Hausdorff dimension estimates for invariant sets

with an equivariant tangent bundle splitting

Astrid Franz

Department of Mathematics, University of Technology Dresden, Mommsenstr. 13, 01062 Dresden

Abstract. Upper bounds for the Hausdorff dimension of compact and invariant sets of diffeomorphisms are given using a singular value function of the tangent map and the topological entropy under the assumption, that there exists an equivariant splitting of the tangent bundle. This improves previous results for compact uniformly hyperbolic sets of diffeomorphisms satisfying an additional pinching condition. Furthermore it is shown that the results can be extended to a special class of noninjective maps.

1. Introduction

In [4] upper bounds for the Hausdorff dimension of compact hyperbolic sets of C^{2}- diffeomorphisms in terms of uniform Lyapunov exponents and the topological entropy of the map are given, which in general are better than those obtained by Fathi in [3].

In the present paper compact invariant sets of diffeomorphisms are considered, for which there exists an equivariant splitting of the tangent bundle. In contrast to hyperbolic sets there nothing is assumed about the stretching rates of the tangent map in the splitting subspaces of the tangent space. It is shown that an upper bound of the Hausdorff dimension of such sets can be given in terms of the singular values of the tangent map reduced to subspaces of the tangent space, as well as in terms of the topological entropy of the given map.

Since the Taylor formula for differentiable maps is used in a different way than in [4], the diffeomorphism considered only needs to be C^{1}. Analogously to [4] uniform Lyapunov exponents can be introduced, and it can be shown that the pinching condition of [4] is unnecessary.

In the final section it is shown that the method of estimating the Hausdorff dimension can be applied not only to diffeomorphisms, but also to a special class of noninjective maps.

Let us first recall the definition of Hausdorff dimension and topological entropy. Let (X, ρ) be a metric space and K ⊂ X a compact subset. For arbitrary real numbers ε > 0 and d ≥ 0 we consider an outer measure at level ε and of order d of K given by

μ(K, d, ε) = inf ∑_{i} r_{i}^{d},

where the infimum is taken over all finite covers of K by balls B(u_{i}, r_{i}) of radii r_{i} ≤ ε centered in u_{i} ∈ X. For fixed d and K the function μ(K, d, ·) is monotone nonincreasing. Hence the limit

μ(K, d) = lim_{ε→0+0} μ(K, d, ε)

exists, although it may be infinite, and is called the outer Hausdorff measure of order d of K. For every compact set K there exists a critical number d^{∗} ≥ 0 with

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

This critical number is called the Hausdorff dimension of K and denoted by dim_{H}(K).

Suppose now that φ : K → K is a continuous map. For numbers ε > 0 and p ∈ ℕ we say that a subset S ⊂ K is a (p, ε)-spanning set for K with respect to φ if for every x ∈ K there exists a y ∈ S such that ρ(φ^{j}(x), φ^{j}(y)) ≤ ε holds for all j = 0, …, p − 1. The compactness of K implies the existence of finite (p, ε)-spanning sets. Let N_{ε}(K, p) denote the least number of points of any set S that (p, ε)-spans K with respect to φ. Then the limit

N_{ε}(K) = lim sup_{p→∞} \frac{1}{p} ln N_{ε}(K, p)

is finite (see [6]). Since N_{ε}(K) is monotone nonincreasing in ε, the limit

h_{top}(φ) = lim_{ε→0+0} N_{ε}(K)

exists, although it may be infinite, and is called the topological entropy of φ on K.

In order to use a singular value function let us briefly recall the definition of singular values. Consider a linear operator L : E → E′ between two n-dimensional Euclidean spaces E and E′ with the scalar products ⟨·, ·⟩_{E} and ⟨·, ·⟩_{E′}, respectively, and let L^{∗} denote the adjoint operator of L, i. e. the unique linear operator L^{∗} : E′ → E satisfying ⟨Lx, y⟩_{E′} = ⟨x, L^{∗}y⟩_{E} for all x ∈ E and y ∈ E′. The singular values of L are defined as the eigenvalues of the positive semi-definite operator \sqrt{L^{∗}L}.

2. Splitting of the tangent bundle

Definition 1. Let (M, g) be an n-dimensional Riemannian C^{1}-manifold, U ⊂ M open, φ : U → M a C^{1}-map and K ⊂ U a compact and φ-invariant set, i.e. φ(K) = K. We say that K has an equivariant tangent bundle splitting T_{K}M = E^{1} ⊕ E^{2} with respect to φ if for any u ∈ K the space E_{u}^{1} := E^{1} ∩ T_{u}M is an n_{1}-dimensional subspace of T_{u}M and E_{u}^{2} := E^{2} ∩ T_{u}M is an n_{2}-dimensional subspace of T_{u}M such that

dφ(E_{u}^{1}) = E_{φ(u)}^{1} and dφ(E_{u}^{2}) = E_{φ(u)}^{2}

are satisfied for any u ∈ K. The natural numbers n_{1} ≥ 1 and n_{2} ≥ 1 are constant for all points u ∈ K and satisfy n_{1} + n_{2} = n.

Hyperbolic sets of diffeomorphisms (see for instance [5]) are a special class of sets with an equivariant splitting of the tangent bundle.

