STABILITY OF ANOSOV DIFFEOMORPHISMS

CLARK ROBINSON AND A. VERJOVSKY

In these notes we give a proof of Anosov’s theorem on structural stability of diffeomorphisms of a compact C^{∞} manifold M without boundary. We also show that the Anosov diffeomorphisms form an open (maybe empty according to M) set in D, where D is the set of C^{r} diffeomorphisms of M with the C^{r} topology, r ≥ 1.

The main references are Moser [1], Mather’s appendix in [2] and Hirsch-Pugh [4].

Definition 1. Let < , > be a C^{∞} Riemannian metric on M and | · | its induced norm on T_{x}M for each x ∈ M. We say that f ∈ D is Anosov if

1. the tangent bundle of M splits in a Whitney direct sum of continuous subbundles T M = E^{s} ⊕ E^{u}, where E^{s} and E^{u} are Df-invariant,

2. there exists constants c, c′ > 0 and 0 < λ < 1 such that

|Df_{x}^{n}v| < c λ^{n}|v|
|Df_{x}^{−n}w| < c′ λ^{n}|w|

for all x ∈ M, v ∈ E_{x}^{s}, and w ∈ E_{x}^{u} and n > 0. M being compact, this definition is independent of the Riemannian metric < , >.

Also, E^{s} and E^{u} are uniquely determined by the above conditions.

A vector bundle π : E → M of class C^{r} is said to be normed if there is a C^{s} (0 ≤ s ≤ r) real function F : E → R such that F |π^{−1}(x) defines a norm on π^{−1}(x) for every x ∈ M. We usually denote such a norm by |v|.

Let π : E → M be a normed vector bundle over M. We denote by Γ(E) the Banach space of continuous sections of E, with norm ∥σ∥ = sup_{x∈M} |σ(x)|, σ ∈ Γ(E).

We denote Γ(T M) simply by Γ(M). If f ∈ D, then f induces a continuous operator f_{∗} : Γ(M) → Γ(M), defined by f_{∗}σ = Dfσ ∘ f^{−1}, σ ∈ Γ(M). That is f_{∗}σ(x) = Df_{f^{−1}(x)}σ(f^{−1}(x)). The linearity of f_{∗} is clear and its continuity follows from the fact that M is compact. In fact, f_{∗} is an isomorphism, where (f_{∗})^{−1} = (f^{−1})_{∗}.

In order to prove that the Anosov diffeomorphisms form an open set, we need the following lemmas.

Lemma 1. f ∈ D is Anosov if and only if f_{∗} is hyperbolic. Also, if f is Anosov then there is a C^{∞} structure of normed vector bundle on T M for which we can take c = c′ = 1 in Definition 1.

Proof. If f is Anosov, then Γ(M) splits in a direct sum of closed subspaces

Γ(M) = Γ(E^{s}) ⊕ Γ(E^{u})

1991 Mathematics Subject Classification. 58F.

Key words and phrases. dynamical systems, stability.

where

σ ∈ Γ(E^{s}) ⇔ σ(x) ∈ E^{s}, ∀ x ∈ M
σ ∈ Γ(E^{u}) ⇔ σ(x) ∈ E^{u}, ∀ x ∈ M.

Since E^{s} and E^{u} are Df-invariant, Γ(E^{s}) and Γ(E^{u}) are f_{∗}-invariant. Let f_{s} = f|Γ(E^{s}) and f_{u} = f|Γ(E^{u}). Then f_{∗} = f_{s} ⊕ f_{u} and f_{s}, f_{u} are (continuous) isomorphisms of Γ(E^{s}), Γ(E^{u}). This implies that

Spectrum(f_{∗}) = Spectrum(f_{s}) ∪ Spectrum(f_{u}).

But f being Anosov,

∥f_{s}^{n}∥ ≤ c λ^{n}
∥f_{u}^{−n}∥ ≤ c′ λ^{n}.

Therefore the spectral radius of f_{s} and f_{u}^{−1} are not bigger than λ < 1. Thus f_{∗} is hyperbolic.

