Hausdorff and fractal dimension estimates of invariant sets of noninjective maps

V. A. Boichenko, G. A. Leonov (State University St. Petersburg, Russia) A. Mirle, V. Reitmann (University of Technology Dresden, Germany)

1 Introduction

In [1] Hausdorff outer measure and dimension estimates for compact sets which are invariant under C^{1}-maps in ℝ^{n} are given. In [4] the conditions of these estimates are weakened using a Lyapunov type function. These results are generalized in [6] to maps on Riemannian manifolds. Using a technique similar to that of Douady and Oesterlé, Temam gave in [7] upper bounds for the fractal dimension of flow invariant sets in a Hilbert space, which is proved in [3] for vector fields on Riemannian manifolds.

For noninjective maps it may be possible to use information about the degree of noninjectivity in order to get measure and dimension estimates under weakened conditions compared with the theorems mentioned above. For the first time Douady-Oesterlé type Hausdorff dimension estimates which use information about the degree of noninjectivity are considered in [5]. There the class of k -1-endomorphisms is described, where the invariant set can be split into k compact subsets and each of these subsets is mapped to the whole invariant set. Then a factor \frac{1}{k} can be used in estimating the Hausdorff premeasure, and so the conditions for finding an upper bound for the Hausdorff dimension can be weakened. But this exact k -1-property is not always given. Therefore we looked for a weaker condition than the k -1-property.

In this paper we consider C^{1}-maps φ on smooth n-dimensional Riemannian manifolds (M, g) and compact sets K ⊂ M. We assume that there exists a number a > 0 such that for any j ∈ ℕ there is a subset K_{j} ⊂ K with the following properties:

1) the ”measure” of K_{j} is at most a^{j} times the ”measure” of K;

2) the ”measures” of the sets φ^{j}(K_{j}) and φ^{j}(K) are equal.

Here ”measure” stands for the Hausdorff premeasure, the Hausdorff measure or the number of balls of a given radius needed to cover a set depending on the quantity we want to estimate. For a = 1 we can take K_{j} = K, i. e. the conditions 1) and 2) are always satisfied. But if we have a noninjective map it may be possible to find a number a < 1 such that 1) and 2) are satisfied. Then this number a can be used to weaken the conditions needed to find dimension estimates.

2 Hausdorff dimension estimates

First let us recall the definition of the Hausdorff outer measure and dimension for compact subsets K of a metric space (X, ρ). For arbitrary real numbers ε > 0 and d ≥ 0 we consider the d-dimensional Hausdorff outer premeasure at level ε of K given by

μ(K, d, ε) = inf ∑_{i} r_{i}^{d},

where the infimum is taken over all finite covers of K by balls B_{r_{i}}(u_{i}) = {v ∈ X | ρ(u_{i}, v) ≤ r_{i}} of radius r_{i} ≤ ε. For fixed d and K the function μ(K, d, ·) is monotone decreasing. Hence the limit

μ(K, d) = lim_{ε→0+0} μ(K, d, ε)

exists and is called d-dimensional Hausdorff outer measure of K. For every compact set K there exists a critical number d^{∗} with

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

This critical number can be characterized as

d^{∗} = sup{d ≥ 0 | μ(K, d) = ∞}

as well. It is called Hausdorff dimension of K and denoted by dim_{H}(K).

To describe how a ball is transformed under the tangent map, the singular value function ω_{d,K}(φ) is used. Let E and E′ be two n-dimensional Euclidean spaces with the scalar products ⟨·, ·⟩_{E} and ⟨·, ·⟩_{E′}, respectively, and L : E → E′ be a linear mapping. Then L^{∗} denotes the adjoint operator, i. e. the unique linear operator L^{∗} : E′ → E satisfying ⟨L^{∗}u, v⟩_{E} = ⟨u, Lv⟩_{E′} for all u ∈ E′, v ∈ E. The singular values α_{1}(L) ≥ … ≥ α_{n}(L) are the eigenvalues of the positive semidefinite operator \sqrt{L^{∗}L}. Then for a number d ∈ [0, n] written in the form d = d_{0} + s with d_{0} ∈ ℕ_{0} and s ∈ [0, 1] we define

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

Let (u_{1}, …, u_{n}) be an orthonormal basis of E such that u_{i} is an eigenvector of L^{∗}L corresponding to the eigenvalue α_{i}(L), i = 1, …, n. Then there exists an orthonormal basis (v_{1}, …, v_{n}) in E′ with v_{i} = \frac{1}{α_{i}} Lu_{i} for any i = 1, …, n with α_{i} ≠ 0. The image of the unit ball B_{1}(0) = {a_{1}u_{1} + … + a_{n}u_{n} ∈ E | (a_{1}, …, a_{n}) ∈ ℝ^{n}, a_{1}^{2} + … + a_{n}^{2} ≤ 1} under the map L is the set \{∑_{1≤i≤n,α_{i}(L)≠0} b_{i}v_{i} ∈ E′ | ∑_{1≤i≤n,α_{i}(L)≠0}(\frac{b_{i}}{α_{i}(L)})^{2} ≤ 1\}, i. e. an ellipsoid where the lengths of the half-axes are α_{1}(L), …, α_{n}(L). Therefore a similar concept for ellipsoids is introduced. Let ℰ be an ellipsoid in an ndimensional Euclidean space E, and let α_{1}(ℰ) ≥ … ≥ α_{n}(ℰ) denote the lengths of the half-axes, i. e. there exists an orthonormal basis (u_{1}, …, u_{n}) in E such that ℰ = \{∑_{1≤i≤n,α_{i}(ℰ)≠0} a_{i}u_{i} ∈ E | ∑_{1≤i≤n,α_{i}(ℰ)≠0}(\frac{a_{i}}{α_{i}(ℰ)} )^{2} ≤ 1\}. Then for an arbitrary number d ∈ [0, n] of the form d = d_{0} + s with d_{0} ∈ ℕ_{0} and s ∈ [0, 1] we have

ω_{d}(ℰ) = α_{1}(ℰ) · . . . · α_{d_{0}}(ℰ) · α_{d_{0}+1}^{s}(ℰ).

Now let d_{u}φ denote the tangent map of φ at the point u ∈ U, then we define

ω_{d,K}(φ) = sup_{u∈K}(ω_{d}(d_{u}φ)),

where K ⊂ U is an arbitrary set.

The following theorem describes the behaviour of the d-dimensional Hausdorff outer measure of a compact set under the iterates of a C^{1}-map.