Analogous to [1], our estimation of dimension is based on a contraction condition of the singular values of the tangent map. But in order to use the splitting structure of the tangent bundle the considered singular value function must have a modified form.

Definition 2. Let (M, g) be an n-dimensional Riemannian C^{1}-manifold, U ⊂ M open, φ : U → M a C^{1}-diffeomorphism and K ⊂ U a compact φ-invariant set with the equivariant splitting T_{K}M = E^{1} ⊕ E^{2}. For a point u ∈ K the singular values of d_{u}φ|_{E_{u}^{1}} are denoted by α_{1}^{1}(u), …, α_{n_{1}}^{1}(u) and the singular values of d_{u}φ^{−1}|_{E_{u}^{2}} are denoted by α_{1}^{2}(u), …, α_{n_{2}}^{2}(u) with respect to their geometric multiplicity. Let the set {α_{1}^{1}(u), …, α_{n_{1}}^{1}(u), α_{1}^{2}(u), …, α_{n_{2}}^{2}(u)} be relabeled as {α_{1}(u), …, α_{n}(u)} with α_{1}(u) ≥ … ≥ α_{n}(u). Let d ∈ [0, n] be a number which is for d > 0 written as d = d_{0}+s with d_{0} ∈ {0, 1, …, n − 1} and s ∈ (0, 1]. Then

ω_{d,K}^{E^{1},E^{2}}(φ) = \{ \begin{matrix}1 & for & d = 0, \\ sup_{u∈K} (α_{1}(u) · . . . · α_{d_{0}}(u)α_{d_{0}+1}(u)^{s}) & for & d ∈ (0, n] \end{matrix}

is the singular value function of order d of φ on K with respect to the splitting E^{1} ⊕ E^{2}.

Now we state a few properties of this singular value function.

(S1) ω_{d,K}^{E^{1},E^{2}}(aφ) = a^{d}ω_{d,K}^{E^{1},E^{2}}(φ) whenever a > 0.
(S2) ω_{d,K}^{E^{1},E^{2}}(φ^{p}) ≤ (ω_{d,K}^{E^{1},E^{2}}(φ))^{p} for any p ∈ ℕ.

Property (S2) means that ω_{d,K}^{E^{1},E^{2}}(φ^{p}) is a subexponential function in p, so the limit

ν_{d} := lim_{p→∞} \frac{1}{p} ln ω_{d,K}^{E^{1},E^{2}}(φ^{p}) exists for any d ∈ [0, n] (see [8]).

Definition 3. The numbers ν_{1}^{u} := ν_{1} and ν_{i}^{u} := ν_{i} − ν_{i−1}, i = 2, …, n, are called uniform Lyapunov exponents of φ on K with respect to the splitting E^{1} ⊕ E^{2}.

In the case of a hyperbolic set K this definition coincides with the Definitions 2.1 and 2.2 of [4] concerning uniform stable and unstable Lyapunov exponents.

An expression similar to the singular value function of maps we define for the direct sum of two ellipsoids. Let E_{1} and E_{2} be arbitrary Euclidean spaces with the dimensions n_{1} ≥ 1 and n_{2} ≥ 1 (n_{1} + n_{2} = n) and E_{1} ⊂ E_{1} and E_{2} ⊂ E_{2} be two ellipsoids where the lengths of the half-axes are a_{1}^{1}, …, a_{n_{1}}^{1} and a_{1}^{2}, …, a_{n_{2}}^{2}, respectively. Relabel the set {a_{1}^{1}, …, a_{n_{1}}^{1}, a_{1}^{2}, …, a_{n_{2}}^{2}} as {a_{1}, …, a_{n}} with a_{1} ≥ … ≥ a_{n}. Then we define

ω_{d}^{E_{1},E_{2}}(E_{1} ⊕ E_{2}) = \{ \begin{matrix}1 & for & d = 0, \\ a_{1} · . . . · a_{d_{0}}a_{d_{0}+1}^{s} & for & d ∈ (0, n], \end{matrix}

where for d > 0 this number is written in the form d = d_{0}+s with d_{0} ∈ {0, 1, …, n−1} and s ∈ (0, 1].

For such sets E_{1} ⊕ E_{2} we have the following lemma analogous to Lemma 3 of [1].

Lemma 1. Let k > 0, m > 0 and d ∈ (0, n_{1} + n_{2}] in the form d = d_{0} + s with d_{0} ∈ {0, 1, …, n_{1} + n_{2} − 1} and s ∈ (0, 1] be numbers satisfying k ≤ m^{d}, and η > 0. Let E_{1} and E_{2} be two Euclidean spaces with the dimensions n_{1} and n_{2} and E_{1} ⊂ E_{1} and E_{2} ⊂ E_{2} two ellipsiods with ω_{d}^{E_{1},E_{2}}(E_{1} ⊕ E_{2}) ≤ k, where the lengths of all half-axes are at most m. Then E_{1} ⊕ E_{2} + B(O, η) is contained in E′_{1} ⊕ E′_{2}, where E′_{1} ⊂ E_{1} and E′_{2} ⊂ E_{2} are ellipsiods with

ω_{d}^{E_{1},E_{2}}(E′_{1} ⊕ E′_{2}) ≤ [1 + (\frac{m^{d_{0}}}{k} )^{\frac{1}{s}} η]^{d} k.