Let us now assume that f_{∗} : Γ(M) → Γ(M) is hyperbolic for f ∈ D and Γ(M) with the norm induced as before by a Riemannian metric on M. As in [3], Γ(M) can be decomposed in a direct sum of f_{∗}-invariant subspaces Γ(M) = Γ^{s} ⊕ Γ^{u}, so that the spectral radius of f_{s} = f_{∗}|Γ^{s} and of f_{u}^{−1} = f_{∗}^{−1}|Γ^{u} are smaller than 1.

For each x ∈ M, define

E_{x}^{s} = {σ(x) | σ ∈ Γ^{s}}
E_{x}^{u} = {σ(x) | σ ∈ Γ^{u}}.

It is not hard to see that E^{s} = ⋃_{x∈M} E_{x}^{s} and E^{u} = ⋃_{x∈M} E_{x}^{u} are continuous subbundles of T M, Df-invariant and T M = E^{s}⊕E^{u}. To see that this sum is direct, we let v ∈ E_{x}^{s} ∩ E_{x}^{u} for some x ∈ M. Since the spectral radius of f_{s} and f_{u}^{−1} are smaller than 1, there is an integer n_{0} so that ∥f_{s}^{n_{0}}∥ < k, ∥f_{u}^{−n_{0}}∥ < k with 0 < k < 1. Define σ^{s} ∈ Γ^{s} and σ^{u} ∈ Γ^{u} such that σ^{s}(f^{−n_{0}}(x)) = Df_{x}^{−n_{0}}v, ∥σ^{s}∥ = |Df_{x}^{−n_{0}}v|, σ^{u}(x) = v, and ∥σ^{u}∥ = |v|. From this we get ∥f_{s}^{n_{0}}(σ^{s})∥ ≤ k |Df_{x}^{−n_{0}}v| and ∥f_{u}^{−n_{0}}(σ^{u})∥ ≤ k |v|. This means that |v| ≤ k |Df_{x}^{−n_{0}}v| and |Df_{x}^{−n_{0}}v| ≤ k |v|, which implies that v = 0 since 0 < k < 1. Thus T M = E^{s} ⊕ E^{u}.

Now set λ = k^{1/n_{0}} < 1,

c = sup_{0≤i<n_{0}}{∥f_{s}∥^{i} λ^{−i} }  and
c′ = sup_{0≤i<n_{0}}{∥f_{u}^{−1}∥^{i} λ^{i} }.

From ∥f_{s}^{n_{0}}∥ < k and ∥f_{u}^{−n_{0}}∥ < k we get

|Df_{x}^{n}v| ≤ c λ^{n} |v|
|Df_{x}^{−n}w| ≤ c′ λ^{n} |w|

for each x ∈ M, v ∈ E_{x}^{s} and w ∈ E_{x}^{u}. This shows that f is Anosov, finishing the proof of the first part of the lemma.

Finally, we prove that if f is Anosov then there is a C^{∞} norm on T M so that we can take c = c′ = 1 in the above inequalities. Following [3], let ρ be such that λ < ρ < 1 and define

|v|_{s} = ∑_{n=0}^{∞} ρ^{−n} |Df_{x}^{n}v|

|w|_{u} = ∑_{n=0}^{∞} ρ^{−n} |Df_{x}^{−n}w|

for v ∈ E_{x}^{s} and w ∈ E_{x}^{u}. For any α ∈ T_{x}M, α can be written as α = v + w, with v ∈ E_{x}^{s} and w ∈ E_{x}^{u}. Define |α|_{1} = |v|_{s} + |w|_{u}. Then | |_{1} is a norm equivalent to the original one and

|Df_{x}v|_{1} ≤ ρ |v|_{1}
|Df_{x}^{−1}w|_{1} ≤ ρ |w|_{1}

for v ∈ E_{x}^{s} and w ∈ E_{x}^{u}. Of course, we can only say that | |_{1} is a C^{0} norm. But now we approximate | |_{1} by a C^{∞} norm so that the above inequalities still hold.