Theorem 1 Let (M, g) be a smooth n-dimensional Riemannian manifold, U ⊂ M be an open set and φ : U → M be a C^{1}-map. Suppose K and K̃ are compact sets for which the relations K ⊂ K̃ ⊂ U and φ^{j}(K) ⊂ K̃ for any j = 1, 2, … are satisfied. Suppose that for some numbers d ∈ [1, n], a > 0 the following conditions are satisfied:

1) ω_{d,K̃}(φ)a < 1;

2) for any natural number j ≥ j_{0} there exists a set K_{j} ⊂ K such that

\begin{matrix}μ (φ^{j}(K_{j}), d) = μ(φ^{j}(K), d), \\ μ (K_{j}, d) ≤ a^{j}μ(K, d); \end{matrix}

3) μ(K, d) < ∞.

Then lim_{j→∞} μ(φ^{j}(K), d) = 0.

Proof The singular value function satisfies the relation

ω_{d,K}(φ^{j}) ≤ ω_{d,K̃}^{j}(φ)

for any j ∈ ℕ. Further for any number δ > 0 we find a number j_{δ} > j_{0} so that for d = d_{0} + s, d_{0} ∈ ℕ_{0}, s ∈ [0, 1] the relation

2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} (ω_{d,K̃}(φ) · a)^{j} ≤ δ

will be true for any j > j_{δ}. Using Lemma 2.3 of [6] we get the following inequalities:

\begin{aligned}μ(φ^{j}(K), d) & = μ (φ^{j}(K_{j}), d) ≤ 2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} ω_{d,K̃}(φ)^{j}μ(K_{j}, d) \\ & ≤ 2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} (ω_{d,K̃}(φ) · a)^{j}μ(K, d) ≤ δμ(K, d). \end{aligned}

Since δ can be chosen arbitrarily small, we get lim_{j→∞} μ(φ^{j}(K), d) = 0. ■

Corollary 1 If the conditions 1) and 2) of Theorem 1 are satisfied for any d ∈ [1, n] and φ(K) ⊃ K, then for such d we have either μ(K, d) = 0, or μ(K, d) = ∞.

Corollary 2 Let the conditions 2) and 3) of Theorem 1 be satisfied. Furthermore let p : K̃ → {u ∈ ℝ | u > 0} be a continuous function such that the following condition is satisfied:

1) sup_{u∈K̃} ω_{d} (\frac{p(φ(u))}{p(u)}d_{u}φ) a < 1.

Then lim_{j→∞} μ(φ^{j}(K), d) = 0.

Proof The proof is based on a technique considering the function p as Lyapunov type function, as described in [4] for M = ℝ^{n}. Resulting from condition 1) there is a number 0 < ν < 1 with a\frac{p(φ(u))}{p(u)}ω_{d}(d_{u}φ) < ν for any u ∈ K̃. Therefore we get

\begin{aligned}a^{j}ω_{d}(d_{u}φ^{j}) & ≤ & a^{j}ω_{d}(d_{φ^{j−1}(u)}φ) … ω_{d}(d_{u}φ) ≤ a^{j} \frac{ν}{a} \frac{p(φ^{j}(u))}{p(φ^{j−1}(u))} … \frac{ν}{a} \frac{p(φ(u))}{p(u)} \\ & = & ν^{j} \frac{p(φ^{j}(u))}{p(u)} ≤ ν^{j} \frac{sup_{u∈K̃} p(u)}{inf_{u∈K̃} p(u)}. \end{aligned}

So for any δ > 0 we find a number j_{δ} > j_{0} so that for d = d_{0} + s, d_{0} ∈ ℕ_{0}, s ∈ [0, 1] the relation

2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} a^{j}ω_{d,K̃}(φ^{j}) ≤ δ

will be true for any j > j_{δ}, and for these numbers j we get, as in the proof of Theorem 1,

\begin{aligned}μ(φ^{j}(K), d) & = μ (φ^{j}(K_{j}), d) ≤ 2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} ω_{d,K̃}(φ^{j})μ(K_{j}, d) \\ & ≤ 2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} ω_{d,K̃}(φ^{j})a^{j}μ(K, d) ≤ δμ(K, d). \end{aligned}

Since δ can be chosen arbitrarily small, we get lim_{j→∞} μ(φ^{j}(K), d) = 0. ■

Example Consider a class of modified horseshoe maps φ which are defined on an open neighbourhood U = (−η, 1 + η) × (−η, 1 + η) ⊂ ℝ^{2} of the unit square Q = [0, 1] × [0, 1], where η > 0 is a sufficiently small number. The map is defined for points (u, v) ∈ Q in such a way that it first contracts Q horizontally with a factor α < \frac{1}{2} and stretches it vertically with a function f, then it is folded along a horizontal line such that the vertical edge of the resulting rectangle is greater than 2, and finally it is formed into an horseshoe (see Figure 1) in such a way that the map can be continuously extended to U and is continuously differentiable on an open neighbourhood Ũ of K = ⋂_{i=−∞}^{∞} φ^{i}(Q), where φ^{i}(·) for negative numbers i means the preimage under the map φ^{−i}.

For example we take α = \frac{1}{3} and let the function f stretch Q with factor 3, if v ≤ \frac{51}{65}, with a factor between 3 and 5, if \frac{51}{65} < v < \frac{4}{5} and with factor 5, if v ≥ \frac{4}{5}, the

[figure: ⟶; ⟶; ⟶; α; hβ_{1}; (1−h)β_{2}; 0; 1; h; 1; x_{2}; x_{1}]

Figure 1: Modified horseshoe map

resulting rectangle is folded on the image of the line v = \frac{51}{65} and then it is formed to an horseshoe in such a way, that the map satisfies

φ(u, v) = \{ \begin{matrix}(\frac{1}{3}u, 3v − \frac{1}{13}) & if & 0 ≤ v ≤ \frac{14}{39}, \\ (1 − \frac{1}{3}u, \frac{148}{65} − 3v) & if & \frac{83}{195} ≤ v ≤ \frac{148}{195}, \\ (1 − \frac{1}{3}u, 5y − 4) & if & \frac{4}{5} ≤ v ≤ 1. \end{matrix}

The set K = ⋂_{i=−∞}^{∞} φ^{i}(Q) is invariant under the map. Therefore if we take K̃ = K, then φ^{j}(K) ⊂ K̃ is satisfied for any j = 1, 2, . . .. The set K can be constructed step by step starting with K^{0} = Q. In every step i > 1 we get K^{i} := K^{i−1} ∩ φ(K^{i−1}) ∩ φ^{−1}(K^{i−1}) and in the limit the invariant set K as K = ⋂_{i=0}^{∞} K^{i}. The set K^{i} consists of 6^{i} rectangles where the lengths of the edges are horizontally \frac{1}{3^{i}} and vertically \frac{1}{3^{i}} and \frac{1}{5^{i}}, respectively (see Figure 2).

