Estimates of the box dimension of map-invariant compact sets on Riemannian manifolds^{1}

Katrin Gelfert^{2}

gelfert@math.tu-dresden.de

Abstract. For a compact and φ-invariant set on a Riemannian manifold and for a C^{1}-map φ it is shown that the global Lyapunov dimension of φ on K bounds the box dimension of K.

Keywords. box dimension, singular value function, Lyapunov exponent, Riemannian manifold AMS subject classification : 58F12, 58F35

1 Introduction

In [1, 8] it has been shown that the global Lyapunov dimension dim_{L}(φ, K) of a C^{1}-map φ on a compact φ-invariant set K ⊂ ℝ^{n} bounds the Hausdorff dimension of K. This result has been generalized to infinite dimensional spaces ([11, 9]) and to Riemannian manifolds ([4, 10]). Il’yashenko conjectured that for a C^{1}-map in ℝ^{n} the global Lyapunov dimension of a compact φ-invariant set bounds not only its Hausdorff dimension from above, but also its box dimension, which itself is an upper bound of the Hausdorff dimension and which seems to be of higher significance for application (see e.g. [6, 7] for motivation). For compact φ-invariant sets on ℝ^{n} and C^{1}-map φ the conjecture was proved by Hunt ([6]). There he used the equivalent definition of the box-counting dimension via a gridcovering. A straight forward generalization of the method of [6] to general spaces does not seem to be obvious. But the result has then been generalized for twice Frechét-differential homeomorphisms on a separable Hilbert space by Il’yashenko himself ([7]). In the present paper we prove the conjecture for φ-invariant compact sets on Riemannian manifolds, for φ of smoothness C^{1}. The proof of the result is closely related to the method in [7].

Let (M, ρ) be a metric space. The box dimension (also called box-counting, fractal, Minkowski, entropic or capacity dimension) of a compact set K ⊂ M is defined as follows. For ε > 0, d ≥ 0 we define, in analogy to the Hausdorff d-measure at niveau ε, the following set-function

μ_{B}(K, d, ε) = inf{∑_{i} ε^{d} : ⋃_{i} B(p_{i}, ε) ⊃ K} = N_{ε}(K) · ε^{d},  (1)

which is also called the capacitive d-measure at niveau ε of K. Here N_{ε}(K) denotes the smallest number of ε-balls which cover K. (Note that μ_{B}(·, d, ε) is an outer measure on subsets of M but no measure.) The upper capacitive d-measure of K is defined by \overline{μ}_{B}(K, d) = lim sup_{ε→0+0} μ_{B}(K, d, ε) and \overline{dim}_{B}K = inf\{d ≥ 0 : \overline{μ}_{B}(K, d) = 0\} is the upper box dimension.

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”

Note that there holds the equality \overline{dim}_{B}K = lim sup_{ε→0+0} − log N_{ε}(K)/ log ε. Further note that for K ⊂ ℝ^{n} the upper box dimension \overline{dim}_{B}K can be equivalently defined in the following way. For ε > 0 take a partition of ℝ^{n} into a grid of n-dimensional boxes of side length ε. Let Ñ_{ε}(K) be the number of boxes which intersect K. Then it holds \overline{dim}_{B}K = lim sup_{ε→0} − log Ñ_{ε}(K)/ log ε, and the limit is independent of the choice of the grid for any ε. Taking the limes inferior in the above procedures defines the lower box-counting dimension \underline{dim}_{B}K. If \overline{dim}_{B}K = \underline{dim}_{B} = dim_{B} K this value is called box dimension of K. There hold the well-known relations dim_{H} K ≤ \underline{dim}_{B}K ≤ dim_{B} K ≤ \overline{dim}_{B}K.

Let us briefly recall the definition of the singular values of a linear map. Consider a linear operator L : E → E′ between two n-dimensional Euclidean spaces E, E′ with scalar products ⟨·, ·⟩_{E} and ⟨·, ·⟩_{E′}, respectively. Then the singular values σ_{i} (i = 1, …, n) of L are the square roots of the eigenvalues of the self-adjoint, positive semidefinite operator L^{∗}L, where L^{∗} denotes the adjoint operator of L.

