}, and q, only and q ∈ K. Further we assume that p_{1} and p_{2} both possess n-dimensional local stable manifolds, that q possesses an (n − 1)-dimensional local stable and an one-dimensional local unstable manifold, W_{loc}^{s}(q) and W_{loc}^{u}(q), respectively, and that points in W_{loc}^{u}(q) are either converging under the flow for t → +∞ to p_{1} or p_{2}. (Other cases can be proved in analogous manner.) We are sketching the construction of a set K_{1} ⊂ K̃, which contains these equilibrium points and which is flow positively invariant. There exist open neighborhoods V_{1} around p_{1} and V_{2} around p_{2} which are positively flow invariant and satisfy

lim_{t→∞} φ^{t}(u) = p_{1} for all u ∈ V_{1} and lim_{t→∞} φ^{t}(u) = p_{2} for all u ∈ V_{2}.

We choose a curve γ : [t_{1}, t_{2}] → W_{loc}^{u}(q) satisfying γ([t_{1}, t_{2}]) ⊃ B(q, ε′) ∩ W_{loc}^{u}(q) for some ε′ > 0. Since q possesses an (n − 1)-dimensional local stable manifold we find a number Δ_{1} > 0 such that the set B(q, Δ_{1}) ∩ W_{loc}^{s}(q) is flow positively invariant. Since points in W_{loc}^{u}(q) converge under the flow for t → +∞ either to p_{1} or to p_{2} we find a number Δ_{2} > 0 such that

lim_{t→∞} φ^{t}(p) ∈ V_{1} ∪ V_{2} for all p ∈ Ω(γ, Δ_{2})\W_{loc}^{s}(q)

holds. For Δ := min{Δ_{1}, Δ_{2}} we define the set U := ⋃_{t≥0} φ^{t}(Ω(γ, Δ)). By construction the set

K_{1} := V_{1} ∪ V_{2} ∪ U

is open and flow positively invariant. The set K\K_{1} is compact, flow negatively invariant and does not contain equilibrium points. Hence, we can apply Theorem 4.1 and we obtain dim_{H}(K\K_{1}) < d + 1.

Let denote Y := ⋃_{t≥0} φ^{t}(γ([t_{1}, t_{2}])) ∪ {p_{1}} ∪ {p_{2}}. By construction lim_{t→∞} φ^{t}(K_{1}) = Y holds. Since Y is a smooth one-dimensional manifold we have dim_{H} Y = 1 and so μ_{H}(Y, d′) = 0 for all d′ > 1. We choose an arbitrary number d′ with dim_{H}(K\K_{1}) < d′ < d + 1. It holds μ_{H}(Y, d′, ε) = 0 for all ε > 0. For every p ∈ K̃ there exists an δ(p) > 0 such that ||d_{v} exp_{p} || ≤ 2 for all v ∈ B(O_{p}, δ(p)). We set δ := min_{p∈K̃} δ(p). Thus, for every p ∈ K̃ there holds ρ(exp_{p} v_{1}, exp_{p} v_{2}) ≤ 2ρ(v_{1}, v_{2}) for all v_{1}, v_{2} ∈ B(O_{p}, δ). We fix an arbitrary ε ∈ (0, δ]. Since Y is compact, we can cover Y for every ε_{1} > 0 by a finite number of balls B(p_{i}, r_{i}) of radius r_{i} ≤ ε satisfying ∑_{i≥1} r_{i}^{d′} < ε_{1}\(2^{n+d′} \sqrt{n}^{d′}). Since the cover is finite we can define \underline{ε} := min_{i≥1} r_{i}. For \underline{ε} there exists a time T (ε_{1}) such that dist(φ^{t}(K_{1}), Y) < \underline{ε} for all t ≥ T (ε_{1}). Thus, the set φ^{t}(K_{1}) can be covered by {B(p_{i}, 2r_{i})}_{i≥1} for all t ≥ T (ε_{1}). Each ball B(p_{i}, 2r_{i}) can be covered already by 2^{n} balls with radius 2r_{i}\sqrt{n} and we get that

μ_{H}(φ^{t}(K_{1}), d′, ε) ≤ 2^{n} ∑_{i≥1} r_{i}^{d′}(2\sqrt{n})^{d′} < ε_{1},

which holds for all t ≥ T (ε_{1}). This implies lim_{t→∞} μ_{H}(φ^{t}(K_{1}), d′, ε) = 0 for all ε > 0.