Let us choose K_{1} := K ∩ [0, 1] × [0, \frac{51}{65}] and K_{j} := K_{j−1} ∩ φ^{−1}(K_{j−1}), j = 2, 3, . . .. Then these sets satisfy φ^{j}(K_{j}) = K = φ^{j}(K), so the first part of condition 2) is true. Furthermore the set K_{j} is contained in 4^{j} rectangles with edges \frac{1}{3^{j}} out of the 6^{j} rectangles forming K^{j}. Let ε_{j} denote half of the minimal distance between two of the different 6^{j} rectangles. If we cover K by balls of radius smaller than ε_{j}, then every ball can contain points of only one of the 6^{j} rectangles. If we consider only the part of K_{j−1} which is contained in one of the 6^{j−1} rectangles of K^{j−1}, then the part of K_{j} which is contained in the same rectangle consists of four linear copies of the part of K_{j−1} with a factor \frac{1}{3}. Therefore for ε < ε_{j} we have

μ(K_{j}, d, ε) = \frac{4}{3^{d}} μ(K_{j−1}, d, 3ε)

Figure 2: Construction of the invariant set K

for any d ∈ [0, 2], in the limit ε → 0 + 0 we get

μ(K_{j}, d) = (\frac{4}{3^{d}} )^{j} μ(K, d).

So the second part of condition 2) is satisfied with a = \frac{4}{3^{d}}. Now we have to check condition 1), i. e. we have to find a number d ∈ [1, 2] such that ω_{d,K}(φ)a < 1 is satisfied. The singular value function here has the form

ω_{d}(d_{u}φ) = \{ \begin{matrix}3 (\frac{1}{3})^{d−1} & for & u ∈ K_{1}, \\ 5 (\frac{1}{3})^{d−1} & for & u ∈ K \ K_{1}, \end{matrix}

so we have ω_{d,K}(φ) = 5 (\frac{1}{3})^{d−1}. It is easy to see that the inequality \frac{4}{3^{d}} 5 (\frac{1}{3})^{d−1} < 1 is equivalent to d > \frac{1}{2} (\frac{ln 20}{ln 3} + 1) ≈ 1.863. For such numbers d the conditions 1) and 2) of Theorem 1 are satisfied, and Corollary 1 yields to μ(K, d) = 0 or μ(K, d) = ∞. □

Now we want to use the same way as above to find an estimate of the Hausdorff dimension of K. Let us additionally assume that K is negative invariant under φ, i. e. K ⊂ φ(K) as in Corollary 1.

If we want to find an upper bound for the Hausdorff dimension K we can not assume μ(K, d) < ∞ as in Theorem 1. Therefore it is necessary to consider the Hausdorff outer premeasure instead of the Hausdorff outer measure, because the outer premeasure of compact sets is always finite.

Theorem 2 Let (M, g) be a smooth n-dimensional Riemannian manifold, U ⊂ M be an open set and φ : U → M be a C^{1}-map. Suppose K and K̃ are compact sets for which the relation K ⊂ φ^{j}(K) ⊂ K̃ ⊂ U for any j = 1, 2, … is satisfied. Suppose that for some numbers a > 0 and d ∈ [1, n] of the form d = d_{0} + s with d_{0} ∈ ℕ_{0} and s ∈ [0, 1] the following conditions are satisfied:

1) ω_{d,K̃}(φ)a < 1;

2) there are numbers l with ω_{d,K̃}(φ) < l < \frac{1}{a} and j_{0} ∈ ℕ such that for any natural number j > j_{0} there exist a set K_{j} ⊂ K and a number ε_{j} > 0 such that

\begin{matrix}μ(φ^{j}(K_{j}), d, ε) = μ(φ^{j}(K), d, ε), \\ μ (K_{j}, d, (d_{0} + 1)^{− \frac{1}{2}} l^{− \frac{j}{d}} ε) ≤ a^{j}μ(K, d, ε) \end{matrix}

holds for any ε ∈ (0, ε_{j}].

Then we have dim_{H}(K) ≤ d.

Proof Like in the proof of Theorem 1 we have ω_{d,K}(φ^{j}) ≤ ω_{d,K̃}(φ)^{j} < l^{j} for any j ∈ ℕ, and we find for any δ > 0 an integer j_{δ} > j_{0} so that the relation 2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} (al)^{j} < δ will be true for any j > j_{δ}. For 0 < ε ≤ max\{ε_{j}, (d_{0} + 1)^{− \frac{1}{2}} l^{− \frac{j}{d}} ε_{0}(l^{j})\}, where ε_{0} is defined in Lemma 2.3 of [6], Lemma 2.3 of [6] results in the following inequalities:

μ(K, d, ε) ≤ μ(φ^{j}(K), d, ε) = μ(φ^{j}(K_{j}), d, ε)
≤ 2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} l^{j}μ(K_{j}, d, (d_{0} + 1)^{− \frac{1}{2}} l^{− \frac{j}{d}} ε)
≤ 2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} (al)^{j}μ(K, d, ε) ≤ δμ(K, d, ε).

Since the number δ can be choosen arbitrarily small, this means μ(K, d, ε) = 0 for any ε ∈ (0, max\{ε_{j}, (d_{0} + 1)^{− \frac{1}{2}} l^{− \frac{j}{d}} ε_{0}(l^{j})\}) and therefore, μ(K, d) = 0. Hence we get dim_{H}(K) ≤ d. ■

Using now a Lyapunov type function we get a corollary of this theorem analogously to Corollary 2.

Corollary 3 Let (M, g), U, K, K̃ and φ be defined as in Theorem 2, and let p : K̃ → {u ∈ ℝ | u > 0} be a continuous function, such that for some numbers a > 0 and d ∈ [1, n] of the form d = d_{0} + s with d_{0} ∈ ℕ_{0} and s ∈ [0, 1] the following conditions are satisfied:

1) sup_{u∈K̃} (\frac{p(φ(u))}{p(u)}ω_{d}(d_{u}φ)) a < 1; 2) there is a number j_{0} ∈ ℕ such that for any natural number j > j_{0} there exist a

set K_{j} ⊂ K and numbers ε_{j} > 0 and l_{j} with

ω_{d,K}(φ^{j}) < l_{j} < (\frac{1}{a} sup_{u∈K̃} (\frac{p(φ(u))}{p(u)}ω_{d}(d_{u}φ)))^{j} \frac{sup_{u∈K̃} p(u)}{inf_{u∈K̃} p(u)} and
μ(φ^{j}(K_{j}), d, ε) = μ(φ^{j}(K), d, ε),
μ (K_{j}, d, (d_{0} + 1)^{− \frac{1}{2}} l_{j}^{− \frac{1}{d}} ε) ≤ a^{j}μ(K, d, ε)

for any ε ∈ (0, ε_{j}].