In the sequel we consider a n-dimensional Riemannian manifold M. Let U ⊂ M be an open set and let φ : U → M be a C^{1}-map. Given p ∈ U we consider the singular values σ_{1}(p) ≥ … ≥ σ_{n}(p) ≥ 0 of the tangential map d_{p}φ : T_{p}M → T_{φ(p)}M. For an arbitrary number d ∈ [0, n] the singular value function of order d of d_{p}φ is defined by

ω_{d}(p) = \begin{cases}1 & for d = 0 \\ σ_{1}(p) · . . . · σ_{⌊d⌋}(p)σ_{⌊d⌋+1}^{d−⌊d⌋}(p) & for d ∈ (0, n]. \end{cases}

Here and in the sequel ⌊d⌋ for d ∈ ℝ denotes the largest integer less than d. In terms of the singular values we define the local Lyapunov dimension dim_{L}(φ, p) to be the largest number d ∈ [0, n] such that ω_{d}(p) = 1. If σ_{1}(p) < 1 let dim_{L}(φ, p) = 0 and if ω_{n}(p) ≥ 1 let dim_{L}(φ, p) = n. Note that for the map φ any singular value σ_{i}(p) (i = 1, …, n) is a continuous function in p, as well as dim_{L}(φ, p) is continuous in p except for σ_{1}(p) = 1, where it is still upper semi-continuous. For a compact set K ⊂ M we introduce the following notation

dim_{L}(φ, K) = max_{p∈K} dim_{L}(φ, p), ω_{d,K}(φ) = max_{p∈K} ω_{d}(p).

We call dim_{L}(φ, K) the global Lyapunov dimension of φ with respect to K.

Theorem 1.1 Let U ⊂ M be an open set on a smooth n-dimensional Riemannian manifold M, let φ : U → M be a C^{1}-map. Suppose that K ⊂ M is a compact and φ-invariant set. Then \overline{dim}_{B}K ≤ dim_{L}(φ, K).

This theorem generalizes results of [6] to Riemannian manifolds. The method of the proof - especially the use of so-called filial coverings - is closely related to [7]. Though some efforts are necessary to apply the method of [7] to maps of smoothness C^{1} which may be non-injective.

2 Preliminary results

One difficulty in obtaining estimates of the box dimension is the fact that the capacitive outer measure μ_{B}(K, d, ε) is not necessarily monotonous with respect to the parameter ε. We formulate and prove the following result which goes back to [5] which will be used for the proof of Theorem 1.1.

Lemma 2.1 Let (M, ρ) be a metric space and let K ⊂ M be a compact set. Suppose that for numbers d ≥ 0, ε′ > 0, 0 < D < 1 the inequality

μ_{B}(K, d, Dε) < μ_{B}(K, d, ε)  (2)

holds for any ε ∈ (0, ε′), then \overline{dim}_{B}K ≤ d.

Proof. Let r < ε′ be chosen arbitrarily. Because of D < 1 there exists a natural number j, for which D^{j}ε′ ≤ r < D^{j−1}ε′ holds. Therefore we have

μ_{B}(K, d, r) = N_{r}(K)r^{d} < N_{D^{j}ε′}(K) · (D^{j−1}ε′)^{d} = D^{−d}μ_{B}(K, d, D^{j}ε′).

Applying (2) for ε := D^{j−1}ε′, …, ε := ε′ we obtain

μ_{B}(K, d, r) ≤ D^{−d}μ_{B}(K, d, D^{j}ε′) ≤ D^{−d}μ_{B}(K, d, D^{j−1}ε′)

≤ D^{−d}μ_{B}(K, d, ε′) < D^{−d}μ_{B}(K, d, ε′).

Since K is compact μ_{B}(K, d, ε′) is finite and μ_{B}(K, d, r) is bounded from above for any r < ε′. Thus, \overline{μ}_{B}(K, d) < ∞ and the lemma is proved. ■

Let for the following E be an n-dimensional Euclidean space and E ⊂ E be an ellipsoid. Let σ_{1}(E) ≥ … ≥ σ_{n}(E) ≥ 0 denote the singular values of E, i.e. the lengths of its halfaxes. Correspondingly we denote by ω_{d}(E) = σ_{1}(E)·. . .·σ_{⌊d⌋}(E)σ_{⌊d⌋+1}(E)^{d−⌊d⌋} for d ∈ (0, n] and ω_{0}(E) = 1 the singular value function of E. We call E degenerated if σ_{i}(E) = 0 for some i ∈ {1, …, n}.