The set K_{1} is positively flow invariant. Applying Lemma 5.1 we conclude dim_{H} K < d + 1. ■

In the following statement we denote for a differentiable function v : M → ℝ by L_{f}v(p) = d_{p}v f(p) the Lie derivative of v in direction of the vector field f.

Corollary 5.1 Suppose that the flow {φ^{t}}_{t∈ℝ} of (2.4) satisfies (2.5), condition (S) with respect to compact sets K and K̃. Denote by Λ the set of equilibrium points in M, let β_{1}(p) ≥ … ≥ β_{n−1}(p) be the eigenvalues of the symmetric part SA(p) of the operator A(p) from (3.4), and let v : M\Λ → ℝ be a C^{1}-function. Suppose that for a number d = d_{0} + d_{1} ∈ (0, n − 1] there exist a number Θ > 0 and a time T > 0 such that

∫_{0}^{T} [β_{1}(φ^{τ}(p)) + … + β_{d_{0}}(φ^{τ}(p)) + d_{1}β_{d_{0}+1}(φ^{τ}(p)) + L_{f}v(φ^{τ}(p))] dτ ≤ −Θ (5.2)

holds for all regular points p ∈ K̃. Then the conclusion of Theorem 4.1 holds.

Proof. As in the proof of Theorem 5.1 we construct an open and positively flow invariant set K_{1} ⊂ K̃ with the following properties : It is K\K_{1} compact, flow negatively invariant and does not contain equilibrium points. And for d′ > 0 with dim_{H}(K\K_{1}) < d′ < d + 1 there holds lim_{t→∞}(φ^{t}(K_{1}), d′, ε) = 0 for all ε > 0.

If h : M\Λ → ℝ_{+} is a function of class C^{1} on M\Λ, we can introduce a new metric tensor on M\Λ by ĝ(p) := h^{2}(p)g(p) for p ∈ M\Λ. It is easy to show that ĝ is a Riemannian metric, on K̃\K_{1} equivalent to g. Since K\K_{1} is compact any change to an equivalent metric does not alter the Hausdorff dimension of K\K_{1}.

Let us fix a regular point p ∈ M. Then the symmetrized covariant derivative ∇f(p) and ˆ∇f(p) at p with respect to the metric g and ĝ, respectively, are related by the relation

\frac{1}{2} [ ˆ∇f(p) + ˆ∇f(p)^{∗}] = \frac{1}{2} [∇f(p) + ∇f(p)^{∗}] + \frac{L_{f}h(p)}{h(p)}id_{T_{p}M}.  (5.3)

Now we consider the linear operators A(p) and Â(p) as defined in (3.4) regarding the metric g and ĝ, respectively. For arbitrary v, w ∈ T_{p}M there holds ⟨v, w⟩ĝ = h^{2}(p)⟨v, w⟩_{g}. Thus, relations (3.8) and (5.3) imply for all v ∈ T^{⊥}(p)

⟨[Â(p) + Â(p)^{∗}] v, v⟩_{g} = ⟨[A(p) + A(p)^{∗}] v, v⟩_{g} + ⟨\frac{L_{f}h(p)}{h(p)}v, v⟩_{g} .  (5.4)

