Embedding strategies for flow-invariant compacta on manifolds ^{1}

Thomas Okon^{2}

okon@math.tu-dresden.de

Abstract. We generalize an embedding result for compacta with finite fractal dimension, which is due to B. R. Hunt and V. Yu. Kaloshin [3] for Banach spaces, to metric spaces and derive estimates for the fractal dimension of the embedded compact set.

AMS subject classification : 60B11, 94A17, and 34C35

1 Introduction

If one tries to estimate the fractal dimension of some flow-invariant compact subset of the phase space of a considered dynamical system experimentally the natural question arises in what way the calculated dimension-estimate is related to the original dimension of the invariant set. More mathematically speaking any N independent measurements of the phase space correspond to a map defined on the phase space with values in the N-dimensional Euclidean space. Under what conditions on the regularity of the measurements and on the number of independent measurements N it is natural to expect a relationship between the calculated and the real dimension?

Embedding theory and their relations to dimensions starts with the fundamental work of H. Whitney [7]. For an overview of the results related to finite dimensional manifolds we refer to [6]. Our essential attention is payed to infinite-dimensional manifolds or purely metric phase spaces. R. Mañé [5] considered a compact subsets X of a Banach space with fractal dimension d. He showed the existence of a topologically generic set of projections from the Banach space to a 2d + 1-dimensional Euclidean space which is injective on X. However this does not allow any statement on the fractal dimension of the projected X. A. Ben-Artzi et. al. [1] showed in the finite dimensional case the existence of such a projection whose inverse is in addition Hölder continuous on the projected X. C. Foias and E. Olson [2] generalized this result to Hilbert spaces. The main starting point of our considerations is the recent work of B. R. Hunt and V. Yu. Kaloshin [3]. They show that the former results can also be expected generically in some probabilistic sense which is called prevalence, see Definition 2 and [4].

The paper is divided into four sections. After the introduction we start with some notation and the definitions of fractal dimension and prevalence. The next part contains a slight generalization of the prevalence embedding results and dimension estimates of B. R. Hunt and V. Yu. Kaloshin [3] and derives related results for metric spaces and differentiable manifolds. Their proofs are given in the last part.

1Based on a lecture hold at the DFG-Workshop “Dimensions of attractors”, St. Petersburg, Russia, October 11-12, 1999

2Supported by the DFG-Schwerpunktprogramm “Dynamik: Analysis, effiziente Simulation und Ergodentheorie”

2 Notation and basic definitions

Let | · | denote the Euclidean norm and λ_{N} the N-dimensional Lebesgue measure in the Euclidean space ℝ^{N}.

For a topological space Y let C_{b}(Y, ℝ^{N}), Lip(Y, ℝ^{N}) be the spaces of all bounded continuous and Lipschitz continuous ℝ^{N}-valued functions on Y respectively. Let ∥ · ∥_{∞} denote the supremums norm and for a fixed y ∈ Y let ∥ · ∥_{y} be a Lipschitz Norm. I.e., ∥f∥_{y} := |f(y)| + L_{f}, f ∈ Lip(Y, ℝ^{N}), where L_{f} is the minimal Lipschitz constant of f. (C_{b}(Y, ℝ^{N}), ∥ · ∥_{∞}) and (Lip(Y, ℝ^{N}), ∥ · ∥_{y}) are Banach spaces. Notice that any two norms ∥ · ∥_{y_{1}} and ∥ · ∥_{y_{2}} are equivalent.

If E is a Banach space the space of all linear ℝ^{N}-valued functions is denoted by L(E, ℝ^{N}) and it is endowed with the operator norm. The topological dual of E is denoted by E′. Notice that for ∅ ≠ Y ⊆ E the restriction r_{Y} : L(E, ℝ^{N}) → Lip(Y, ℝ^{N}) is Lipschitz continuous.

We consider C^{∞}-manifolds M modeled on a Banach space E such that there exists one chart φ : M → U, U ⊆ E open. Notice that the mapping φ is an isometric homeomorphism if M is endowed with the metric ρ_{φ} which is induced by the chart. For k ∈ ℕ ∪ {0} let C_{b}^{k}(M, ℝ^{N}) denote the space of those k-times differentiable ℝ^{N}-valued functions on M which have bounded derivatives up to the kth order. Denote ∥f∥_{k}^{φ} := ∑_{i=0}^{k} ∥(fφ^{−1})^{(i)}∥_{∞}, f ∈ C_{k}(M, ℝ^{N}). Then ∥ · ∥_{k}^{φ} is a norm and since Φ : (C_{k}(U, ℝ^{N}), ∥ · ∥_{k}^{id}) → (C_{k}(M, ℝ^{N}), ∥ · ∥_{k}^{φ}) given by Φ(f) := f ∘ φ, f ∈ C_{k}(U, ℝ^{N}) is an isometric isomorphism (C_{k}(M, ℝ^{N}), ∥ · ∥_{k}^{φ}) is a Banach space. Notice that in case of U is bounded the restriction r_{Y} : L(E, ℝ^{N}) → C_{k}(U, ℝ^{N}) is Lipschitz continuous for any k ∈ ℕ ∪ {0} and that in case of a convex U we have C_{k}(M, ℝ^{N}) ⊆ Lip(M, ℝ^{N}) if k ≥ 1.