3. Dimension estimate in the case of diffeomorphisms

Theorem 1. Let (M, g) be an n-dimensional Riemannian C^{1}-manifold, U ⊂ M open, φ : U → M a C^{1}-diffeomorphism and K ⊂ U a compact and φ-invariant set for which there exists an equivariant splitting T_{K}M = E^{1} ⊕ E^{2}. Let λ = h_{top}(φ|_{K}) denote the topological entropy of φ on K and assume that there exists a number d ∈ [0, n] such that ω_{d,K}^{E^{1},E^{2}}(φ) < e^{−2λ} holds. Then dim_{H}(K) ≤ d.

Proof. The condition ω_{d,K}^{E^{1},E^{2}}(φ) < e^{−2λ} can only be satisfied for d > 0, because in the case d = 0 we have ω_{0,K}^{E^{1},E^{2}}(φ) = 1 and the topological entropy is always nonnegative, i. e. e^{−2λ} ≤ 1. We consider d in the form d = d_{0} + s with d_{0} ∈ {0, 1, …, n − 1} and s ∈ (0, 1]. With ν := ω_{d,K}^{E^{1},E^{2}}(φ) we have ν < e^{−2λ}. Therefore there exists a number l with ν < l < e^{−2λ}. Because of property (S2) we have ω_{d,K}^{E^{1},E^{2}}(φ^{p}) ≤ ν^{p} < l^{p} for any p ∈ ℕ.

Let p ∈ ℕ be fixed. There exists a sufficiently large number m > 0 such that the inequalities

ν^{p} < m^{d}, sup_{u∈K} |||d_{u}φ^{p}||| ≤ m, sup_{u∈K} |||d_{u}φ^{−p}||| ≤ m  (1)

are satisfied, where ||| · ||| denotes the operator norm for the given linear operator. Let η > 0 be the number satisfying the equation

[1 + (\frac{m^{d_{0}}}{ν^{p}} )^{\frac{1}{s}} η]^{d} ν^{p} = l^{p}.  (2)

Let r_{1} be so small that there exists an open set V ⊂ U containing K such that

|||τ_{φ^{p}(v)}^{φ^{p}(u)}d_{v}φ^{p}τ_{u}^{v} − d_{u}φ^{p}||| ≤ η, |||τ_{φ^{−p}(v)}^{φ^{−p}(u)}d_{v}φ^{−p}τ_{u}^{v} − d_{u}φ^{−p}||| ≤ η  (3)

for any u, v ∈ V with ρ(u, v) ≤ r_{1} are satisfied, where ρ(·, ·) denotes the geodesic distance between the points of M and τ_{u}^{v} 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(O_{u}, r_{u}) is true, where O_{u} denotes the origin of the tangent space T_{u}M. Because of the compactness of K there is a number r_{2} = min_{u∈K} r_{u} > 0 such that |||d_{v} exp_{u} ||| ≤ 2 is satisfied for any u ∈ K and any v ∈ B(O_{u}, r_{2}). This means

ρ(exp_{u} v, exp_{u} w) ≤ 2ρ(v, w)  (4)

for any u ∈ K and any v, w ∈ B(O_{u}, r_{2}). Now we take r_{0} = min\{r_{1}, \frac{1}{m}(\frac{ν}{l})^{\frac{p}{d}} r_{2}\} and let ε ∈ (0, r_{0}) be fixed. Let

\begin{aligned}B^{p}(u, r) & := & \{u ∈ U | ρ(φ^{q}(u), φ^{q}(u)) ≤ r, q = −p, …, p\} \\ & = & ⋂_{q=−p}^{p} φ^{q}(B(φ^{−q}(u), r)). \end{aligned}

denote a Bowen ball of order p with centre u ∈ K and radius r > 0 and let {B^{p}(u_{i}, r_{i})} be a finite cover of the set K with Bowen balls of order p with centre u_{i} ∈ K and radii r_{i} ≤ ε. The Bowen ball B^{p}(u_{i}, r_{i}) satisfies B^{p}(u_{i}, r_{i}) ⊂ φ^{p}(B(φ^{−p}(u_{i}), r_{i})) and the Taylor formula for the differentiable map φ^{p} gives for every v ∈ B(φ^{−p}(u_{i}), r_{i}) the relation

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

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

Together with (3) we get

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

Analogously we have

exp_{u_{i}}^{−1}(B^{p}(u_{i}, r_{i})) ⊂ d_{φ^{p}(u_{i})}φ^{−p}(B(O_{φ^{p}(u_{i})}, r_{i})) + B(O_{u_{i}}, ηr_{i}).