Now we choose the function h(p) = exp(\frac{v(p)}{d}) (p ∈ M). By L_{f}h(p) = h(p)\frac{L_{f} v(p)}{d} and (5.4) we obtain that the eigenvalues β̂_{i}(p) of the operator \frac{1}{2}[Â(p) + Â(p)^{∗}]|_{T^{⊥}(p)} are related to the eigenvalues β_{i}(p) (i = 1, …, n − 1) with respect to the original metric g by β̂_{i}(p) = β_{i}(p) + \frac{L_{f} v(p)}{d}. Therefore,

β̂_{1}(p) + … + β̂_{d_{0}}(p) + d_{1}β̂_{d_{0}+1}(p) = β_{1}(p) + … + β_{d_{0}} + d_{1}β_{d_{0}+1}(p) + L_{f}v(p)

guarantees (5.2) and thus (4.1) of Theorem 4.1. Hence dim_{H}(K\K_{1}) < d + 1. Applying Lemma 5.1 we conclude dim_{H} K < d + 1. ■

6 Examples

The considered outer measures defined via tube covers show in many cases a better contraction behavior under the positive semi-flow of a vector field than conventional outer measures defined via a covering of balls do. Thus, for a certain class of flows on ndimensional Riemannian manifolds we can improve upper Hausdorff dimension estimates which are obtained with methods from [14] or from [17] for the ℝ^{n} which we show in Example 6.1. For certain vector fields in ℝ^{3} we show in Example 6.3 that the methods of the present paper provide always more effective conditions for upper Hausdorff bounds than those from [14, 17]. In addition, we improve for these systems results about the structure of ω-limit sets, which are closely related to results in [2, 7, 8].

Example 6.1 Consider the vector field f : M → T M from (2.4) on the smooth ndimensional Riemannian manifold (M, g) and let K, K̃ ⊂ M be two compact sets satisfying (2.5). Suppose that K̃ does not contain equilibrium points of (2.4) and suppose that for the vector field f the property

⟨S∇f(p)z, f(p)⟩ = 0 holds for all p ∈ K̃ and all z ∈ T^{⊥}(p).

Therefore by definition (3.4) it follows that for any p ∈ K̃ the operators SA(p)|_{T^{⊥}(p)} and S∇f(p)|_{T^{⊥}(p)} are identical. By α_{1}(p) ≥ … ≥ α_{n}(p) we denote the eigenvalues of S∇f(p). Theorem 3.1 from [14] states that if for some d = d_{0} + d_{1} ∈ (0, n] with d_{0} ∈ {0, …, n − 1} and d_{1} ∈ (0, 1] the inequality

α_{1}(p) + … + α_{d_{0}}(p) + d_{1}α_{d_{0}+1}(p) < 0

holds for all p ∈ K̃, the estimate dim_{H} K < d = d_{0} + d_{1} is true. With the help of Lemma 3.1 one shows that the (n−1) eigenvalues of SA(p)|_{T^{⊥}(p)} coincide with (n−1) eigenvalues of S∇f(p). The subspace T^{⊤}(p) = {αf(p) : α ∈ ℝ} is the eigenspace of the remaining nth eigenvalue \overline{α}(p) ∈ \{α_{1}(p), …, α_{n}(p)\}. For the C^{1}-function v : K̃ → ℝ given by v(p) = \frac{1}{2} ln ||f(p)||^{2} we have L_{f}v(p) = \frac{⟨∇f(p)f(p),f(p)⟩}{||f(p)||^{2}} = \overline{α}(p) for each p ∈ K̃. If \overline{α}(p) ≥ 0 holds for all p ∈ K̃ then

α_{1}(p) + … α_{d_{0}}(p) + d_{1}α_{d_{0}+1}(p) = β_{1}(p) + … β_{d_{0}−1}(p) + d_{1}β_{d_{0}}(p) + L_{f}v(p).