Let E be a Banach space and E_{1}, E_{2} closed subspaces so that E = E_{1} ⊕ E_{2}. Given 0 < τ < 1, we denote by L_{τ} the hyperbolic isomorphisms L of E leaving E_{1}, E_{2} invariant such that ∥L|E_{1}∥ < τ and ∥L^{−1}|E_{2}∥ < τ.

The following lemma, due to Hirsch and Pugh, was proved in [4]. The proof we present here was suggested by Palis.

Lemma 2. Given τ, 0 < τ < 1, there exists ϵ > 0 such that if the isomorphism

T : E → E with respect to the splitting E = E_{1} ⊕ E_{2} has the form (\begin{matrix}A & B \\ C & D \end{matrix}), with

L = (\begin{matrix}A & 0 \\ 0 & D \end{matrix}) ∈ L_{τ} and ∥B∥ < ϵ, ∥C∥ < ϵ, then T is hyperbolic.

Proof. First we notice that there exists ϵ > 0 (which depends only on τ) such that if ∥B∥ < ϵ, ∥C∥ < ϵ, then T is locally conjugate to L. (See [3].) In fact, we get a global uniformly continuous conjugacy h between T and L, i.e., T̃ h = h L, where T̃ = T near the origin. It is easy to see that the local images of E_{1} and E_{2}, h(E_{1}) and h(E_{2}), generate closed linear subspaces Ẽ_{1} and Ẽ_{2}, invariant by T and Ẽ_{1} ∩ Ẽ_{2} = 0. Also, ∥T^{n}|Ẽ_{1}∥ < 1 and ∥T^{−n}|Ẽ_{2}∥ < 1 for some integer n, which imply that the spectral radii of T |Ẽ_{1} and of T^{−1}|Ẽ_{2} are less than one. Notice that Ẽ_{1} and Ẽ_{2} are characterized by the fact that T^{n}v → 0 and T^{−n}w → 0 as n → ∞ for any v ∈ Ẽ_{1} and w ∈ Ẽ_{2}.

Finally, we show that E = Ẽ_{1} ⊕ Ẽ_{2}. To see this, it is enough to show that h(v + w) − h(v) ∈ Ẽ_{2} for small v ∈ Ẽ_{1} and w ∈ Ẽ_{2}. In fact,

∥L^{−n}(v + w) − L^{−n}v∥ = ∥L^{−n}w∥ < λ^{n} ∥w∥.

Therefore,

h(L^{−n}(v + w)) − h(L^{−n}v) = T^{−n}(h(v + w) − h(v))

converges to the origin as n → ∞ for h uniformly continuous. Thus E = Ẽ_{1}⊕Ẽ_{2} and since the spectral radii of T |Ẽ_{1} and of T^{−1}|Ẽ_{2} are less than one, T is hyperbolic.

We can now prove the following theorem.

Theorem 1. The Anosov diffeomorphisms form an open set in Diff(M).

Proof. Let f be an Anosov diffeomorphism. Then f_{∗} is a hyperbolic isomorphism of Γ(M). Thus Γ(M) = Γ_{s} ⊕ Γ_{u}, where Γ_{s} and Γ_{u} are given by Lemma 1. Γ_{s} and Γ_{u} are f_{∗}-invariant and ∥f_{∗}|Γ_{s}∥ < τ, ∥f_{∗}^{−1}|Γ_{u}∥ < τ for some τ such that 0 < τ < 1. It is immediate that given ϵ > 0, there is a neighborhood N(f) ⊂ Diff(M) with

the property that for any g ∈ N(f), g_{∗} = (\begin{matrix}A & B \\ C & D \end{matrix}) with respect to the splitting

Γ(M) = Γ_{s} ⊕ Γ_{u}, where ∥A∥ < τ, ∥D^{−1}∥ < τ, ∥B∥ < ϵ, and ∥C∥ < ϵ. Thus taking ϵ as in Lemma 2, g_{∗} is hyperbolic, and by Lemma 1, g is Anosov.