Therefore we obtain

\begin{aligned}exp_{u_{i}}^{−1}(B^{p}(u_{i}, r_{i})) ⊂ & (d_{φ^{p}(u_{i})}φ^{−p}(B(O_{φ^{p}(u_{i})}, r_{i})) ∩ d_{φ^{−p}(u_{i})}φ^{p}(B(O_{φ^{−p}(u_{i})}, r_{i}))) \\ & +B(O_{u_{i}}, ηr_{i}) \end{aligned}

with

\begin{matrix}d_{φ^{p}(u_{i})}φ^{−p}(B(O_{φ^{p}(u_{i})}, r_{i})) ∩ d_{φ^{−p}(u_{i})}φ^{p}(B(O_{φ^{−p}(u_{i})}, r_{i})) \\ ⊂ π_{2} (d_{φ^{p}(u_{i})}φ^{−p}(B(O_{φ^{p}(u_{i})}, r_{i}))) ⊕ π_{1} (d_{φ^{−p}(u_{i})}φ^{p}(B(O_{φ^{−p}(u_{i})}, r_{i}))) , \end{matrix}

where π_{1} denotes the projection of T_{K}M onto E^{1} and π_{2} denotes the projection of T_{K}M onto E^{2}. The set E_{i}^{1} := π_{1} (d_{φ^{−p}(u_{i})}φ^{p}(B(O_{φ^{−p}(u_{i})}, r_{i}))), as the projection of an ellipsoid, is an ellipsoid and lies in E_{u_{i}}^{1}. Analogously E_{i}^{2} := π_{2} (d_{φ^{p}(u_{i})}φ^{−p}(B(O_{φ^{p}(u_{i})}, r_{i}))) is an

ellipsoid in E_{u_{i}}^{2}. Because of ω_{d,K}^{E^{1},E^{2}}(φ^{p}) ≤ ν^{p} we have ω_{d}^{E_{u_{i}}^{1},E_{u_{i}}^{2}}(E_{i}^{1} ⊕ E_{i}^{2}) ≤ ν^{p}r_{i}^{d}, and from (1) it follows that the lengths of the half-axes of E_{i}^{1} and E_{i}^{2} are less than or equal mr_{i}. Therefore with Lemma 1 and (2) there exist ellipsoids E_{i}^{1′} ⊂ E_{u_{i}}^{1} and E_{i}^{2′} ⊂ E_{u_{i}}^{2} with

exp_{u_{i}}^{−1}(B^{p}(u_{i}, r_{i})) ⊂ E_{i}^{1′} ⊕ E_{i}^{2′}

and

ω_{d}^{E_{u_{i}}^{1},E_{u_{i}}^{2}}(E_{i}^{1′} ⊕ E_{i}^{2′}) ≤ l^{p}r_{i}^{d}.  (5)

The number of Bowen balls needed to cover K is equal to the least number N_{ε}(K, 2p+1) of points of any set that (2p + 1, ε)-spans K with respect to φ. Thus K is covered by N_{ε}(K, 2p + 1) sets of the form exp_{u_{i}}(E_{i}^{1′} ⊕ E_{i}^{2′}) with ω_{d}^{E_{u_{i}}^{1},E_{u_{i}}^{2}}(E_{i}^{1′} ⊕ E_{i}^{2′}) ≤ l^{p}r_{i}^{p}.

Let i ∈ {1, …, N_{ε}(K, 2p + 1)} be fixed, and let a_{1}^{1} ≥ a_{2}^{1} ≥ … ≥ a_{n_{1}}^{1} denote the lengths of the half-axes of E_{i}^{1′} and a_{1}^{2} ≥ a_{2}^{2} ≥ … ≥ a_{n_{2}}^{2} the lengths of the half-axes of E_{i}^{2′}. Relabeling and ordering yields the set {a_{1}, …, a_{n}}. Furthermore, let d_{1} and d_{2} denote the number of lengths of {a_{1}, …, a_{d_{0}}} belonging to half-axes of E_{i}^{1′} and E_{i}^{2′}, respectively, and let ζ := a_{d_{0}+1}. If a_{d_{0}+1} is the length of an axis of E_{i}^{1′} we put s_{1} := s and s_{2} := 0, else s_{2} := s and s_{1} := 0. Using (5) we get

ζ ≤ (ω_{d}^{E_{u_{i}}^{1},E_{u_{i}}^{2}}(E_{i}^{1′} ⊕ E_{i}^{2′}))^{\frac{1}{d}} ≤ (l^{p}r_{i}^{d})^{\frac{1}{d}} < r_{i} ≤ ε.  (6)

Similarly to the proof of Lemma 1 of [1], E_{i}^{1′} can be covered by N_{1} := Π_{j=1}^{d_{1}}([\frac{a_{j}^{1}}{ζ}] + 1)

balls of radius ζ\sqrt{d_{1} + 1}, and E_{i}^{2′} can be covered by N_{2} := Π_{j=1}^{d_{2}}([\frac{a_{j}^{2}}{ζ}] + 1) balls with radius ζ\sqrt{d_{2} + 1}. Therefore E_{i}^{1′} ⊕ E_{i}^{2′} is covered by N_{1}N_{2} balls of radius 2ζ\sqrt{d_{0} + 1}. Hence from (4) the set exp_{u_{i}}(E_{i}^{1′} ⊕ E_{i}^{2′}) is covered by N_{1}N_{2} balls of radius 4ζ\sqrt{d_{0} + 1}. Using (6) we obtain

μ(exp_{u_{i}}(E_{i}^{1′} ⊕ E_{i}^{2′}), d, 4ε\sqrt{d_{0} + 1}) ≤ N_{1}N_{2}4^{d}\sqrt{d_{0} + 1}^{d}ζ^{d}
≤ 2^{2d+d_{0}}(d_{0} + 1)^{\frac{d}{2}} a_{1} … a_{d_{0}}ζ^{s}
≤ l^{p}r_{i}^{d}2^{2d+d_{0}}(d_{0} + 1)^{\frac{d}{2}}
≤ l^{p}ε^{d}2^{2d+d_{0}}(d_{0} + 1)^{\frac{d}{2}} .