Thus Theorem 4.1 gives an upper bound of dim_{H} K which is less than or equal to the upper bound we would get applying Theorem 3.1 from [14]. If d_{0} = 1 then Theorem 4.1 gives the better estimate dim_{H} K ≤ 1. □

Example 6.2 As a special case of the vector field described in Example 6.1 consider the vector field in ℝ^{2} given by

θ̇ = a sin θ, ẋ = −x + b  (6.1)

(with parameters a ≥ 1, b ≠ 0), being in the first coordinate periodic with period 2π. The arising dynamical system can be interpreted as a dynamical system on the cylinder Z = ℝ^{2}\G with respect to the subgroup G = {k·(2π, 0)|k ∈ ℤ} of ℝ^{2}. Let τ : ℝ^{2} → ℝ^{2}\G denote the canonical projection which associates to any z ∈ ℝ^{2} the equivalence class [z].

The family of maps {τ_{|_{τ(U)}}^{−1} : τ(U) ↦ U | U ⊂ ℝ^{2} open, τ : U → Z injective} is a twodimensional atlas of class C^{∞} on Z. With the standard metric for factor manifolds given by (g_{ij}) = I in any chart, Z becomes a smooth 2-dimensional Riemannian manifold which is called flat cylinder. Every solution of (6.1) is bounded in the second coordinate. Obviously, the set K = {u ∈ Z|u = [z], z = (θ, 0), θ ∈ ℝ} is compact and flow invariant with respect to (6.1). The variational system (3.2) with respect to any solution (θ(t), 0) in K is given by

ẏ = (\begin{matrix}a cos θ(t) & 0 \\ 0 & −1 \end{matrix}) y.

The vectors in T^{⊥}(θ(t), 0) are of the form (0, x) with x ∈ ℝ. It follows that the system in normal variations (3.3) is given by

ż = (\begin{matrix}−a cos θ(t) & 0 \\ 0 & −1 \end{matrix}) z.

Thus β_{1}(x) ≡ −1 and condition (4.1) is satisfied with d_{0} = 1, d_{1} = 0 and Θ = T = 1. By Theorem 4.1 we conclude that dim_{H} K ≤ 1. Note that in the present situation other available theorems [14, 17] are not applicable since the divergence of the right-hand side of (6.1) gives the expression a cos θ − 1 which is, in contrast to the assumptions of Theorem 3.1 from [14], not always negative. □

Example 6.3 Consider an arbitrary C^{2}-vector field f in ℝ^{3}, e.g., the differential equation

ẋ = f(x) .  (6.2)

Suppose that for (6.2) the global flow {φ^{t}}_{t∈ℝ} exists. Let K, K̃ ⊂ ℝ^{3} be two compact sets and suppose that for the flow {φ^{t}}_{t∈ℝ} of (6.2) it holds K ⊂ φ^{t}(K) ⊂ K̃ for all t ≥ 0. Denote the Jacobi matrix of f in x by Df(x). Suppose that K̃ possesses a finite number of equilibrium points and for any equilibrium point x ∈ K̃ of (6.2) all eigenvalues of Df(x) have negative real part. Introduce the symmetric part of Df(x) by Q(x) = SDf(x) = \frac{1}{2} (Df(x) + Df(x)^{∗}). As in the previous sections for any regular x of f define the hyperplanes T^{⊥}(x) = {z ∈ ℝ^{3}| f(x)^{∗}z = 0} and the linear operator SA(x) : T^{⊥}(x) → T^{⊥}(x), given by

SA(x) = SDf(x) − \frac{f(x)f(x)^{∗}}{||f(x)||^{2}} SDf(x) = (I − \frac{f(x)f(x)^{∗}}{||f(x)||^{2}} ) Q(x).