Finally in any metric space (M, ρ) we denote by B_{M}(x, r) the closed ball of radius r ≥ 0 around x ∈ M.

Definition 2.1 Let (M, ρ) be a metric space. The fractal dimension dim_{F} M of M is defined by

dim_{F} M = lim sup_{ε→0+} \frac{log N_{ε}}{− log ε}

where N_{ε} is the minimal cardinality of a covering of M consisting of closed balls of radius ε.

Definition 2.2 (of Prevalence, see [4]) Let V be a complete metric linear space. A Borel measure μ on V is said to be transverse to a Borel set S ⊆ V if it has compact support and μ(S + v) = 0 for any v ∈ V.

A set S′ ⊆ V is said to be shy if there exists a Borel set S such that S′ ⊆ S ⊆ V and a Borel measure μ on V which is transverse to S.

A set S ⊆ V is said to be prevalent if V \ S is shy.

The motivation of shyness and prevalence comes from the Fubini theorem. We remark that any prevalent set is dense and that a subset in an Euclidean space is shy iff its Lebesgue measure vanishes. For further information on prevalence see [4]. Notice also that if a Dirac measure is transverse to a Borel set S than S vanishes.

3 Main statements

Theorem 3.1 Let E be a Banach space, and ∅ ≠ X ⊆ Y ⊆ E with compact X. Let dim_{F} X ≤ d < ∞, N > 2d, and

0 < α < \frac{N − 2d}{N(1 + d)}.  (1)

Let H ⊆ Lip(Y, ℝ^{N}) and (H, ∥ · ∥_{H}) be a Banach space such that r_{Y} : L(E, ℝ^{N}) → (H, ∥ · ∥_{H}) is continuous.

Then the set of all f ∈ H such that

∃ C > 0 ∀ x, y ∈ X C|f(x) − f(y)|^{α} ≥ ∥x − y∥  (2)

is prevalent in H.

It is possible to prove this theorem is the same manner as in B. R. Hunt and V. Yu. Kaloshin [3] where Y = E. To hold the paper self-contained we present the proof at the end of the paper. Notice also that it is not possible to get our results by a direct application of the Hunt/Kaloshin theorem since in general r_{Y} (Lip(E, ℝ^{N})) ≠ Lip(Y, ℝ^{N}).

Corollary 3.1 Let (M, ρ) be a metric space and ∅ ≠ X ⊆ M be compact. Let dim_{F} X ≤ d < ∞, N > 2d, and 0 < α < (N − 2d)/(N(1 + d)).

Then the set of all f ∈ Lip(M, ℝ^{N}) such that

∃ C > 0 ∀ x, y ∈ X : C|f(x) − f(y)|^{α} ≥ ρ(x, y)

is prevalent in Lip(M, ℝ^{N}).

Corollary 3.2 Let M be a C^{∞}-manifold modeled on one chart φ : M → U where U is bounded and convex. Let ∅ ≠ X ⊆ M be compact with dim_{F} X ≤ d < ∞, N > 2d, and 0 < α < (N − 2d)/(N(1 + d)).

Then the set of all f ∈ C_{k}(M, ℝ^{N}) such that

∃ C > 0 ∀ x, y ∈ X : C|f(x) − f(y)|^{α} ≥ ρ_{φ}(x, y)

is prevalent in C_{k}(M, ℝ^{N}).

Corollary 3.3 Under the hypotheses of Theorem 3.1 respectively Corollary 3.1 or Corollary 3.2 for prevalent almost all f the estimate

\frac{N − 2d}{N(1 + d)} dim_{F} (X) ≤ dim_{F} f(X) ≤ dim_{F} X

holds for the fractal dimension of the embedded X.

It has to be remarked that there cannot exist an estimate as above for the Hausdorff dimension. B. R. Hunt and V. Yu. Kaloshin [3] constructed compact subsets X_{d} of the sequence space l^{2} with Hausdorff dimension d such that for all bounded linear functions f : l^{2} → ℝ^{N} the Hausdorff dimension of f(X_{d}) is bounded by d/(1 + d/2).