We reformulate the Lemma 3 [1], which is true for non-degenerated ellipsoid E, in order to take into consideration possibly degenerated ellipsoids. (Note that the case of degeneracy is of interest if the considered map φ introduced in Chapter 1 is non-injective.)

Lemma 2.2 Let m, k, ς be positive real numbers, d ∈ (0, n], k ≤ m^{d}. Suppose that there hold ω_{s}(E)ς^{d−s} ≤ l for any s = 0, …, ⌊d⌋, ω_{d}(E) ≤ k, and σ_{1}(E) ≤ m hold. Then for any η > 0 the set E + B(O, η) is contained in an ellipsoid E′ ⊂ E satisfying

ω_{d}(E′) ≤ (1 + (m^{⌊d⌋}/k)^{1/(d−⌊d⌋)} η)^{d} k,
σ_{⌊d⌋+1}(E′) ≤ (1 + (m^{⌊d⌋}/k)^{1/(d−⌊d⌋)} η) max{ς, σ_{⌊d⌋+1}(E)}.

Proof. The idea is to “fatten” the ellipsoid E a bit in the following way : If for the chosen value ς it holds σ_{⌊d⌋+1}(E) < ς we replace the singular values σ_{i}(E) (i = ⌊d⌋ + 1, …, n) by ς. These singular values determine a non-degenerated ellipsoid, which contains E, and for which Lemma 3 in [1] can be applied. ■

Applying the covering results in [1, 11] and Lemma 2.2 we obtain an estimation of the number of balls needed to cover the set E + B(O, η).

Corollary 2.1 Under the assumptions of Lemma 2.2 for any η > 0 the sum E + B(O, η) can be covered by [2^{d}k/ς̃^{d}] balls of radius (1 + (m^{⌊d⌋}/k)^{1/(d−⌊d⌋)} η) ς̃\sqrt{⌊d⌋ + 1}, where ς̃ = max{ς, σ_{⌊d⌋+1}(E)}.

3 Proof of Theorem 1.1

For the setting on the manifold we use the following connection between the manifold and the tangent space which is enabled by the exponential map. Remark 3.1 For the compact set K we can choose δ > 0 such that for any p ∈ K the exponential map exp_{p} maps B(O_{p}, δ) ⊂ T_{p}M, the δ-ball centered in the origin O_{p} of T_{p}M, diffeomorphically onto the ball B(p, δ) ⊂ M. And, by compactness, exp_{p} uniformly C^{1}-approaches isometries on B(O_{p}, δ) as δ → 0. For sufficiently small δ > 0 therefore it follows, that for any p ∈ K the inequality ρ(exp_{p} v_{1}, exp_{p} v_{2}) < 2||v_{1} − v_{2}||_{T_{p}M} is satisfied for arbitrary v_{1}, v_{2} ∈ B(O_{p}, δ). Let this number δ be fixed in the sequel. Proof of Theorem 1.1. The proof can be sketched as follows. For ε small enough we fix an arbitrary covering U_{0} of the compact set K consisting of finitely many ε-balls which intersect K. We choose for U_{0} a number γ ∈ (0, 1). With this we construct a covering G of finitely many balls of constant radius R ≈ γε satisfying ∑_{S∈G} μ_{B}(S, d, R) < ∑_{S∈U_{0}} μ_{B}(S, d, ε). Then we will apply Lemma 2.1.

Considering higher iterates of the map φ we take into account that ω_{d,K}(φ^{i}) is a sub-exponential function in i. Without loss of generality we can therefore assume that

2\sqrt{⌊d⌋ + 1}(ω_{d,K}(φ))^{1/d} < \frac{1}{2} and (4\sqrt{⌊d⌋ + 1})^{d} ω_{d,K}(φ) < \frac{1}{2}

hold. Let l > ω_{d,K}(φ) be any number satisfying

2\sqrt{⌊d⌋ + 1}l^{1/d} < \frac{1}{2}, and (4\sqrt{⌊d⌋ + 1})^{d} l < \frac{1}{2}  (3)