Denote the eigenvalues of Q(x), ordered with respect to size and algebraic multiplicity, by α_{1}(x) ≥ α_{2}(x) ≥ α_{3}(x). Suppose that β_{1}(x) ≥ β_{2}(x) are the eigenvalues of SA(x) restricted to the subspace T^{⊥}(x) and suppose further that β_{1}(x) and β_{2}(x) are not eigenvalues of S∇f(x). Since

det (βI − (I − \frac{f(x)f(x)^{∗}}{||f(x)||^{2}} ) Q(x))

= det(βI − Q(x)) (1 + \frac{f(x)^{∗}Q(x)(βI − Q(x))^{−1}f(x)}{||f(x)||^{2}})

= \frac{β det(βI − Q(x))}{||f(x)||^{2}}f(x)^{∗}(βI − Q(x))^{−1}f(x)

we see that β_{1} and β_{2} are solutions of the equation f(x)^{∗} (β_{i}(x)I − Q(x))^{−1} f(x) = 0. We introduce the notation

det(βI − Q(x)) ≡ β^{3} + δ_{2}(x)β^{2} + δ_{1}(x)β + δ_{0}(x).

Note that δ_{2}(x) = −(α_{1}(x)+α_{2}(x)+α_{3}(x)), δ_{1}(x) = α_{1}(x)α_{2}(x)+α_{2}(x)α_{3}(x)+α_{1}(x)α_{3}(x) and δ_{0}(x) = −α_{1}(x)α_{2}(x)α_{3}(x). A direct calculation shows that

lim_{β→∞} \frac{β}{||f(x)||^{2}} f(x)^{∗}(βI − Q(x))^{−1}f(x) = 1,  (6.3)
lim_{β→∞}β (\frac{β}{||f(x)||^{2}} f(x)^{∗}(βI − Q(x))^{−1}f(x) − 1)

= lim_{β→∞} \frac{β}{||f(x)||^{2}} f(x)^{∗}Q(x)(βI − Q(x))^{−1}f(x)

= \frac{1}{||f(x)||^{2}} f(x)^{∗}Q(x)f(x) =: Δ_{1}(x),  (6.4)
and lim_{β→∞} \frac{β}{||f(x)||^{2}} (βf(x)^{∗}Q(x)(βI − Q(x))^{−1}f(x) − f(x)^{∗}Q(x)f(x))
= \frac{1}{||f(x)||^{2}} f(x)^{∗}Q(x)^{2}f(x) =: Δ_{2}(x).  (6.5)

From (6.3)-(6.5) it follows that the eigenvalues β_{i}(x) (i = 1, 2) of SA(x) are the zeros of the polynomial

β^{2} + [δ_{2}(x) + Δ_{1}(x)] β + [δ_{1}(x) + δ_{2}(x)Δ_{1}(x) + Δ_{2}(x)] .

Using this fact one sees immediately that the assumptions of Corollary 5.1 are satisfied for (6.2) if we suppose for the auxiliary function v(x) = \frac{1}{2} ln ||f(x)||^{2}, defined on the set of all regular points of ℝ^{3}, the following conditions : There exists a continuous function s : K̃ → [0, d_{1}] with d_{1} ∈ (0, 1] such that with h(x) := \frac{1−s(x)}{1+s(x)} the inequalities

δ_{2}(x) − h(x)Δ_{1}(x) > 0

and \frac{1}{4h(x)^{2}} (δ_{2}(x) − h(x)Δ_{1}(x))^{2} > \frac{1}{4} (δ_{2}(x) − Δ_{1}(x))^{2} − δ_{1}(x) − Δ_{2}(x)

hold for all regular points x ∈ K̃.