4 Proofs of the main statements

The proof of Corollary 3.3 is, in view of the Hölder continuity of prevalent almost all (f|_{X})^{−1}, obvious.

Proof. (of Corollary 3.1) Choose a point ∗ ∈ M. Define

ϕ : M → C_{b}(M, ℝ)

by ϕ(m)(x) := d(x, m) − d(x, ∗), x ∈ M, m ∈ M.

ϕ is an isometric homeomorphism onto Y := ϕ(M) and induces an isometric isomorphism

Φ : Lip(Y, ℝ^{N}) ⟶ Lip(M, ℝ^{N})

by Φ(f) := fϕ, f ∈ Lip(Y, ℝ^{N}). Precisely we have ∥Φ(f)∥_{m} = ∥f∥_{ϕ(m)}, f ∈ Lip(Y, ℝ^{N}) for any m ∈ M. X_{Y} := ϕ(X) is compact, dim_{F} X_{Y} ≤ d, and since ∅ ≠ X_{Y} ⊆ Y ⊆ C_{b}(M, ℝ) we can apply Theorem 3.1 with (H, ∥ · ∥_{H}) := (Lip(Y, ℝ^{N}), ∥ · ∥_{y}) and arbitrary y ∈ Y. So there exists a Borel measure μ on Lip(Y, ℝ^{N}) with compact support Q which is transversal to Lip(Y, ℝ^{N}) \ Z_{Y} where

Z_{Y} := {f ∈ Lip(Y, ℝ^{N}) ; ∃ C > 0 ∀ x, y ∈ X_{Y} : C|f(x) − f(y)|^{α} ≥ ∥x − y∥_{∞}}.

We claim that the distribution μ_{Φ} on Lip(M, ℝ^{N}) which is induced by Φ is transversal to Lip(M, ℝ^{N}) \ Z where

Z := {f ∈ Lip(M, ℝ^{N}) ; ∃ C > 0 ∀ x, y ∈ X : C|f(x) − f(y)|^{α} ≥ ρ(x, y)}.

Indeed, the support of μ_{Φ} is contained in the compact set Φ(Q) and it remains to show that μ_{Φ}(g + Lip(M, ℝ^{N}) \ Z) = 0, g ∈ Lip(M, ℝ^{N}). This follows from Φ(Z_{Y}) = Z, which can be proved straightforward, and the transversality of μ. ■

Proof. (of Corollary 3.2) We can prove in the same manner as above starting with the isometric homeomorphism φ : M → U instead of ϕ which induces an isometric isomorphism Φ : C_{k}(U, ℝ^{N}) → C_{k}(M, ℝ^{N}) as above by composition. To apply Theorem 3.1 use (C_{k}(U, ℝ^{N}), ∥ · ∥_{k}^{id}) for the space (H, ∥ · ∥_{H}). ■

Proof. (of Theorem 3.1) Denote by S′ the elements of H such that (2) does not hold, i.e.,

S′ := {f ∈ H; ∀C > 0 ∃x, y ∈ X : C|f(x) − f(y)|^{α} < ∥x − y∥}.

We have to show the existence of a compactly supported Borel measure on H which is transversal to S′.

We divide the existence-proof into five steps.

Step 1: Definition of the Borel set S of Definition 2.

We claim that if

∃ j_{0} ∈ ℕ ∀ j ≥ j_{0} ∀ x, y ∈ X : ∥x − y∥ ≥ 2^{−jα} ⇒ |f(x) − f(y)| ≥ 2^{−j}.

holds for some f ∈ H then (2) holds for f, too.

To see this fix x, y ∈ X. If ∥x − y∥ ≤ 2^{−j_{0}α} there exists a unique j ∈ ℕ, j ≥ j_{0} such that 2^{−(j+1)α} < ∥x−y∥ ≤ 2^{−jα} and we have 2^{α}|f(x)−f(y)|^{α} ≥ ∥x−y∥. If ∥x−y∥ > 2^{−j_{0}α} then (diam X)2^{j_{0}α}|f(x) − f(y)|^{α} ≥ ∥x − y∥.

Define

S := {f ∈ H; ∃x_{j}, y_{j} ∈ X : ∥x_{j} − y_{j}∥ ≥ 2^{−jα} and |f(x_{j}) − f(y_{j})| < 2^{−j} for infinitely many j ∈ ℕ}. (3)

Then S is a Borel set in H and S′ ⊆ S.

Step 2: A sufficient condition for transverseness.