Then we have dim_{H}(K) ≤ d.

Example For the modified horseshoe map described before, the first part of condition 2) of Theorem 2 is satisfied for arbitrary numbers ε > 0 and d ∈ [1, 2]. Furthermore we know that for any d > \frac{1}{2} (\frac{ln 20}{ln 3} + 1) the relation ω_{d,K}(φ)a < 1 holds with a = \frac{4}{3^{d}}, and for ε < 3^{j}ε_{j} we have

μ(K_{j}, d, 3^{−j}ε) = a^{j}μ(K, d, ε).

Because of \frac{ω_{d,K}(φ)}{3^{d}} = \frac{5}{3^{2d−1}} < \frac{1}{4} for d > \frac{1}{2} (\frac{ln 20}{ln 3} + 1), we have 3^{−j}ωd,K^{\frac{j}{d}}(φ) < \frac{1}{2} and

therefore, \frac{1}{2}ω_{d,K}^{− \frac{j}{d}}(φ) > 3^{−j} for any j ∈ ℕ. So there is a number l > ω_{d,K}(φ) with al < 1 and l < \sqrt{2}ω_{d,K}(φ) , and for this number l the relation 2^{− \frac{1}{2}} l^{− \frac{j}{d}} > 3^{−j} holds. Thus we get with the inequality stated above μ(K_{j}, d, 2^{− \frac{1}{2}} l^{− \frac{j}{d}} ε) ≤ a^{j}μ(K, d, ε). So all the conditions of Theorem 2 are satisfied and we get dim_{H}(K) ≤ d for any number d > \frac{1}{2} (\frac{ln 20}{ln 3} + 1); in the limit this yields to dim_{H}(K) ≤ \frac{1}{2} (\frac{ln 20}{ln 3} + 1) ≈ 1.863.

In this example the dimension estimate can be improved using a Lyapunov type function as described in Corollary 3 to reduce ω_{d}(d_{u}φ). Since here the singular value function is constant on K_{1} and K \ K_{1}, respectively, the simplest type of Lyapunov function is of the same kind, i. e. p(u) = 1 for u ∈ K_{1} and p(u) = P > 0 for u ∈ K\K_{1}. The constant P has to be chosen in such a way that sup_{u∈K} (\frac{p(φ(u))}{p(u)}ω_{d}(d_{u}φ)) becomes minimal. We have φ(u) ∈ K_{1} for any u ∈ K \ K_{1}, but the images of points of K_{1} are in K_{1} and K \ K_{1}. Therefore we have

\frac{p(φ(u))}{p(u)}ω_{d}(d_{u}φ) = \{ \begin{matrix}\frac{1}{P} 5 (\frac{1}{3})^{d−1} & if & u ∈ K \ K_{1}, \\ P 3 (\frac{1}{3})^{d−1} & if & u ∈ K_{1}, φ(u) ∈ K \ K_{1}, \\ 3 (\frac{1}{3})^{d−1} & if & u ∈ K_{1}, φ(u) ∈ K_{1}, \end{matrix}

this becomes minimal if \frac{5}{P} = 3P, i. e. P = \sqrt{\frac{5}{3}}. This function p can be continuously extended to ℝ^{2} in the form

p(u, v) = \{ \begin{matrix}1 & if & v ≤ \frac{148}{195}, \\ 1 + (\sqrt{\frac{5}{3}} − 1) (\frac{195v}{8} − \frac{37}{2}) & if & \frac{148}{195} < v ≤ \frac{4}{5}, \\ \sqrt{\frac{5}{3}} & if & v > \frac{4}{5}. \end{matrix}

With this function p we get for d ∈ [1, 2]

sup_{u∈K} (\frac{p(φ(u))}{p(u)}ω_{d}(d_{u}φ)) = \sqrt{15} (\frac{1}{3})^{d−1} .

In order to satisfy condition 1) we have to look for numbers d satisfying the property \frac{4}{3^{d}} \sqrt{15} (\frac{1}{3})^{d−1} < 1, which is equivalent to d > \frac{1}{2} (^{\frac{ln(4\sqrt{15})}{ln 3}} + 1). Because of \frac{ω_{d,K}(φ^{j})}{3^{jd}} =

(\frac{5}{3^{2d−1}} )^{j} < (\frac{1}{4} \sqrt{\frac{5}{3}})^{j} for d > \frac{1}{2} (^{\frac{ln(4\sqrt{15})}{ln 3}} + 1) and j ∈ ℕ, we have ^{\frac{ω \frac{1}{d} ; d,K(φ^{j})}{3^{j}}} < \frac{1}{2} \sqrt[4]{\frac{5}{3}} and

therefore, 2\sqrt[4]{\frac{3}{5}}ωd,K^{\frac{1}{d}}(φ^{j}) < 3^{j} for any j ∈ ℕ. So we find a number l_{j} with ω_{d,K}(φ^{j}) <

l_{j} < (\frac{1}{a} sup_{u∈K̃} (\frac{p(φ(u))}{p(u)}(ω_{d}(d_{u}φ)))^{j} \frac{sup_{u∈K̃} p(u)}{inf_{u∈K̃} p(u)} and l_{j} < \sqrt{2}\sqrt[4]{\frac{3}{5}}ω_{d,K}(φ^{j}), and for this number l_{j} the second part of condition 2) is satisfied. Therefore Corollary 3 can be applied and gives dim_{H}(K) ≤ d for any d > \frac{1}{2} (^{\frac{ln(4\sqrt{15})}{ln 3}} + 1). In the limit we get

dim_{H}(K) ≤ \frac{1}{2} (^{\frac{ln(4\sqrt{15})}{ln 3}} + 1) ≈ 1.747, this is a better result than before.

In this example we would get the same improved result if we had considered the Lyapunov function p as changing the metric on ℝ^{2} by multiplying the metric tensor with p. But in general this does not need to be the case. Using a Lyapunov function does not influence the number a, but changing the metric can also change a. □

Since condition 2) of Theorem 2 is not easy to be checked, especially if the map is not piecewise linear, we now give some stronger conditions which can be checked more easily.

Corollary 4 Let (M, g) be a smooth n-dimensional Riemannian manifold, U ⊂ M be an open set,φ : U → M be a C^{1}-map and K ⊂ U be compact. Suppose that for numbers a > 0 and d ∈ [1, n] of the form d = d_{0} + s with d_{0} ∈ ℕ_{0} and s ∈ [0, 1] the following conditions are satisfied:

1) ω_{d,K}(φ)a < 1;

2) there is a number j_{0} ∈ ℕ such that for any natural number j ≥ j_{0} there exist a set K_{j} ⊂ K with

φ^{j}(K_{j}) = K,

a natural number N_{j}, a number l_{0} and C^{1}-maps f_{i,j} : U → M, i = 1, …, N_{j}, with

K_{j} = ⋃_{i=1}^{N_{j}} f_{i,j}(K),

max_{i=1,... ,N_{j}} ω_{d,K}(f_{i,j}) < l_{0}^{j} and

N_{j} < 2^{−d_{0}}(d_{0} + 1)^{− \frac{d}{2}} a^{j}l_{0}^{−j}.