Since this inequality holds for any i ∈ {1, …, N_{ε}(K, 2p + 1)}, we have

\begin{aligned}μ(K, d, 4ε\sqrt{d_{0} + 1}) & ≤ & ∑_{i=1}^{N_{ε}(K,2p+1)} μ(exp_{u_{i}}(E_{i}^{1′} ⊕ E_{i}^{2′}), d, 4ε\sqrt{d_{0} + 1}) \\ & ≤ & N_{ε}(K, 2p + 1)l^{p}ε^{d}2^{2d+d_{0}}(d_{0} + 1)^{\frac{d}{2}} . \end{aligned}  (7)

The definition of the topological entropy of φ on K implies that for arbitrary α > 0 there exist numbers ε_{0}(α) > 0 and p_{0}(α, ε) ∈ ℕ such that

N_{ε}(K, 2p + 1) < e^{p(2λ+α)}

holds for 0 < ε < ε_{0}(α) and p > p_{0}(α, ε). Therefore we have

μ(K, d, 4ε\sqrt{d_{0} + 1}) < e^{p(2λ+α+ln l)}ε^{d}2^{2d+d_{0}}(d_{0} + 1)^{\frac{d}{2}} .

Because of l < e^{−2λ} we have 2λ + ln l < 0. We can choose α > 0 sufficiently small such that 2λ + ln l + α < 0 is also true. Then for ε < ε_{0}(α) and p > p_{0}(α, ε) we have

μ(K, d, 4ε\sqrt{d_{0} + 1}) < e^{p(2λ+α+ln l)}ε^{d}2^{2d+d_{0}}(d_{0} + 1)^{\frac{d}{2}} ⟶^{p→∞} 0,

i. e. μ(K, d, 4ε\sqrt{d_{0} + 1}) = 0 and therefore, μ(K, d) = 0. This means dim_{H} K ≤ d. ■

If we consider the long-time behaviour of the mapping, then the singular values

used in Theorem 1 can be replaced by the uniform Lyapunov exponents of Definition 3.

Corollary 1. Let (M, g) be an n-dimensional Riemannian C^{1}-manifold, U ⊂ M open and φ : U → M a C^{1}-diffeomorphism. Furthermore let K ⊂ U be a compact and φ-invariant set for which there exists an equivariant splitting T_{K}M = E^{1} ⊕ E^{2}, and let ν_{1}^{u} ≥ … ≥ ν_{n}^{u} be the uniform Lyapunov exponents of φ on K with respect to the splitting T_{K}M = E^{1} ⊕ E^{2}. Let D < n be the smallest natural number with the property

2h_{top}(φ|_{K}) + ν_{1}^{u} + … + ν_{D}^{u} + ν_{D+1}^{u} < 0.  (8)

Then the Hausdorff dimension of K can be estimated by

dim_{H}(K) ≤ D + \frac{2h_{top}(φ|_{K}) + ν_{1}^{u} + … + ν_{D}^{u}}{|ν_{D+1}^{u}|} .

Proof. Let us again denote λ = h_{top}(φ|_{K}). Because of λ ≥ 0 condition (8) implies ν_{D+1}^{u} < 0 and therefore, ν_{m}^{u} < 0 for any m ∈ {D + 1, …, n}. Let d ∈ (D +

\frac{2λ+ν_{1}^{u}+...+ν_{d_{0}}^{u}}{|ν_{d_{0}+1}^{u}|} , n] be an arbitrary number in the form d = d_{0} +s with d_{0} ∈ \{0, …, n−1\} and s ∈ (0, 1]. Then we have 2λ + ν_{1}^{u} + … + ν_{d_{0}}^{u} + sν_{d_{0}+1}^{u} = 2λ + ν_{d} < 0. Choose ε > 0 so small that ν_{d} + ε < −2λ is satisfied. Since ν_{d} = lim_{p→∞} \frac{1}{p} ln ω_{d,K}^{E^{1},E^{2}}(φ^{p}) there is a

natural number N with \frac{1}{N} ln ω_{d,K}^{E^{1},E^{2}}(φ^{N}) < ν_{d} + ε. This means

ω_{d,K}^{E^{1},E^{2}}(φ^{N}) < e^{N(ν_{d}+ε)} < e^{−2Nλ}.

Since the topological entropy of φ^{N} on K is Nλ the assumptions of Theorem 1 are satisfied for the map φ^{N} and the invariant set K. Therefore with Theorem 1 we get dim_{H}(K) ≤ d for any d ∈ (D + \frac{2λ+ν_{1}^{u}+...+ν_{d_{0}}^{u}}{|ν_{d_{0}+1}^{u}|} , n]. In the limit this yields dim_{H}(K) ≤

D + \frac{2λ+ν_{1}^{u}+...+ν_{d_{0}}^{u}}{|ν_{d_{0}+1}^{u}|}. ■

Corollary 1 is an upper Hausdorff dimension estimate analogously to Theorem 1.2 of [4], but with weakened conditions: The C^{2}-property of the considered diffeomorphism is weakened to C^{1} and the invariant set does not need to be hyperbolic, only an equivariant splitting of the tangent bundle has to exist. Furthermore, the pinching condition AB^{2} < 1 is not necessary. This condition ensures that the splitting of the tangent bundle is smooth (see [5]), but this smoothness is not needed in the proof of Corollary 1.