As a corollary we get that if the inequalities

\begin{aligned}δ_{2}(x) − Δ_{1}(x) & > 0 and \\ δ_{1}(x) + Δ_{2}(x) & > 0 \end{aligned}  (6.6)

are satisfied for all regular points x of f on K̃ by Corollary 5.1 dim_{H} K ≤ 1 holds. Further, the set K consists of a finite number of equilibrium points and closed trajectories. This can be easily shown using coverings of appropriated tubular neighborhoods. Note that this result is closely related to results in [2, 7, 8]. If in addition to this the set K̃ is positively invariant with respect to the flow of (6.2), connected, and if K̃ contains exactly one equilibrium point being asymptotically stable, then K̃ is contained in the basin of attraction of this equilibrium point. One of the most effective sufficient condition which guarantees that in the present situation the set K̃ is contained in the basin of attraction of an equilibrium is the Hartmann-Olech condition ([8]) requiring that α_{1}(x) + α_{2}(x) < 0 for all regular points x ∈ K̃. Note that this is always sufficient for the condition (6.6)

Let us formulate a further corollary from Theorem 5.1 for the case M = ℝ^{3}. Suppose now that for the function δ_{2}(x) we have δ_{2}(x) > 0 for all regular points x ∈ K̃ and there exists a continuous function s : K̃ → [0, d_{1}) with d_{1} ∈ (0, 1] such that the inequalities

\begin{matrix}\frac{1 + s(x)}{1 − s(x)} δ_{2}(x) − Δ_{1}(x) ≥ 0 and \\ \frac{s(x)}{(1 − s(x))^{2}} δ_{2}(x)^{2} − \frac{s(x)}{1 − s(x)}δ_{2}(x)Δ_{1}(x) + δ_{1}(x) + Δ_{2}(x) ≥ 0 \end{matrix}  (6.7)

hold for all regular x ∈ K̃. It follows from Corollary 5.1 that dim_{H} K < 2 + d_{1}. It is well-known (see [17, 14]) that a sufficient condition for the dimension estimate dim_{H} K < 2 + d_{1} is the inequality

α_{1}(x) + α_{2}(x) + d_{1}α_{3}(x) < 0 for all x ∈ K̃.  (6.8)

We want to show that our condition (6.7) is always satisfied supposed that (6.8) is satisfied. We choose the function s on K̃ as

s(x) = \begin{cases}0 & if α_{1}(x) + α_{2}(x) ≤ 0 \\ −(α_{1}(x) + α_{2}(x))/α_{3}(x) & if α_{1}(x) + α_{2}(x) > 0. \end{cases}

It is easy to see that for all regular x ∈ K̃

\frac{1 + s(x)}{1 − s(x)}δ_{2}(x) − Δ_{1}(x) ≥ \frac{1 + s(x)}{1 − s(x)}(−α_{1}(x) − α_{2}(x) − α_{3}(x)) − α_{1}(x)

= \frac{1}{1 − s(x)} [(−α_{1}(x) − α_{2}(x) − s(x)α_{3}(x)) − (α_{1}(x) + s(x)α_{2}(x) + α_{3}(x))] ≥ 0.

Furthermore, it holds for the considered x

Δ_{2}(x) − \frac{s(x)}{1 − s(x)}δ_{2}(x)Δ_{1}(x) =

\frac{f(x)^{∗} (Q(x) − \frac{s(x)}{2(1−s(x))}δ_{2}(x)I)^{2} f(x)}{||f(x)||^{2}} − \frac{s(x)^{2}}{4(1 − s(x))^{2}} δ_{2}(x)^{2}.

Now we determine the minimal eigenvalue of the matrix

(Q(x) − \frac{s(x)}{2(1 − s(x))}δ_{2}(x)I)^{2} .  (6.9)

By definition of the function s(x) it follows that

α_{1}(x) − \frac{s(x)}{2(1 − s(x))}δ_{2}(x) ≤ \frac{1}{2} (α_{1}(x) − α_{2}(x)) ,  (6.10)
α_{2}(x) − \frac{s(x)}{2(1 − s(x))}δ_{2}(x) ≤ −\frac{1}{2} (α_{1}(x) − α_{2}(x)) ,  (6.11)
α_{3}(x) − \frac{s(x)}{2(1 − s(x))}δ_{2}(x) ≤ α_{3}(x) − α_{2}(x) − \frac{1}{2} (α_{1}(x) − α_{2}(x))  (6.12)

for all regular x ∈ K̃. Therefore, for any regular x ∈ K̃ the value (α_{1}(x) − \frac{s(x)}{2(1−s(x))}δ_{2}(x))^{2} is the minimal eigenvalue of the matrix (6.9) and we conclude that