Then we have dim_{H}(K) ≤ d.

Proof Using Lemma 2.3 of [6] for any j ∈ ℕ with j > j_{0} there exists a number ε_{j} such that

μ(K_{j}, d, \sqrt{d_{0} + 1}l0^{\frac{j}{d}} ε) ≤ ∑_{i=1}^{N_{j}} 2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} l_{0}^{j}μ(K, d, ε)
= N_{j}2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} l_{0}^{j}μ(K, d, ε)
≤ a^{j}μ(K, d, ε)

holds for any ε ∈ (0, ε_{j}]. Because of N_{j} ≥ 1, we have 2^{−d_{0}}(d_{0} + 1)^{− \frac{d}{2}} a^{j}l_{0}^{−j} ≥ 1 and therefore, 2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} l_{0}^{j}ω_{d,K}^{j}(φ) < 1 for any j ≥ j_{0}. This means l_{0}ω_{d,K}(φ) < 1, and because of aω_{d,K}(φ) < 1, there are numbers l > ω_{d,K}(φ) and j_{0} ∈ ℕ such that al < 1 and (l_{0}l)^{\frac{j}{d}} < \frac{1}{d_{0}+1} for any j > j_{0} are satisfied. For these numbers j we have

μ(K_{j}, d, \frac{1}{\sqrt{d_{0} + 1}l^{\frac{j}{d}}} ε) ≤ μ(K_{j}, d, \sqrt{d_{0} + 1}l0^{\frac{j}{d}} ε) ≤ a^{j}μ(K, d, ε).

Applying Theorem 2 we get dim_{H}(K) ≤ d. ■

Example Since in our example of the modified horseshoe map the set K_{j} consists of 4^{j} linear copies of K by a factor \frac{1}{3^{j}}, we have N_{j} = 4^{j} and l_{0} > \frac{1}{3^{d}}. So with a > 4l_{0} > \frac{4}{3^{d}} condition 2) of Corollary 4 is satisfied. Condition 1) results in d > 1 + \frac{ln 5−ln a}{ln 3}. So we get dim_{H}(K) ≤ 1+ \frac{ln 5−ln a}{ln 3}, in the limit a → \frac{4}{3^{d}} this means dim_{H}(K) ≤ \frac{1}{2} (\frac{ln 20}{ln 3} + 1) ≈ 1.863. This is the same result as before without a Lyapunov function, but we could reach it with less expense. □

3 Fractal dimension estimates

First let us recall the definition of the fractal dimension for compact subsets K of a metric space (X, ρ). For a given number ε > 0 let N_{ε}(K) denote the smallest number of balls of radius ε needed to cover K. Then the limit

dim_{F} (K) = lim sup_{ε→0} \frac{ln N_{ε}(K)}{− ln ε}

is called upper capacity dimension or fractal dimension of the set K.

In order to estimate the fractal dimension of an invariant set of a C^{1}-map on a Riemannian manifold we need two lemmata, which are formulated and proved analogously to assertions of [1].

Lemma 1 Let E be an n-dimensional Euclidean space, (u_{1}, …, u_{n}) an orthonormal basis and ℰ = \{a_{1}u_{1} + … + a_{n}u_{n} ∈ E | (\frac{a_{1}}{α_{1}(ℰ)} )^{2} + … + (\frac{a_{n}}{α_{n}(ℰ)} )^{2} ≤ 1\} with α_{1}(ℰ) ≥ … ≥ α_{n}(ℰ) > 0 an ellipsoid. Then for any η > 0, the set ℰ + B_{η}(0), where B_{η}(0) denotes the ball with radius η centered at the origin, is contained in the ellipsoid ℰ′ = (1 + \frac{η}{α_{n}(ℰ)} ) ℰ.

Proof We have B_{α_{n}(ℰ)}(0) = \{a_{1}u_{1}+…+a_{n}u_{n} ∈ E | (\frac{a_{1}}{α_{n}(ℰ)} )^{2}+…+(\frac{a_{n}}{α_{n}(ℰ)} )^{2} ≤ 1\} ⊂ ℰ and therefore,

ℰ + B_{η}(0) ⊂ ℰ + \frac{η}{α_{n}(ℰ)}ℰ = (1 + \frac{η}{α_{n}(ℰ)}) ℰ. ■

Lemma 2 Let E be an n-dimensional Euclidean space, (u_{1}, …, u_{n}) an orthonormal basis, ℰ = \{a_{1}u_{1} + … + a_{n}u_{n} ∈ E | (\frac{a_{1}}{α_{1}(ℰ)} )^{2} + … + (\frac{a_{n}}{α_{n}(ℰ)} )^{2} ≤ 1\} with α_{1}(ℰ) ≥ … ≥ α_{n}(ℰ) > 0 an ellipsoid and 0 < r < α_{n}(ℰ). Then

N_{\sqrt{n}r}(ℰ) ≤ \frac{2^{n}ω_{n}(ℰ)}{r^{n}}.

Proof The ellipsoid ℰ is contained in the parallelepiped

P = {a_{1}u_{1} + … + a_{n}u_{n} ∈ E | |a_{1}| ≤ α_{1}(ℰ), …, |a_{n}| ≤ α_{n}(ℰ)},

whereas the edges have the lengths 2α_{1}(ℰ), …, 2α_{n}(ℰ). This parallelepiped can be covered by N cubes C_{r}(v_{i}), v_{i} ∈ E, i = 1, …, N, with edges 2r, where with v_{i} = v_{i1}u_{1} + … v_{in}u_{n}, (v_{i1}, …, v_{in}) ∈ ℝ^{n}, we have

C_{r}(v_{i}) = {a_{1}u_{1} + … + a_{n}u_{n} ∈ E | |a_{j} − v_{ij}| ≤ r}.

The number N can be estimated by

N = ∏_{j=1}^{n} ([\frac{α_{j}(ℰ)}{r}] + 1) ≤ 2^{n} ∏_{j=1}^{n} \frac{α_{j}(ℰ)}{r}.