Choose an open set V ⊂ M which contains K and is contained in a compact set K̃ ⊂ M for which

k′ := ω_{d,K̃}(φ) < l

holds. Choose m > 0 satisfying k′ ≤ m^{d} and σ_{1,K̃}(φ) ≤ m. Then equation

(1 + Cη)^{d} k′ = l, where C = (\frac{m^{⌊d⌋}}{k′} )^{1/(d−⌊d⌋)} ,  (4)

uniquely defines a number η > 0. Further, because of σ_{⌊d⌋+1,K̃}(φ) ≤ ω_{d,K̃}(φ)^{1/d} ≤ (k′)^{1/d},

(1 + Cη)σ_{⌊d⌋+1,K̃}(φ) ≤ l^{1/d}  (5)

holds. We take ε_{0} > 0 small enough such that

ε_{0} ≤ min \{\sqrt{⌊d⌋ + 1}(1 + Cη)δ, \frac{1}{2} dist\{K, M\V \}\} and ||τ_{φ(p)}^{φ(u)}d_{p}φτ_{u}^{p} − d_{u}φ|| ≤ η

(6) are satisfied for all u, p ∈ V with ρ(u, p) ≤ ε_{0}. Any ball B(p, ε) with ε ≤ ε_{0} which intersects K is entirely contained in the open set V. And Taylors formula applied to the differentiable map φ provides that for each y ∈ B(p, ε)

|| exp_{φ(p)}^{−1} φ(y) − d_{p}φ(exp_{p}^{−1}(y))||_{T_{φ(p)}M} ≤

sup_{w∈B(p,ε)} ||τ_{φ(w)}^{φ(p)}d_{w}φτ_{p}^{w} − d_{p}φ||_{T_{p}M} · || exp_{p}^{−1}(w)||_{T_{φ(p)}M} (7)

holds. And the inclusion

φ(B(p, ε)) ⊂ exp_{φ(p)} (d_{p}φB(O_{p}, ε) + B(O_{φ(p)}, ηε))

follows.

Initial covering U_{0} of K : Suppose ε has been chosen small enough. Fix an arbitrary cover

U_{0} = ⋃_{j=1}^{J} B(p_{j}, ε)  (8)

of K consisting of finitely many balls of radius ε which intersect K.

Covering of φ(B(p,ϵ)) : Choose some number ς satisfying ω_{s,K}(φ)ς^{d−s} ≤ k′ for any s = 0, …, ⌊d⌋. Then by Corollary 2.1 for any ball B(p, ε) intersecting K the set d_{p}φB(p, ε)+B(O_{φ(p)}, ηε) ⊂ T_{φ(p)}M can be covered by [2^{d}k/ς̃^{d}] balls of radius \sqrt{⌊d⌋ + 1}(1+ Cη) ς̃ε, where ς̃ = ς if σ_{⌊d⌋+1}(p) ≤ ς and ς̃ = σ_{⌊d⌋+1}(p) otherwise. We project the balls of this covering from the tangent space T_{φ(p)}M onto the manifold M via the exponential map. By Remark 3.1 we get that φ(B(p, ε)) is covered by

N(p) = [\frac{2^{d}k′}{σ(p)^{d}} ((1 + Cη)2\sqrt{⌊d⌋ + 1})^{d}]

balls of radius σ(p)ε. Where here and in the sequel we denote for any p ∈ M

σ(p) = \begin{cases}(1 + Cη)2\sqrt{⌊d⌋ + 1}ς & if σ_{⌊d⌋+1}(p) < ς \\ (1 + Cη)2\sqrt{⌊d⌋ + 1}σ_{⌊d⌋+1}(p) & otherwise. \end{cases}

The above defined function σ : K̃ → ℝ : p ↦ σ(p) is continuous since the singular value σ_{⌊d⌋+1}(p) of the linear map d_{p}φ depends continuously on p. By assumptions, (5), and (3) there exist numbers α, β > 0 satisfying

0 < α < σ(p) < β < 1 for any p ∈ K̃.  (9)

Filial coverings : Now we fix for any ball B(p, ε) ∩ K ≠ ∅ a family of balls which cover φ(B(p, ε)) (which can be done in a unique way by construction) : For B(p, ε) let

F^{(1)}(B(p, ε)) = {B(q_{j}, σ(p)ε)}_{j=1}^{N(p)}.