4. Example of a diffeomorphism

We want to illustrate the dimension estimation by Theorem 1 and Corollary 1 for a horseshoe map which stretches the unit square [0, 1] × [0, 1] ⊂ ℝ^{2} horizontally with a factor μ_{1} < \frac{1}{2} and vertically with a factor μ_{2} > 2 and then folds it to a horseshoe (see Figure 1).

[figure: ⟶; ⟶; 1; 1; μ_{1}; μ_{2}]

Figure 1. Horseshoe map

This map can be extended to a C^{1}-diffeomorphism φ on an open neighbourhood U of the unit square. The maximal invariant set K := ⋂_{i=−∞}^{∞} φ^{i}([0, 1] × [0, 1]) is hyperbolic in the sense of [4] if we consider the equivariant splitting

\begin{aligned}T_{K}M & = & \{[u, v] | u ∈ K, v ∈ ℝ^{2}\} \\ & = & \{[u, (a, 0)] | u ∈ K, a ∈ ℝ\} ⊕ \{[u, (0, b)] | u ∈ K, b ∈ ℝ\}. \end{aligned}

For this set it is possible to compare the results of Theorem 1 and Corollary 1 with those of Theorem 1.2 of [4]. The invariant set K is the product of two Cantor sets with Hausdorff dimensions _{\frac{ln 2}{ln \frac{1}{μ_{1}}}} and \frac{ln 2}{ln μ_{2}} . Since Cantor sets have equal Hausdorff and fractal dimensions, Corollary 7.4 of [2] gives dim_{H}(K) = _{\frac{ln 2}{ln \frac{1}{μ_{1}}}} + \frac{ln 2}{ln μ_{2}} as the exact Hausdorff

dimension of K. We can thus compare the dimension estimates with the exact value.

The topological entropy of this map on K is λ = ln 2 because the horseshoe map is topologically conjugate to the Bernoulli shift on the set of infinite sequences with two symbols. Applying Theorem 1 we obtain the singular value function of the described C^{1}-diffeomorphism φ on the invariant set K

ω_{d,K}^{E^{1},E^{2}}(φ) = \{ \begin{matrix}(max\{μ_{1}, \frac{1}{μ_{2}} \})^{d} , & for 0 ≤ d ≤ 1, \\ max\{μ_{1}, \frac{1}{μ_{2}} \} (min\{μ_{1}, \frac{1}{μ_{2}} \})^{d−1} , & for 1 < d ≤ 2. \end{matrix}

Let us consider two cases.

Case 1: max\{μ_{1}, \frac{1}{μ_{2}} \} < \frac{1}{4}.

Then the condition ω_{d,K}^{E^{1},E^{2}}(φ) < e^{−2λ} = \frac{1}{4} can be satisfied for numbers d < 1. We have dim_{H}(K) < d for any d with max\{μ_{1}, \frac{1}{μ_{2}} \}^{d} < \frac{1}{4} and, therefore, d > _{\frac{2 ln 2}{− ln(max\{μ_{1}, \frac{1}{μ_{2}} \})}}.

Thus dim_{H}(K) ≤ _{\frac{2 ln 2}{− ln(max\{μ_{1}, \frac{1}{μ_{2}} \})}}. Hence Theorem 1 gives a sharp estimation in the

case μ_{1}μ_{2} = 1, otherwise it gives only an upper bound.

Case 2: max\{μ_{1}, \frac{1}{μ_{2}} \} ≥ \frac{1}{4}. Here we have to consider d > 1. Theorem 1 gives dim_{H}(K) ≤ d for any d with

max\{μ_{1}, \frac{1}{μ_{2}} \} (min\{μ_{1}, \frac{1}{μ_{2}} \})^{d−1} < \frac{1}{4}, which is equivalent to d > 1 + \frac{ln(4 max\{μ_{1}, \frac{1}{μ_{2}} \})}{− ln(min\{μ_{1}, \frac{1}{μ_{2}} \})}.

Hence we get dim_{H}(K) ≤ 1 + \frac{ln(4 max\{μ_{1}, \frac{1}{μ_{2}} \})}{− ln(min\{μ_{1}, \frac{1}{μ_{2}} \})}, which is also a sharp estimation in the

case μ_{1}μ_{2} = 1. Since the singular values are constant on K, Corollary 1 yields the same result as Theorem 1.

Now let us consider Theorem 1.2 of [4]. The uniform Lyapunov exponents are ln μ_{1} and − ln μ_{2}, the numbers A and B considered by Gu in [4] are A = max\{\frac{1}{μ_{1}} , μ_{2}\} = _{\frac{1}{min\{μ_{1}, \frac{1}{μ_{2}} \}}}

and B = max\{μ_{1}, \frac{1}{μ_{2}} \}. Therefore the pinching condition AB^{2} < 1 made by Gu has the form max\{μ_{1}, \frac{1}{μ_{2}} \}^{2} < min\{μ_{1}, \frac{1}{μ_{2}} \}, i. e. \frac{1}{μ_{1}^{2}} < μ_{2} < \frac{1}{\sqrt{μ_{1}}} . This is an additional condition compared with Theorem 1 and Corollary 1.