Each of the cubes C_{r}(v_{i}) is contained in the ball B_{\sqrt{n}r}(v_{i}) of radius \sqrt{n}r. ■

Theorem 3 Let (M, g) be a smooth n-dimensional Riemannian manifold, U ⊂ M be an open set, φ : U → M be a C^{1}-map and K ⊂ U be a compact set which is invariant under φ. Assume that 0 < ρ_{K}(φ) := min_{u∈K} α_{n}(d_{u}φ) < \sqrt{n}^{−1} and there exists a number d ∈ (0, n] such that

ω_{n,K}(φ)ρ_{K}^{d−n}(φ) ≤ 8^{−n}n^{− \frac{d}{2}}

is satisfied. Then dim_{F} (K) ≤ d.

Proof Let η ∈ (0, ρ_{K}(φ)) be an arbitrary number and r_{1} > 0 be so small that there exists an open set V ⊂ M containing K which itself lies inside a compact set K̃ ⊂ U such that

|||τ_{φ(v)}^{φ(u)}d_{v}φτ_{u}^{v} − d_{u}φ||| ≤ η

for any u, v ∈ V with ρ(u, v) ≤ r_{1} is satisfied, where ||| · ||| denotes the operator norm for the given linear operator. By ρ(·, ·) we mean the geodesic distance between the points of M and by τ_{u}^{v} we denote the isometry between the tangent spaces T_{u}M and T_{v}M defined by parallel transport along the geodesic for points lying sufficiently near each other.

Let exp_{u} : T_{u}M → M denote the exponential map at an arbitrary point u ∈ M. Since exp_{u} is a smooth map satisfying |||d_{O_{u}} exp_{u} ||| = 1 we find for any point u ∈ M a number r_{u} > 0 such that |||d_{v} exp_{u} ||| ≤ 2 for any v ∈ B_{r_{u}}(O_{u}) is true, where O_{u} denotes the origin of the tangent space T_{u}M. Because of the compactness of K̃ containing V there is a number r_{2} = min_{u∈K̃} r_{u} > 0 such that |||d_{v} exp_{u} ||| ≤ 2 is satisfied for any u ∈ V and any v ∈ B_{r_{2}}(O_{u}). Furthermore there is a number α > 0 such that α_{1}(d_{u}φ) < α for any u ∈ V is satisfied.

Now we can find a number r_{0} ≤ min\{r_{1}, \frac{r_{2}}{2+α+η}\} such that any ball B_{r_{0}}(u) containing points of K is entirely contained in V. Let r ∈ (0, r_{0}) be fixed. Since K is compact and invariant under φ, there are a finite number of points u_{j} ∈ V, j = 1, …, N_{r}(K), such that

K = ⋃_{j=1}^{N_{r}(K)} B_{r}(u_{j}) ∩ K = ⋃_{j=1}^{N_{r}(K)} φ(B_{r}(u_{j}) ∩ K)

is satisfied. The Taylor formula for differentiable maps provides that for every v ∈ B_{r}(u_{j}) the relation

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

≤ sup_{w∈B_{r}(u_{j})} |||τ_{φ(w)}^{φ(u_{j})}d_{w}φτ_{u_{j}}^{w} − d_{u_{j}}φ||| · ∥ exp_{u_{j}}^{−1}(w)∥

holds. Thus the image of every ball B_{r}(u_{j}) satisfies the inclusion

φ(B_{r}(u_{j})) ⊂ exp_{φ(u_{j})}(d_{u_{j}}φ(B_{r}(O_{u_{j}})) + B_{ηr}(O_{φ(u_{j})})).

With ℰ_{j} := d_{u_{j}}φ(B_{1}(O_{φ(u_{j})}))) and ℰ′_{j} = (1 + \frac{η}{α_{n}(ℰ_{j})} ) ℰ_{j} we get with Lemma 1

φ(B_{r}(u_{j})) ⊂ exp_{φ(u_{j})} (r(ℰ_{j} + B_{η}(O_{φ(u_{j})}))) ⊂ exp_{φ(u_{j})} (rℰ′_{j}) .

With σ := \sqrt{n}ρ_{K}(φ) we have

N_{σr}(K) ≤ N_{r}(K) max_{j=1,... ,N_{r}(K)} N_{σr}(exp_{φ(u_{j})}(rℰ′_{j})).

Every ball B_{σr}(v), v ∈ M, containing points of exp_{φ(u_{j})}(rℰ′_{j}) is contained in the ball B_{(2+α_{n}(ℰ′_{j}))r}(u_{j}) ⊂ B_{r_{2}}(u_{j}), and so we have B_{σr}(v) ⊃ exp_{φ(u_{j})}(B_{\frac{1}{2} σr}(exp_{φ(u_{j})}^{−1} v)). This means

N_{σr}(exp_{φ(u_{j})}(rℰ′_{j})) ≤ N_{\frac{1}{2} σr}(rℰ′_{j}).

Since ρ_{K}(φ) ≤ α_{n}(d_{u_{j}}φ) = α_{n}(ℰ_{j}) ≤ α_{n}(ℰ′_{j}) is satisfied, Lemma 2 yields to

N_{\frac{1}{2} σr}(rℰ′_{j}) ≤ \frac{2^{n}ω_{n}(rℰ′_{j})}{(\frac{1}{2}rρ_{K}(φ))^{n}} = \frac{4^{n}ω_{n}(ℰ′_{j})}{ρ_{K}^{n}(φ)} ≤ \frac{4^{n} (1 + \frac{η}{ρ_{K}(φ)})^{n} ω_{n}(ℰ_{j})}{ρ_{K}^{n}(φ)} ≤ \frac{8^{n}ω_{n}(d_{u_{j}}φ)}{ρ_{K}^{n}(φ)}.

Thus we get

N_{σr}(K) ≤ N_{r}(K)\frac{8^{n}ω_{n,K}(φ)}{ρ_{K}^{n}(φ)} = N_{r}(K)\frac{8^{n}ω_{n,K}(φ)}{ρ_{K}^{n}(φ)} \frac{ρ_{K}^{d}(φ)n^{\frac{d}{2}}}{σ^{d}} ≤ σ^{−d}N_{r}(K).

Because of σ < 1 for any ε ∈ (0, r_{0}) we can find a number l ∈ ℕ_{0} such that σ^{l+1}r_{0} ≤ ε < σ^{l}r_{0} is satisfied. So we get

N_{ε}(K) ≤ σ^{−ld}N_{σ^{−l}ε}(K) ≤ (\frac{r_{0}}{ε} )^{d} N_{σ^{−l}ε}(K) ≤ (\frac{r_{0}}{ε} )^{d} N_{σr_{0}}(K)

and therefore,

dim_{F} (K) = lim sup_{ε→0} \frac{ln N_{ε}(K)}{− ln ε} ≤ d. ■

Corollary 5 Let (M, g), U, K and φ be as in Theorem 3 and ρ_{K}(φ) > 0. Suppose that there exists a number d ∈ [0, n] with

ω_{n,K}(φ)ρ_{K}^{d−n}(φ) < 1.