We call this family F^{(1)}(B(p, ε)) the family of filial balls of B(p, ε) (of order 1). Denote in the following by r(S) the radius of a ball S. With the above the following relations hold for the constructed covering :

∑_{S∈F^{(1)}(B(p,ε))} r(S)^{d} ≤ (4\sqrt{⌊d⌋ + 1})^{d}(1 + Cη)^{d}k′ r(B(p, ε))^{d}

by (4) and (3) ≤ \frac{1}{2}r(B(p, ε))^{d}.  (10)

Further we define a sequence of filial coverings by setting recursively for l = 1, 2, …

F^{(l+1)}(B(p, ε)) = ⋃{F^{(1)}(S) : S ∈ F^{(l)}(B(p, ε))}

and call this the family of filial balls of B(p, ε) of order (l + 1). As in (10) we have for each family of filial balls of order (i + 1) (i = 0, 1, …)

∑_{S∈F^{(i+1)}(B(p,ε))} r(S)^{d} = ∑_{S∈F^{(i)}(B(p,ε))} ∑_{S′∈F^{(1)}(S)} r(S′)^{d}

≤ \frac{1}{2} ∑_{S∈F^{(i)}(B(p,ε))} r(S)^{d}

≤ (\frac{1}{2})^{i+1} ε^{d}.  (11)

Construction of a homogeneous covering of K : For any point p ∈ K̃ we choose a sequence s(p) = {s_{0}(p), s_{1}(p), … } of pre-images of p with respect to φ :

s_{0}(p) = p,
define recursively for i = 1, 2, … s_{i}(p) = q for q ∈ {u : φ(u) = s_{i−1}(p)}.

Let γ ∈ (0, 1) now be a fixed number for which

\frac{2^{− log(αγ)/ log β}}{α^{d}} < \frac{1}{4}  (12)

holds. For any point p ∈ K̃ and the associated sequence s(p) we can uniquely define a natural number I(p) for which

αγ < σ(s_{1}(p)) · . . . · σ(s_{I(p)}(p)) ≤ γ  (13)

holds. We assume that γ has been chosen small enough such that I(p) > 1 holds for any p ∈ K̃. We set

I = sup_{p∈K̃} I(p).

Because of (9) obviously I is finite. By (9) and (13) we get αγ < β^{I(p)} and α^{I(p)} < γ and therefore

\frac{log(αγ)}{log β} < I(p) < \frac{log γ}{log α}  (14)

for any p ∈ K̃.

For any point p ∈ K̃ we construct a ball which contains p and which radius is approximately γε, namely with the associated sequence of pre-images s(p) of p we fix a sequence of balls {B_{i}(p)}_{i=0,... ,I(p)} for which r(B_{0}(p)) ≈ γε. This is done in the following way : B_{I(p)}(p) belongs to the initially fixed covering U_{0} of K and contains s_{I(p)}(p). For PSfrag replacements

B_{I(p)}(p)

B_{I(p)−1}(p) B_{1}(p) B_{0}(p) q_{I(p)}(p)

q_{I(p)−1}(p) q_{1}(p) q_{0}(p)

s_{I(p)}(p) s_{I(p)−1}(p)

s_{1}(p) s_{0}(p) = p

…

i = 0, …, I(p) − 1 we proceed then recursively. The union of the family of filial balls F^{(1)}(B_{I(p)−i}(p)) covers φ(B_{I(p)−i}(p)). For B_{I(p)−(i+1)}(p) we choose one ball in this covering which contains s_{I(p)−(i+1)}(p). Let us denote by q_{i}(p) the center point of the ball B_{i}(p), (i = 0, … I(p)). With this construction we have

p = s_{0}(p) ∈ B_{0}(p), …, s_{I(p)}(p) ∈ B_{I(p)}(p).

For the radius r(B_{0}(p)) of the ball it holds

r(B_{0}(p)) = σ(q_{1}(p)) · . . . · σ(q_{I(p)}(p))r(B_{I(p)}(p))
= σ(q_{1}(p)) · . . . · σ(q_{I(p)}(p))ε,

since for any p ∈ K̃ the ball B_{I(p)}(p) belongs to the ε-covering U_{0}.