The function τ used in Theorem 1.2 of [4] has the form

τ(d) = \{ \begin{matrix}d ln max\{μ_{1}, \frac{1}{μ_{2}} \}, & for 0 ≤ d ≤ 1, \\ ln max\{μ_{1}, \frac{1}{μ_{2}} \} + (d − 1) ln min\{μ_{1}, \frac{1}{μ_{2}} \}, & for 1 < d ≤ 2. \end{matrix}

It is the natural logarithm of the singular value function used in Theorem 1. Hence Theorem 1.2 of [4] yields the same dimension estimate as Theorem 1 and Corollary 1, but with the additional restriction \frac{1}{μ_{1}^{2}} < μ_{2} < \frac{1}{\sqrt{μ_{1}}} on the parameters.

In the dissipative case μ_{1}μ_{2} < 1 we can apply the theorem of Douady and Oesterlé (see [1]) to the horseshoe map as well. This leads to the estimate dim_{H}(K) ≤ 1 − \frac{ln μ_{2}}{ln μ_{1}} which is always greater than 1. Thus in case 1 considered above (max\{μ_{1}, \frac{1}{μ_{2}} \} ≤ \frac{1}{4}) the Douady-Oesterlé estimation is worse than the estimation by Theorem 1. In case 2 (max\{μ_{1}, \frac{1}{μ_{2}} \} > \frac{1}{4}) we have μ_{1} < \frac{1}{μ_{2}} because of μ_{1}μ_{2} < 1, therefore the estimation by

Theorem 1 has the form dim_{H}(K) ≤ 1 − \frac{ln \frac{4}{μ_{2}}}{ln μ_{1}} . From the assumption μ_{2} > 2 it follows

that \frac{4}{μ_{2}} < μ_{2} holds, so 1 − \frac{ln \frac{4}{μ_{2}}}{ln μ_{1}} is always less than 1 − \frac{ln μ_{2}}{ln μ_{1}} .

In this example Theorem 1 thus gives better upper bounds for the Hausdorff dimension of the invariant set than the theorem of Douady and Oesterlé. This is remarkable, because in Theorem 1 we assume not only that the singular value function has to be less than 1, but that it is also less than e^{−2h_{top}(φ|_{K})}. But the horseshoe map shows that this stronger assumption can be compensated for by using some singular values of the inverse map.

5. Estimate in the case of k -1-endomorphisms

The upper Hausdorff dimension estimation of Theorem 1 and Corollary 1 is not restricted to diffeomorphisms. It can also be applied to a special class of noninjective maps, the k -1-endomorphisms.

Definition 4. Let (M, g) be an n-dimensional Riemannian manifold. A k -1-endo-morphism system (φ, K, K_{1}, …, K_{k}) is a (k + 2)-tuple (k ∈ ℕ) consisting of a map φ : K → M and (k + 1) compact sets K ⊂ M, K_{1} ⊂ M, …, K_{k} ⊂ M with K = ⋃_{i=1}^{k} K_{i}, φ(K_{i}) = K for all i = 1, …, k and K_{i} ∩ K_{j} = ∅ for any 1 ≤ i < j ≤ k.

The map φ is then called a k -1-endomorphism. The definition implies φ(K) = K, i.e. K is invariant under φ. Furthermore, every point of the set K has at least k different preimages under the map φ. If the manifold M is C^{1} and, additionally, the following property (C1) is satisfied then every point of K has exactly k preimages.

(C1) Every partial map φ|_{K_{i}} can be extended to a C^{1}-diffeomorphism φ_{i} on an open neighbourhood U_{i} ⊂ M of K_{i} (i = 1, …, k).

Since the sets K_{1}, …, K_{k} are compact and pairwise disjoint, neighbourhoods U_{i} (i = 1, …, k) can be chosen pairwise disjoint and the map φ can be extended to a C^{1}- map defined on U = ⋃_{i=1}^{k} U_{i}, which coincides with φ_{i} on U_{i} (i = 1, …, k) and is a C^{1}-map. This extension will be denoted by φ as well. Since in this way φ is a C^{1}-map defined on a neighbourhood U of K, Definition 1 can be applied to φ. If there exists an equivariant splitting of the tangent bundle in the sense of Definition 1, then analogously to Definitions 2 and 3 a modified singular value function and uniform Lyapunov exponents for k -1-endomorphisms can be introduced, if we consider φ_{i}^{−1} (i = 1, …, k) instead of φ^{−1}. The following theorem and corollary give, analogously to Theorem 1 and Corollary 1, upper bounds for the Hausdorff dimension of invariant sets of k -1-endomorphisms with an equivariant tangent bundle splitting.

Theorem 2. Suppose that (M, g) is an n-dimensional Riemannian C^{1}-manifold and (φ, K, K_{1}, …, K_{k}) is a k -1-endomorphism system satisfying (C1) such that there exists an equivariant splitting T_{K}M = E^{1} ⊕ E^{2}. Let λ = h_{top}(φ|_{K}) denote the topological entropy of φ on K and assume that there exists a number d ∈ [0, n] such that ω_{d,K}^{E^{1},E^{2}}(φ) < ke^{−2λ} holds. Then dim_{H}(K) ≤ d.