Then dim_{F} (K) ≤ d.

Proof Since K is invariant under φ it is also invariant under φ^{i} for any i ∈ ℕ. The iterates of φ satisfy the relations

\begin{matrix}ω_{n,K}(φ^{i}) ≤ ω_{n,K}^{i}(φ), \\ ρ_{K}(φ^{i}) ≥ ρ_{K}^{i}(φ), \end{matrix}

and therefore,

ω_{n,K}(φ^{i})ρ_{K}^{d−n}(φ^{i}) ≤ (ω_{n,K}(φ)ρ_{K}^{d−n}(φ))^{i} .

Furthermore we have

ω_{n,K}(φ)ρ_{K}^{d−n}(φ) ≥ ρ_{K}^{n}(φ)ρ_{K}^{d−n}(φ) = ρ_{K}^{d}(φ).

So we can assume without loss of generality that

\begin{matrix}ω_{n,K}(φ)ρ_{K}^{d−n}(φ) ≤ 8^{−n}n^{− \frac{d}{2}} & and \\ ρ_{K}(φ) < \sqrt{n}^{−1} \end{matrix}

are satisfied; in the opposite case we consider the map φ^{i} with sufficiently large i. So all the conditions of Theorem 3 are satisfied, and we get dim_{F} (K) ≤ d. ■

Corollary 6 Let (M, g), U, K and φ be defined as in Theorem 3 and p : U → {u ∈ ℝ | u > 0} be a continuous function. Suppose that the following conditions are satisfied:

1) ω_{n}(d_{u}φ) = const ≠ 0 ∀u ∈ K;

2) there exists a number s ∈ [0, 1] such that \frac{p(φ(u))}{p(u)}ω_{n−1+s}(d_{u}φ) < 1 ∀u ∈ K.

Then dim_{F} (K) ≤ n − 1 + s.

Proof Resulting from condition 2) there exists a positive number ν < 1 with \frac{p(φ(u))}{p(u)}ω_{n−1+s}(d_{u}φ) ≤ ν for any u ∈ K. Therefore we have

ω_{n−1+s,K}(φ^{i}) ≤ max_{u∈K} \frac{p(u)}{p(φ^{i}(u))} \frac{p(φ^{i}(u))}{p(φ^{i−1}(u))}ω_{n−1+s}(d_{φ^{i−1}(u)}φ) … \frac{p(φ(u))}{p(u)}ω_{n−1+s}(d_{u}φ)
≤ \frac{max_{u∈K} p(u)}{min_{u∈K} p(u)}ν^{i}.

Furthermore the relation ω_{n−1+s,K}(φ) ≥ ρ_{K}^{n−1+s}(φ) holds. Therefore we can assume without loss of generality that ω_{n−1+s,K}(φ) < 8^{−n}n^{\frac{n−1+s}{2}} and ρ_{K}(φ) < \sqrt{n}^{−1} is satisfied, in the opposite case we consider φ^{i} with sufficiently large i. We take u_{0} ∈ K such that α_{n}(d_{u_{0}}φ) = ρ_{K}(φ). Resulting from condition 1) we get for arbitrary u ∈ K the relation

ω_{n}(d_{u}φ)ρ_{K}^{s−1}(φ) = ω_{n}(d_{u_{0}}φ)α_{n}^{s−1}(d_{u_{0}}φ) = ω_{n−1+s}(d_{u_{0}}φ) < 8^{−n}n^{\frac{n−1+s}{2}} .

So with Theorem 3 we get dim_{F} (K) ≤ n − 1 + s. ■

Let us remark that conditions analogous to 1) are considered in [8]. There fractal dimension estimates are given in terms of Lyapunov exponents.

Now we want to take into consideration the degree of noninjectivity in the method of estimating the fractal dimension developed in Theorem 3.

Theorem 4 Let (M, g) be a smooth n-dimensional Riemannian manifold, U ⊂ M be an open set, φ : U → M be a C^{1}-map, K ⊂ U be a compact set which is invariant under φ. Suppose ρ_{K}(φ) > 0, and let a, b > 0 be numbers such that the following conditions are satisfied:

1) for any j ∈ ℕ there is a compact set K_{j} ⊂ K and a number ε_{j} > 0 such that

\begin{matrix}φ^{j}(K_{j}) = K, \\ N_{ε}(K_{j}) ≤ a^{j}N_{b^{j}ε}(K) \end{matrix}

are satisfied;

2) there exists a number d ∈ [0, n] with a^{\frac{d}{n}} b^{−d}ω_{n,K}(φ)ρ_{K}^{d−n}(φ) ≤ 8^{−n}n^{− \frac{d}{2}} .

Then we have dim_{F} (K) ≤ d.

Proof Analogous to the proof of Theorem 3 let η ∈ (0, ρ_{K}(φ)) be an arbitrary number and r_{1}, r_{2} > 0 be so small that there exists an open set V ⊂ M containing K such that the closure of V is a compact subset of U and ∥τ_{φ(v)}^{φ(u)}d_{v}φτ_{u}^{v} − d_{u}φ∥ ≤ η for any u, v ∈ V with ρ(u, v) ≤ r_{1} and |||d_{v} exp_{u} ||| ≤ 2 for any u ∈ V and any v ∈ B_{r_{2}}(O_{u}) is satisfied. With α defined in the proof of Theorem 3 we can find a number r_{0} ≤ min\{r_{1}, \frac{r_{2}}{2+α+η}, \frac{ε_{1}}{b} \} such that any ball B_{r_{0}}(u) containing points of K is entirely contained in V. Let r ∈ (0, r_{0}) be fixed. Since K_{1} is compact and φ(K_{1}) = K, there are a finite number of points u_{j} ∈ V, j = 1, …, N_{r}(K_{1}), such that

K = ⋃_{j=1}^{N_{r}(K_{1})} φ(B_{r}(u_{j}) ∩ K)

is satisfied. Using the Taylor formula we get that the image of every ball B_{r}(u_{j}) satisfies the inclusion

φ(B_{r}(u_{j})) ⊂ exp_{φ(u_{j})}(d_{u_{j}}φ(B_{r}(O_{u_{j}})) + B_{ηr}(O_{φ(u_{j})})),

and with ℰ_{j} := d_{u_{j}}φ(B_{1}(O_{φ(u_{j})})) and ℰ′_{j} = (1 + \frac{η}{α_{n}(ℰ_{j})} ) ℰ_{j} we get with Lemma 1

φ(B_{r}(u_{j})) ⊂ exp_{φ(u_{j})} (r(ℰ_{j} + B_{η}(O_{φ(u_{j})}))) ⊂ exp_{φ(u_{j})} (rℰ′_{j}) .