Remark 1. Notice that the map ρ : Diff(M) → Isom(Γ(M)), defined by ρ(g) = g_{∗}, is not continuous. What we used in the proof above was the continuity of the norm of the operators corresponding to the decomposition of g_{∗} with respect to the splitting Γ(M) = Γ_{s} ⊕ Γ_{u}.

As before, we denote by D the space of diffeomorphisms on M with the C^{1} topology. We denote by H the space of homeomorphisms on M with the C^{0} topology.

Theorem 2. (Anosov) If f is an Anosov diffeomorphism then f is structurally stable. In particular, there exists a neighborhood V of f in D, a neighborhood U of the identity id : M → M in H, and a continuous function h : V → U such that if g ∈ V then h = h(g) is the unique solution in U of the functional equation

h ∘ g = f ∘ h.

Before proving the theorem we need several definitions, constructions, and lemmas.

Definition 2. Let K_{1} and K_{2} be compact metric spaces, U an open subset of a Banach space F_{1}, and V an open subset of a Banach space F_{2}. Suppose that we have f : K_{1} → K_{2} and f̄: K_{1} × U → K_{2} × V continuous such that the following diagram is commutative:

K_{1} × U ⟶^{f̄} K_{2} × V ⟶^{p_{2}} V
π↓  ↓^{p_{1}}
K_{1}  ⟶^{f} K_{2}

where π, p_{1}, and p_{2} are projections. We say that f̄ is vertically of class C^{r} (r ≥ 0) if p_{2} ∘ f has r partial derivatives with respect to the variable in U and the partials are continuous mappings

D_{2}^{k} (p_{2} ∘ f̄) : K_{1} × U → L_{s}^{k}(F_{1}, F_{2})

for k = 0, …, r. Here L_{s}^{k}(F_{1}, F_{2}) are symmetric k-multilinear mappings from F_{1} to F_{2}. In particular, for each fixed x ∈ K_{1}, p_{2} ∘ f̄(x, ·) : U → V is of class C^{r}.

Definition 3. Let π_{1} : E^{1} → M and π_{2} : E^{2} → N be two Riemannian vector bundles of class C^{0} over compact metric spaces M and N. Let f̄: E^{1} → E^{2} be a continuous map that preserves fibers, i.e., there exists a map f : M → N such that f ∘ π_{1} = π_{2} ∘ f̄. We say that f̄ is vertically of class C^{r} or f̄ is of class C^{r} along the fibers, 0 ≤ r ≤ ∞, if the local representatives of f̄ in local vector bundle charts are vertically of class C^{r} (using Definition 2).

Let f : M → N be a continuous function and π : E → M a Riemannian vector bundle. f^{∗}(E) is the subset of M × E of pairs (x, v) such that f(x) = π(v). Let π(f) be the projection on the first factor of M × E. π(f) : f^{∗}(E) → M is a vector bundle. There is a Riemannian metric induced on f^{∗}(E) by the inclusion in M ×E.

Let π_{i} : E^{i} → M i = 1, 2 be two Riemannian vector bundles. Let U ⊂ E^{1} be an open subset such that π_{1}|U : U → M is a surjection. Let Γ(U) ⊂ Γ(E^{1}) be the open subset of sections with images in U. We assume U is connected enough so the Γ(U) is nonempty.

Let f̄: U → E^{2} be a continuous function that preserves fibers covering f : M → M. We denote by

Ωf̄: Γ(U) → Γ(f^{∗}E^{2})

the map induced by composition on the left by f̄,

Ωf̄: γ ↦ f̄∘ γ.

Lemma 3. If f̄ is vertically of class C^{r}, 0 ≤ r ≤ ∞, then Ωf̄ : Γ(U) → Γ(f^{∗}E^{2}) is of class C^{r}.

Proof. For r = 0, Ωf̄ corresponds to the composition of continuous functions on a compact set. It is a standard result that Ωf̄ is continuous.