Corollary 2. Suppose that (M, g) is an n-dimensional Riemannian C^{1}-manifold and (φ, K, K_{1}, …, K_{k}) is a k -1-endomorphism system satisfying (C1) such that there exists an equivariant splitting T_{K}M = E^{1} ⊕ E^{2}. Let ν_{1}^{u} ≥ … ≥ ν_{n}^{u} be the uniform Lyapunov exponents of φ on K with respect to the splitting T_{K}M = E^{1} ⊕ E^{2} and let D < n be the smallest natural number with the property

2h_{top}(φ|_{K}) − ln k + ν_{1}^{u} + … + ν_{D}^{u} + ν_{D+1}^{u} < 0.

Then the Hausdorff dimension of K can be estimated by

dim_{H}(K) ≤ D + \frac{2h_{top}(φ|_{K}) − ln k + ν_{1}^{u} + … + ν_{D}^{u}}{|ν_{D+1}^{u}|} .

These statements can be proven analogously to Theorem 1 and Corollary 1, if we consider the partial maps φ_{i} (i = 1, …, k) instead of the map φ. Furthermore the number k reduces the number of Bowen balls needed to cover K. Here the inequality (7) takes the form

k^{p}μ(K, d, 4ε\sqrt{d_{0} + 1}) ≤ N_{ε}(K, 2p + 1)l^{p}ε^{d}2^{2d+d_{0}}(d_{0} + 1)^{\frac{d}{2}} .

The proof of Corollary 2 is based in addition on the fact that the topological entropy of the

considered k -1-endomorphisms can be estimated by

h_{top}(φ|_{K}) ≥ ln k.

6. Example of a k -1-endomorphism

Let us consider the flat cylinder, namely, the product of the real line with the unit circle M = ℝ × S^{1} = {u = (x, y) | x ∈ ℝ, y = e^{iθ}, θ ∈ [0, 2π)}. This is a Riemannian manifold, for which x and the inverse function to y = e^{iθ} define a local coordinate system. Define a mapping φ on the sets A_{1} = \{(x, y) ∈ M | x ∈ [0, \frac{1}{3}]\} and A_{2} = \{(x, y) ∈ M | x ∈ [\frac{2}{3}, 1]\} by

φ(x, y) = \{ \begin{matrix}(3x, y) & for (x, y) ∈ A_{1}, \\ (3x − 2, y) & for (x, y) ∈ A_{2}. \end{matrix}

Then (φ, K = ⋂_{j=0}^{∞} φ^{−j}(A_{1} ∪ A_{2}), K_{1} = K ∩ A_{1}, K_{2} = K ∩ A_{2}) is a 2-1-endomorphism-system satisfying (C1). The singular values of the tangent map are the constants

α_{1} = 1 and α_{2} = 3. For any point u ∈ K let E_{u}^{i} ⊂ T_{u}M denote the eigenspace related to the eigenvalue α_{i} of d_{u}φ (i = 1, 2). Then T_{K}M = E^{1} ⊕E^{2} is an equivariant splitting of the tangent bundle. The singular values with respect to the splitting are 1 and \frac{1}{3}, the topological entropy of this map on K is ln 2. Hence

the condition of Theorem 2 can not be satisfied for a number d ∈ [0, 1]. For d ∈ (1, 2] it has the form 1 · (\frac{1}{3})^{d−1} < 2e^{−2 ln 2} = \frac{1}{2}, which is satisfied for any d > 1 + \frac{ln 2}{ln 3}, and we get the estimate dim_{H}(K) ≤ 1+ \frac{ln 2}{ln 3}. Since K is the product of the circle with the standard Cantor set, this estimate is sharp. The same estimate is obtained using Corollary 2 since the singular values with respect to the splitting are constant over the whole set K and, therefore, the uniform Lyapunov exponents coincide with the logarithms of these singular values.

Let us remark that for this example Theorem 1.2 of [4] is not applicable because the map is not invertible. Furthermore one of the singular values with respect to the splitting is 1 which means that the corresponding uniform Lyapunov exponent is zero. Therefore the eigenspace corresponding to this exponent is contained neither in the stable nor in the unstable part of the tangent space, i.e. we can not split the tangent bundle in a stable and an unstable bundle.

References

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

[2] Falconer K J 1990 Fractal Geometry: Mathematical Foundations and Applications (Chichester: John Wiley & Sons)

[3] Fathi A 1989 Expansiveness, Hyperbolicity and Hausdorff Dimension Commun. Math. Phys. 126: 249 - 262

[4] Gu X 1991 An upper bound for the Hausdorff dimension of a hyperbolic set Nonlinearity 4: 927 - 934

[5] Hirsch M W and Pugh C C 1970 Stable manifolds and hyperbolic sets Proc. Symp. in Pure Mathematics 14: 133 - 163

[6] Mañé R 1987 Ergodic Theory and Differentiable Dynamics (Berlin: Springer)

[7] Noack A and Reitmann V 1996 Hausdorff dimension of invariant sets of timedependent vector fields ZAA 15/2: 457 - 473

[8] Temam R 1988 Infinite-Dimensional Dynamical Systems in Mechanics and Physics (New York: Springer)