With σ := a^{\frac{1}{n}} b^{−1}\sqrt{n}ρ_{K}(φ) we have

N_{σr}(K) ≤ N\frac{r}{b} (K_{1}) max_{j=1,... ,N_{r}(K_{1})} N_{σr}(exp_{φ(u_{j})}(\frac{r}{b}ℰ′_{j}))
≤ aN_{r}(K) max_{j=1,... ,N_{r}(K_{1})} N_{σr}(exp_{φ(u_{j})}(\frac{r}{b}ℰ′_{j}))

and, analogous to the proof of Theorem 3, N_{σr}(exp_{φ(u_{j})}(\frac{r}{b}ℰ′_{j})) ≤ N_{\frac{1}{2} σr}(\frac{r}{b}ℰ′_{j}). Since ρ_{K}(φ) ≤ α_{n}(d_{u_{j}}φ) = α_{n}(ℰ_{j}) ≤ α_{n}(ℰ′_{j}) is satisfied, Lemma 2 yields to

\begin{aligned}N_{\frac{1}{2} σr}(\frac{r}{b}ℰ′_{j}) & ≤ & \frac{2^{n}ω_{n}(\frac{r}{b}ℰ′_{j})}{(\frac{1}{2}a^{\frac{1}{n}} b^{−1}rρ_{K}(φ))^{n}} = \frac{4^{n}ω_{n}(ℰ′_{j})}{aρ_{K}^{n}(φ)} ≤ \frac{4^{n} (1 + \frac{η}{ρ_{K}(φ)})^{n} ω_{n}(ℰ_{j})}{aρ_{K}^{n}(φ)} \\ & ≤ & \frac{8^{n}ω_{n}(d_{u_{j}}φ)}{aρ_{K}^{n}(φ)}. \end{aligned}

Thus we get

N_{σr}(K) ≤ aN_{r}(K)\frac{8^{n}ω_{n,K}(φ)}{aρ_{K}^{n}(φ)} = N_{r}(K)\frac{8^{n}ω_{n,K}(φ)}{ρ_{K}^{n}(φ)} \frac{a^{\frac{d}{n}} ρ_{K}^{d}(φ)n^{\frac{d}{2}}}{b^{d}σ^{d}} ≤ σ^{−d}N_{r}(K).

Because of σ < 1 for any ε ∈ (0, r_{0}) we can find a number l ∈ ℕ_{0} such that σ^{l+1}r_{0} ≤ ε < σ^{l}r_{0} is satisfied. So we get

N_{ε}(K) ≤ σ^{−ld}N_{σ^{−l}ε}(K) ≤ (\frac{r_{0}}{ε} )^{d} N_{σ^{−l}ε}(K) ≤ (\frac{r_{0}}{ε} )^{d} N_{σr_{0}}(K)

and therefore,

dim_{F} (K) = lim sup_{ε→0} \frac{ln N_{ε}(K)}{− ln ε} ≤ d. ■

Corollary 7 Let condition 1) of Theorem 4 and the following condition be satisfied:

2) there exists a number d ∈ [0, n] with a^{\frac{d}{n}} b^{−d}ω_{n,K}(φ)ρ_{K}^{d−n}(φ) < 1.

Then we have dim_{F} (K) ≤ d.

Proof Since K is invariant under φ it is also invariant under φ^{i} for any i ∈ ℕ. The iterates of φ satisfy the relations

\begin{matrix}ω_{n,K}(φ^{i}) ≤ ω_{n,K}^{i}(φ), \\ ρ_{K}(φ^{i}) ≥ ρ_{K}^{i}(φ), \end{matrix}

and therefore,

a^{\frac{di}{n}} b^{−di}ω_{n,K}(φ^{i})ρ_{K}^{d−n}(φ^{i}) ≤ (a^{\frac{d}{n}} b^{−d}ω_{n,K}(φ)ρ_{K}^{d−n}(φ))^{i} .

So we can assume without loss of generality that

a^{\frac{d}{n}} b^{−d}ω_{n,K}(φ)ρ_{K}^{d−n}(φ) ≤ 8^{−n}n^{− \frac{d}{2}}

and, because of

a^{\frac{d}{n}} b^{−d}ω_{n,K}(φ)ρ_{K}^{d−n}(φ) ≥ a^{\frac{d}{n}} b^{−d}ρ_{K}^{n}(φ)ρ_{K}^{d−n}(φ) = (a^{\frac{1}{n}} b^{−1}ρ_{K}(φ))^{d} ,

the relation

a^{\frac{1}{n}} b^{−1}ρ_{K}(φ) < \sqrt{n}^{−1}

is satisfied; in the opposite case we consider the map φ^{i} with sufficiently large i and substitute a by a^{i}.

So all the conditions of Theorem 4 are satisfied, and we get dim_{F} (K) ≤ d. ■

Example Let us again consider the modified horseshoe map. For the sets K_{j} defined before, we know that for sufficiently small ε the equality N_{ε}(K_{j}) = 4^{j}N_{3^{j}ε}(K) is valid, i. e. with a = 4 and b = 3 condition 1) is satisfied. Then condition 2) of Corollary 7 results in 4^{\frac{d}{2}} 3^{−d} \frac{5}{3} (\frac{1}{3})^{d−2} < 1, which is equivalent to d > \frac{ln 15}{ln 4.5}. So Corollary 7 can be applied for any d > \frac{ln 15}{ln 4.5} and gives dim_{F} (K) ≤ d. In the limit we get dim_{F} (K) ≤ \frac{ln 15}{ln 4.5} ≈ 1.800.

By changing the metric with the Lyapunov type function p used in estimating the Hausdorff dimension we do not change a and b, because we do not change the metric on K_{1}. So we have N_{ε}(K_{j}) = 4N_{3ε}(K_{j_{1}}) for any j > 1 and only the relation between K_{1} and K is changed, because the balls on K \ K_{1} become smaller. This means N_{ε}(K_{1}) ≤ 4N_{3ε}(K_{j_{1}}). So condition 1) is again satisfied with a = 4 and b = 3. Condition 2) now results in 4^{\frac{d}{2}} 3^{−d} \frac{\sqrt{15}}{3} (\frac{1}{3})^{d−2} < 1, which means d > \frac{ln 3\sqrt{15}}{ln 4.5}. So we get dim_{F} (K) ≤ \frac{ln 3\sqrt{15}}{ln 4.5} ≈ 1.631. □

This work was supported by the Russian Research Fund 96 - 01 - 01172 and the DFG-Schwerpunktprogramm ”‘Dynamik: Analysis, effiziente Simulation und Ergodentheorie”’.

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