Let γ ∈ Γ(U). Let σ ∈ Γ(E^{1}) be small enough in norm so that γ + σ ∈ Γ(U). For each x ∈ M, we apply Taylor’s Theorem to the function f̄_{x} : E_{x}^{1} → E_{f(x)}^{2} at the point γ(x). We obtain

(1)  Ωf̄(γ + σ)(x) = Ωf̄(γ)(x) + ∑_{k=1}^{r} \frac{1}{k!} D^{k}f̄_{x}(γ(x))(σ(x))^{k}
+ R(γ(x), σ(x))(σ(x))^{r}.

Here (σ(x))^{k} = (σ(x), …, σ(x)), and R(x, y) ∈ L_{x}^{r}(E_{x}^{1}, E_{f(x)}^{2}). L_{x}^{r}(E_{x}^{1}, E_{f(x)}^{2}) are symmetric r-multilinear functions from E_{x}^{1} to E_{f(x)}^{2}. Writing formula (1) without evaluation at x we obtain

Ωf̄(γ + σ) = Ωf̄(γ) + ∑_{k=1}^{r} \frac{1}{k!} D^{k}f̄_{x}(γ)(σ)^{k} + R(γ, σ)(σ)^{r} (2)

where we are only taking the derivative of f̄ along the fiber and

R(γ, σ) ∈ L_{x}^{r}(Γ(E^{1}), Γ(f^{∗}E^{2})).

We leave it to the reader to check that R( , ) is continuous and that R(γ, 0) = 0. By the converse to Taylor’s Theorem, [7,2.1], it follows that Ωf̄ is of class C^{r} and that

D^{k}Ωf̄(γ)(σ_{1}, …, σ_{k}) = D^{k}f̄_{x}(γ)(σ_{1}, …, σ_{k})

for σ_{1}, …, σ_{k} ∈ Γ(E^{1}). Then D^{k}Ωf̄: Γ(U) → L_{x}^{k}(Γ(E^{1}), Γ(f^{∗}E^{2})).

The following lemma is obvious.

Lemma 4. Let π : E → N be a Riemannian vector bundle of class C^{0}. Let M and N be compact metric spaces. Let f : M → N be a continuous function. Let A_{f} : Γ(E) → Γ(f^{∗}E) be defined by γ ↦ γ ∘ f. Then for fixed f, A_{f} is a continuous linear function in γ and hence C^{∞}.

Let C be the space of continuous functions from M to M. We give M a C^{∞} Riemannian metric. The topology of C is given by the metric d̄:

d̄(f, g) = sup{d(f(x), g(x)) : x ∈ M}

where d is the distance between points of M induced by the Riemannian structure on M. In Theorem 2 we have

H = {h ∈ C : h is a homeomorphism }.

We take this opportunity to give the construction that makes C into a Banach manifold. To prove Theorem 2 we only use the local coordinate chart at the identity given by the following lemma.

Lemma 5. C admits the structure of a C^{∞} manifold modeled on a Banach space.

Proof. Let U be an open cover of M by convex neighborhoods. (Convex with respect to the Riemannian structure.) Let δ > 0 be a Lebesgue number associated to the open cover, i.e., given a ball B of radius less than or equal to δ there exists a U ∈ U such that B ⊂ U.

Let f ∈ C. Let Γ(f) denote the Banach space of continuous sections of f^{∗}(T M), Γ(f^{∗}(T M)). Let U(f) = U_{δ}(f) be the open ball in Γ(f) of radius δ centered at the zero section. Let B(f) = B_{δ}(f) be the open ball in C centered at f or radius δ. We parameterize B(f) by U(f) as follows. Let ϕ_{f} : U(f) → B(f) be given by

(ϕ_{f}(σ))(x) = exp_{f(x)}(σ(x))

for σ ∈ U(f). We have that

d̄(ϕ_{f}(σ_{1}), ϕ_{f}(σ_{2})) = sup_{x∈M} d(exp_{f(x)} σ_{1}(x), exp_{f(x)} σ_{2}(x) )

≤ sup_{x∈M}{|σ_{1}(x) − σ_{2}(x)|}

≤ ∥σ_{1} − σ_{2}∥.