Now we cover the compact set K by a finite family of balls G̃ = {B_{0}(p_{l})}_{l=1}^{L}. We set

R = max_{l=1,... ,L} r(B_{0}(p_{l})).

Each ball B_{0}(p_{l}) in G̃ we replace by the ball B_{l} := B(q_{0}(p_{l}), R) and obtain a uniform covering G = {B_{l}}_{l=1}^{L} of K with balls of equal radius R in the interval [αγε, γε).

Estimate of the oscillation of the radii in G̃ : We choose some Δ > 1 such that

Δ^{2dI} < 2  (15)

is satisfied. As remarked above σ(p) depends continuously on p. We can choose a sufficiently small ε′ > 0 such that for any p, q within the compact set K with ρ(p, q) < ε′

\frac{σ(p)}{σ(q)} ≤ Δ  (16)

holds. By r(B_{0}(p_{l})) = εσ(q_{1}(p_{l})) · . . . · σ(q_{I(p_{l})}(p_{l})), (16), and (13) we have

\frac{r(B_{0}(p_{l}))}{ε} ≤ σ(s_{1}(p_{l})) · . . . · σ(s_{I(p_{l})}(p_{l}))Δ^{I(p_{l})} ≤ Δ^{I(p_{l})}γ

and analogously

\frac{ε}{r(B_{0}(p_{l}))} ≤ \frac{Δ^{I(p_{l})}}{σ(s_{1}(p_{l})) · . . . · σ(s_{I(p_{l})}(p_{l}))} ≤ Δ^{I(p_{l})}(αγ)^{−1}.

Therefore, for any two balls B_{0}(p_{l}) and B_{0}(p_{k}) in the covering G̃ we obtain

\frac{r(B_{0}(p_{l}))}{r(B_{0}(p_{k}))} ≤ \frac{Δ^{I(p_{k})}}{αγ} Δ^{I(p_{l})}γ ≤ Δ^{2I} \frac{1}{α}.

Which means that

R ≤ \frac{Δ^{2I}}{α} r(B_{0}(p_{l})).  (17)

holds for any ball B_{0}(p_{l}) ∈ G̃.

Estimation of the measures : Suppose now that the initial covering U_{0} has been chosen to be a covering of balls of radius ε ≤ min{ε′, ε_{0}}. We now estimate μ_{B}(K, d, R). Since K is covered by ⋃_{l=1}^{L} B_{l}

μ_{B}(K, d, R) ≤ ∑_{l=1}^{L} μ_{B}(B_{l}, d, R) = ∑_{l=1}^{L} R^{d}

with (17) ≤ \frac{Δ^{2dI}}{α^{d}} ∑_{l=1}^{L} r(B_{0}(p_{l}))^{d}.  (18)

By construction for any B_{0}(p_{l}) (l = 1, …, L) there exists a ball B(p, ε) ∈ U_{0} such that B_{0}(p_{l}) belongs to the family of filial balls of B(p, ε) of order I(p). Thus, each item in the sum (18) appears at most once as item in the following sum

∑_{j=1}^{J} ∑_{i=I^{∗}}^{I} ∑_{S∈F^{(i)}(B(p_{j},ε))} r(S)^{d}.

Here we denote I^{∗} = min_{p∈K̃} I(p). By (14) and the definition of I we have

\frac{log αγ}{log β} ≤ I^{∗} ≤ I(p) ≤ I for any p ∈ K̃.

Therefore, we get

μ_{B}(K, d, R) ≤ \frac{Δ^{2dI}}{α^{d}} ∑_{j=1}^{J} ∑_{i=I^{∗}}^{I} ∑_{S∈F^{(i)}(B(p_{j},ε))} r(S)^{d}

with (11) ≤ \frac{Δ^{2dI}}{α^{d}} ∑_{j=1}^{J} ∑_{i=I^{∗}}^{∞} (\frac{1}{2})^{i} ε^{d}

Because of (14) we yield

μ_{B}(K, d, R) ≤ \frac{Δ^{2dI}}{α^{d}} ∑_{j=1}^{J} 2^{− \frac{log(αγ)}{log β}} \frac{1}{1 − 2^{−1}} ε^{d} = 2Δ^{2dI} \frac{2^{− \frac{log(αγ)}{log β}}}{α^{d}} ∑_{j=1}^{J} ε^{d}

and with (15) and (12) < ∑_{j=1}^{J} ε^{d}.