Consider a Borel measure μ on H which has compact support Q ⊆ H. μ is transverse to S if μ(f + S) = 0, f ∈ H which is equivalent to

μ({π ∈ Q; f + π ∈ S}) = 0,  (4)

f ∈ H. Let f ∈ H be fixed. For any j ∈ ℕ define

A_{f}^{j} := {π ∈ Q; ∃x, y ∈ X : ∥x − y∥ ≥ 2^{−jα} and

|(f + π)(x) − (f + π)(y)| < 2^{−j}}.

Then by (3), (4) we have to show that μ(lim sup_{j→∞} A_{f}^{j}) = 0. By the Borel-Cantelli lemma a sufficient condition for this is the convergence of ∑_{j=1}^{∞} μ(A_{f}^{j}).

Step 3: A sufficient estimate for μ(A_{f}^{i}).

Assume there exists a compactly supported Borel measure on H such that

∀ d′ > d ∃ C > 0 ∀ f ∈ H : μ(A_{f}^{j}) ≤ Cj^{2N}2^{−j(N(1−α(1+d′)−2d′)}.  (5)

Then for small |d′ − d| (1) holds for d′ instead of d, too. I.e., Nα + d′α < N − 2d′ and the exponent in (5) is negative.

Step 4: Construction of μ.

Since dim_{F} X ≤ d for any d′ > d there exists C_{d′} > 0 such that for any j ∈ ℕ there exists a covering of X of at most

d_{j} := ⌊C_{d′}2^{jαd′}⌋  (6)

balls of radius 2^{−jα}/5.

Fix j ∈ ℕ. Let V_{j} be the linear hull of the centers of the balls of the jth covering. By the Hahn-Banach theorem we can fix an isometric embedding ι_{j} : (V_{j})′ ↪ E′. Let S′_{j} := ι_{j}B_{(V_{j})′}(0, 1). S′_{j} is a convex compact subset of E′. By choosing a basis in (V_{j})′ it is also possible to consider S′_{j} as an compact convex set in the Euclidean space ℝ^{d_{j}} where d_{j} is the dimension of V_{j}.

Define

Q := {r_{Y} (π_{1}, …, π_{N}) ; π_{l} := ∑_{j=1}^{∞} j^{−2}ϕ_{lj}, ϕ_{lj} ∈ S′_{j}, j ∈ ℕ, l = 1, …, N}.  (7)

Q is a continuous image of the compact cube Π_{1}^{N}Π_{j=1}^{∞}S′_{j} since by assumption r_{Y} : L(E, ℝ^{N}) → (H, ∥ · ∥_{H}) is continuous. Therefore Q is compact, too. Let μ be the distribution on H generated by the product of the uniform distributions on S′_{j}.

Notice at this point for later purpose that there exists a commom Lipschitz constant for Q since our embeddings ι_{j} are isometric.

Step 5: Verification of (5).

Let Z_{j} := {(x, y) ∈ X × X; ∥x − y∥ ≥ 2^{−jα}}, j ∈ ℕ. Then A_{f}^{j} = {π ∈ Q; ∃(x, y) ∈ Z_{j} : |(f + π)(x) − (f + π)(y)| < 2^{−j}}, j ∈ ℕ. Since dim_{F} Z_{j} ≤ dim_{F} (X × X) ≤ (dim_{F} X)^{2} for any fixed d′ > d there exists a constant C_{1} > 0 such that for any j ∈ ℕ there exists a covering ∪_{i=1}^{N_{j}}B_{Z_{j}}((x_{i}^{j}, y_{i}^{j}), 2^{−j}) of Z_{j} with (x_{i}^{j}, y_{i}^{j}) ∈ Z_{j}, i = 1, …, N_{j}, and N_{j} ≤ C_{1}2^{2jd′}. So we have the estimate

μ(A_{f}^{j}) ≤ C_{1}2^{2jd′} max_{i=1}^{N_{j}} μ({π ∈ Q; ∃(x, y) ∈ B_{Z_{j}}((x_{i}^{j}, y_{i}^{j}), 2^{−j}) :

|(f + π)(x) − (f + π)(y)| < 2^{−j}}). (8)

Fix some of the above i, j and let

(x, y) ∈ B_{Z_{j}}((x_{i}^{j}, y_{i}^{j}), 2^{−j}) such that |(f + π)(x) − (f + π)(y)| < 2^{−j}.  (9)

Let L be a common Lipschitz constant for f + Q. By the triangle-inequality

|(f + π)(x_{i}^{j}) − (f + π)(y_{i}^{j})| < (2L + 1)2^{−j}

is necessary for (9). Denote for i = 1, …, N_{j}, j ∈ ℕ

A_{f}^{ji} := {π ∈ Q; |(f + π)(x_{i}^{j}) − (f + π)(y_{i}^{j})| < (2L + 1)2^{−j}}.