Therefore ϕ_{f} is continuous. On the other hand, ϕ_{f} has an inverse ϕ_{f}^{−1} : B(f) → U(f) defined by

ϕ_{f}^{−1}(g)(x) = (x, (exp_{f(x)})^{−1}(g(x))).

Because the neighborhoods in U are convex, the expression (exp_{f(x)})^{−1}(g(x)) is well defined. By the uniform continuity of the exponential on M, it follow there is a constant e such that

∥ϕ_{f}^{−1}(g_{1}) − ϕ_{f}^{−1}(g_{2})∥ = sup_{x∈M}{|(exp_{f(x)})^{−1}(g_{1}(x)) − (exp_{f(x)})^{−1}(g_{2}(x))|}

≤ e sup_{x∈M}{d(g_{1}(x), g_{2}(x))}

≤ e d̄(g_{1}, g_{2}).

Thus ϕ_{f}^{−1} is continuous.

We have defined an atlas for C, whose local charts are modeled on the Banach spaces f^{∗}(T M) where f ∈ C. To complete the proof, it suffices to show the changes of coordinates are C^{∞}.

Let ϕ_{f} : U(f) → B(f) and ϕ_{g} : U(g) → B(g) be two charts. We need to prove that

ϕ_{g}^{−1}ϕ_{f} : U(f) → U(g)

is a diffeomorphism of class C^{∞} on it domain of definition.

Let V (f) = {v ∈ f^{∗}(T M) : |v| < δ} and V (g) = {v ∈ g^{∗}(T M) : |v| < δ}. Then U(f) = Γ(V (f)), U(g) = Γ(V (g)). Define the homeomorphism G : V (f) → V (g) by

G(x, v) = (x, (exp_{g(x)})^{−1} ∘ exp_{f(x)} v).

G is well defined by the convexity of the neighborhoods. We have that ϕ_{g}^{−1}ϕ_{f}(v) = G ∘ v = Ω_{G}(v). G preserves fibers. G is vertically of class C^{∞} since along a fixed fiber

G(x, ·) = (x, (exp_{g(x)})^{−1} ∘ exp_{f(x)} ·).

By Lemma 3, ϕ_{g}^{−1}ϕ_{f} is of class C^{∞}. In the same way, (Ω_{G})^{−1} = Ω_{G^{−1}} = ϕ_{f}^{−1}ϕ_{g} is of class C^{∞}.

Remark 2. The tangent space of C at f, T_{f}C, can be identified with Γ(f^{∗}T M). In particular, T_{id}C = Γ(T M) = Γ(M).

Remark 3. Let Λ ⊂ M be a compact subset. Let B(Λ, M) be the topological space of bounded functions from Λ to M. Then we can give B(Λ, M) the structure of a manifold of class C^{∞} modeled on bounded sections of T M|Λ.

Proof of Theorem 2: We want to look at the map D × D × C → C given by (g_{1}, g_{2}, h) ↦ g_{1} ∘ h ∘ g_{2}^{−1}. If g_{1} ∘ h ∘ g_{2}^{−1} = h then g_{1} ∘ h = h ∘ g_{2}. Thus fixed points of the map give a semiconjugacy between g_{1} and g_{2}. (To be a conjugacy, we need h to be a homeomorphism.) Also g ∘ id ∘ g^{−1} = id. We want to prove the stability of this fixed point.

We take local coordinates in C near id, ϕ : U ⊂ Γ(M) → C with ϕ(σ)(x) = exp_{x} σ(x). For neighborhoods V of f in D and U of 0 in Γ(M),

A : V × V × U → Γ(M)

is well defined by

A(g_{1}, g_{2}, h) = ϕ^{−1}(g_{1} ∘ ϕ(h) ∘ g_{2}^{−1}),  or
A(g_{1}, g_{2}, h)(x) = exp_{x}^{−1}(g_{1} ∘ exp_{g_{2}^{−1}(x)} ∘(h ∘ g_{2}^{−1}(x))).