Since the initial covering (8) of K by balls of radius ε has been chosen arbitrarily we can replace in the last expression ∑_{j=1}^{J} ε^{d} by μ_{B}(K, d, ε).

Therefore, the assumptions of Lemma 2.1 are satisfied and it follows \overline{dim}_{B}K ≤ d. ■

4 Application and further results

Example. We want to illustrate the use of a Riemannian metric in the case of the Hénon map, although these dimension estimates have been obtained in [3] by consideration of map-invariant sets in ℝ^{2}. We consider the Hénon map f : ℝ^{2} → ℝ^{2} given by

\binom{x}{y} ↦ \binom{a + by − x^{2}}{x}  (19)

with parameters 0 < a, 0 < b < 1. The map has two fixed points given by y_{±} = x_{±} = \frac{1}{2}(b − 1 ± \sqrt{(b − 1)^{2} + 4a}). In view of [3] we consider on ℝ^{2} a Riemannian metric given by the metric tensor g, which is in any point p = col(x, y) ∈ ℝ^{2} given by the matrix

g(p) = V (p) · ( \begin{matrix}c & 0 \\ 0 & 1 \end{matrix} ) , where V (p) = e^{(2−d)γ(x+cy)}

where d ∈ (1, 2], γ = 1/(−2x_{−} + b − 1)\sqrt{x_{−}^{2} + b}, and c > 0 is a parameter. The singular values of the linearized map of (19),

(\begin{matrix}−2x & b \\ 1 & 0 \end{matrix}) ,  (20)

with respect to the Riemannian metric g are given by

σ_{1,2}(p) = \frac{V (p)}{2} (\sqrt{4x^{2} + (\frac{1}{c} + cb)^{2}} ± \sqrt{4x^{2} + (\frac{1}{c} − cb)^{2}}) .

And as the parameter c varies σ_{1}(p) takes on its minimal value at c = 1/\sqrt{b} with σ_{1}(p) = \frac{V (p)}{2} (\sqrt{4x^{2} + b} + |x|). Consider the case b = 0.3, a = 1.4. Let K ⊂ ℝ^{2} be a f-invariant compact set. For c = 1/\sqrt{b} the estimates of [3] and Theorem 1.1 give \overline{dim}_{L}K ≤ 1.496 compared to \overline{dim}_{L}K ≤ 1.523 in [6]. While the numerically determined box-counting dimension of an f-invariant set within the square −1.8 ≤ x, y ≤ 1.8 is approximately 1.28. □

As an immediate consequence of the proof of Theorem 1.1 we obtain box dimension estimates also for compact sets which are negatively invariant with respect to the map.

Theorem 4.1 Let U ⊂ M be an open set on a smooth n-dimensional Riemannian manifold M, let φ : U → M be a C^{1}-map. Suppose that K, K′ ⊂ M are compact sets satisfying K ⊂ φ(K) ⊂ K′. Then \overline{dim}_{B}K ≤ dim_{L}(φ, K′).

Scetch of the proof. We fix any cover U_{0} of K of finitely many ε-balls (ε small) with non-empty intersection with K. We fix some appropriate value γ ∈ (0, 1). For any B ∈ U_{0} we consider the family of filial balls of B of order I(B) such that the radius r(S) of the ball S in the filial family F^{(I(B))}(B) is approximately γε. With the sets {S : S ∈ F^{(I(B))}(B), B ∈ U_{0}} we construct as above a family {S} of equal radius R, R ≈ γε. And we pick from {S} a family G of finitely many such balls which covers K′. As above we show then μ_{B}(K, d, R) ≤ μ_{B}(K′, d, R) < μ_{B}(K, d, ε). From which the statement follows. ■ Related results. Theorem 4.1 generalizes Corollary 6 in [2] where for the above situation the bounds on the box dimension where obtained only for map negatively-invariant sets with constant singular value function ω_{n}(d_{p}φ) ≡ const. Further it immediately follows that the bounds of Theorem 3.1 in [10] hold not only for the Hausdorff but also for the box dimension.

References

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