By (8) we have

μ(A_{f}^{j}) ≤ C_{1}2^{2jd′} max_{i=1}^{N_{j}} μ(A_{f}^{ji})  (10)

By construction of μ, recall (7), we arrive at the estimate

μ(A_{f}^{ji}) ≤ (sup_{z∈ℝ}μ_{S′_{j}}({ϕ_{lj} ∈ S′_{j}; j^{−2}ϕ_{lj}(x_{i}^{j} − y_{i}^{j}) ∈ [z, z + (4L + 2)2^{−j}]}))^{N}  (11)

for μ(A_{f}^{ji}) and independent of l = 1, …, N.

Choose x_{0}, y_{0} ∈ V_{j} such that ∥x_{0} − x_{i}^{j}∥ ≤ 2^{−jα}/5 and ∥y_{0} − y_{i}^{j}∥ ≤ 2^{−jα}/5. There exists ψ ∈ S′_{j} with ∥ψ∥ = ψ(x_{0} − y_{0}) ≥ (3/5)2^{−jα}. Since (x_{i}^{j}, y_{i}^{j}) ∈ Z_{j} we have ψ(x_{i}^{j} − y_{i}^{j}) ≥ 2^{−jα}/5. Let P ⊆ E′ be the annihilator of {x_{i}^{j} − y_{i}^{j}}. Then the right hand side of (11) is bounded by

(sup_{p∈ℝ}μ_{S′_{j}}({ϕ_{lj} ∈ tψ + P ; t ∈ [p, p + 5(4L + 2)j^{2}2^{−j(1−α)}]}))^{N} .  (12)

We consider ψ, S′_{j}, and P now as lying in ℝ^{d_{j}}. Let θ denote the angle between ψ and P. Let R_{t} := S′_{j} ∩ (tψ + P), t ∈ ℝ. By convexity and compactness of S′_{j} there exists t_{0} ∈ ℝ such that R_{t_{0}} is maximal with respect to inclusion and so all ϕ_{lj} belong to a cylinder with basis R_{t_{0}} and height 5(4L + 2)j^{2}2^{−j(1−α)}|ψ| sin θ. I.e., the right hand side of (12) is bounded from above by the d_{j}-dimensional volume of this cylinder divided by λ_{d_{j}}(S′_{j}) and to the power N. Since ψ and −ψ both belong to S′_{j} and S′_{j} is convex the cones with base R_{t_{0}} and peaks ψ and −ψ respectively are contained in S′_{j}, too. So λ_{d_{j}}(S′_{j}) is bounded from below by 2d_{j}^{−1} sin θ|ψ|λ_{d_{j}−1}(R_{t_{0}}).

I.e., the right hand side of (12) is bounded by

(5(2L + 1)j^{2}2^{−j(1−α)}d_{j})^{N} .

In combination with (10) we get

μ(A_{f}^{j}) ≤ C_{1}2^{2jd′} (5(2L + 1)j^{2}2^{−j(1−α)}d_{j})^{N}

and in view of (6) we arrive at

μ(A_{f}^{j}) ≤ C_{d′}2^{jαd′}C_{1}2^{2jd′} (5(2L + 1)j^{2}2^{−j(1−α)}d_{j})^{N} .

I.e., C := (C_{1}C_{d′}5^{N}(2L + 1)^{N}) is a possible constant for (5). ■

References

[1] A. Ben-Artzi, A. Eden, C. Foias, and B. Nicolaenko. Hölder continuity for the inverse of the Mañé projection. J. Math. Ana. Appl., 178:22–29, 1993.

[2] C. Foias and E. Olsen. Finite fractal dimension and Hölder-Lipschitz parametrization. Ind. Univ. J. Math., 45(3):603–616, 1996.

[3] B. R. Hunt and V. Yu. Kaloshin. Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces. Nonlinearity, 12:1263–1275, 1999.

[4] B. R. Hunt, T. Sauer, and J. A. James. Prevalence: A translation-invariant ”almost every” on infinite-dimensional spaces. Bull. AMS, 27(2):217–238, 1992.

[5] R. Mañé. On the dimension of the compact invariant sets of certain non-linear maps, in Dynamical Systems and Turbulence, volume 898. Springer, New York, 1981.

[6] T. Sauer, J. A. Yorke, and M. Casdagli. Embedology. J. Stat. Phy., 65(3/4):579–616, 1991.

[7] H. Whitney. Differentiable manifolds. Annals Math., 37(3):645–680, 1936.