\frac{s(x)}{(1 − s(x))^{2}} δ_{2}(x)^{2} − \frac{s(x)}{1 − s(x)}δ_{2}(x)Δ_{1}(x) + δ_{1}(x) + Δ_{2}(x) ≥

\frac{s(x)}{(1 − s(x))^{2}} δ_{2}(x)^{2}− \frac{s(x)}{1 − s(x)}δ_{2}(x)α_{1}(x)+α_{1}(x)^{2}+(α_{1}(x)+α_{2}(x))α_{3}(x)+α_{1}(x)α_{2}(x) ≥ 0

for all regular x ∈ K̃ of f. This shows that (6.7) is satisfied supposed that (6.8) is satisfied. □

References

[1] V. Afraimowich: Pesin’s dimension for Poincaré recurrences. Chaos 7 (1997) 1, 12-20.

[2] G. Borg: A condition for existence of orbitally stable solutions of dynamical systems. Kungl. Tekn. Högsk. Handl. Stockholm. 153 (1960), 16-23.

[3] C. Carathéodory : Über das lineare Maß für Punktmengen - eine Verallgemeinerung des Längenbegriffs. Göttinger Nachrichten (1914), 406-426.

[4] A. Douady et J. Oesterlé: Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris, Ser. A290 (1980), 1135-1138.

[5] F. Federer: Geometric Measure Theory. Springer, New-York, Berlin (1969).

[6] K. Gelfert: Dimension estimates of flow-invariant sets on manifolds using adapted Carathéodory outer measures. Diploma Thesis, University of Technology Dresden (1997).

[7] F. Hartmann: Ordinary Differential Equations. John Wiley, New York, London, Sydney (1964).

[8] P. Hartman and C. Olech: On global asymptotic stability of solutions of ordinary differential equations. Trans. Amer. Math. Soc. 104 (1962), 154-178.

[9] W. Klingenberg: Riemannian Geometry. De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, New York (1982).

[10] G. A. Leonov: Estimation of the Hausdorff dimension of attractors of dynamical systems. Leningrad State University, Translated from Differentsial’nye Uravneniya, 27(1991) 5, 767-771. (Russian)

[11] G. A. Leonov: Construction of a special outer Carathéodory measure for the estimation of the Hausdorff dimension of attractors. Vestnik of St. Petersburg University, 1(1995) 22, 24-31. (Russian)

[12] G. A. Leonov, A. Noack and V. Reitmann: Orbital stability conditions of vector fields by estimates of higher order characteristic exponents. To appear in Nonlinear Analysis TMA.

[13] G. A. Leonov, D. V. Ponomarenko and V. B. Smirnova: Local instability and localization of attractors. From stochastic generators to Chua’s system. Acta Applicandae Mathematical. 40 (1995), 179-243.

[14] A. Noack and V. Reitmann: Hausdorff dimension estimates for invariant sets of timedependent vector fields. Journal for Analysis and its Applications, 15 (1996) 2, 457-473.

[15] Ya. B. Pesin: Dimension Type Characteristics for Invariant Sets of Dynamical Systems. Russian Math. Surveys, 43 (1988) 4, 111-151. (Russian)

[16] Ya. B. Pesins: Dimension Theory in Dynamical Systems : Contemporary Views and Applications. Chicago Lectures in Mathematics, The University of Chicago Press, Chicago and London (1997).

[17] R. A. Smith: Some applications of Hausdorff dimension inequalities for ordinary differential equations. Proc. Roy. Soc. Edinburgh. 104A (1986) 3-4, 235-259.