For g_{1}, g_{2} ∈ V, define G(g_{2}, g_{2}) : T M → T M by

G(g_{1}, g_{2})(v_{x}) = exp_{g_{2}(x)}^{−1}(g_{2} ∘ exp_{x} v_{x}).

Then

(Ω_{G(g_{1},g_{2})}A′_{g_{2}^{−1}}h)(x) = G(g_{1}, g_{2}) ∘ h ∘ g_{2}^{−1}(x)
= exp_{x}^{−1}(g_{1} ∘ exp_{g_{2}^{−1}(x)}(h ∘ g_{2}^{−1}(x)))
= A(g_{1}, g_{2}, h)(x).

Here A′_{g_{2}^{−1}} is the map given by Lemma 4.

Lemma 6. A has a partial derivative with respect to the third variable. When g_{1} = g_{2} = g we have

D_{3}A(g, g, 0)k = Dg(g^{−1})k ∘ g^{−1} = g_{∗}k.

D_{3}A(g_{1}, g_{2}, h) is continuous in the first and third variables, uniformly in the second variable, i.e., given (g_{1}, h) and ϵ > 0 there exists neighborhoods V′ of g_{1} and U′ of h such that for f_{11}, f_{12} ∈ V′, f_{2} ∈ V, and h_{1}, h_{2} ∈ U′

∥D_{3}A(f_{11}, f_{2}, h_{1}) − D_{3}A(f_{12}, f_{2}, h_{2})∥ < ϵ.

In particular, given ϵ > 0, there exist neighborhoods V′ of f and U′ of 0 in Γ(M) such that the Lipschitz constant

L(A(f_{11}, f_{2}, ·)|U′ − D_{3}A(f_{11}, f_{2}, 0)|U′) < ϵ

for g_{1}, g_{2} ∈ V′.

Proof. By Lemmas 3 and 4, the partial derivative of A with respect to the third variable exists. Since D(exp_{x})(0_{x}) = id : T_{x}M → T_{x}M, it follows that

D_{3}A(g, g, 0)k = Dg(g^{−1})k ∘ g^{−1}.

Let G_{1} = G(f_{11}, f_{2}) and G_{2} = G(f_{12}, f_{2}). Then

∥D_{3}A(f_{11}, f_{2}, h_{1}) − D_{3}A(f_{12}, f_{2}, h_{2})∥

= sup_{k∈Γ(M) with ∥k∥=1}{∥DG_{1}(h_{1} ∘ f_{2}^{−1})k ∘ f_{2}^{−1} − |DG_{2}(h_{2} ∘ f_{2}^{−1})k ∘ f_{2}^{−1}∥}

≤ sup_{x∈M}{∥DG_{1}(h_{1} ∘ f_{2}^{−1}(x)) − DG_{2}(h_{2} ∘ f_{2}^{−1}(x))∥}.

Using the uniformity in the exponential, and letting f_{11}, f_{12} → g and h_{1}, h_{2} → h, we get that

∥DG_{1}(h_{1} ∘ f_{2}^{−1}(x)) − DG_{2}(h_{2} ∘ f_{2}^{−1}(x))∥ → 0

uniformly in f_{2} and x. Remember that G_{i}(f_{2}^{−1}(x), v) = exp_{x}^{−1}(f_{1i} ∘ exp_{f_{2}^{−1}(x)} v). This proves the desired continuity of D_{3}A.

The Lipschitz constant follows from the above results using the Mean Value Theorem. See [5, 8.6.2] for example.

Remark 4. (Caution) D_{3}A(g_{1}, g_{2}, h) is not continuous in g_{2}. To see this consider the case of a map defined in the plane so we can disregard the exponentials. Let h = 0 and take g′_{2} arbitrarily near g_{2} in the C^{1} topology but with (g_{2})^{−1}(x_{0}) ≠ (g′_{2})^{−1}(x_{0}). For each such g′_{2} there exists a k ∈ Γ(M) such that ∥k∥_{0} = 1 and |k ∘ (g_{2})^{−1}(x_{0}) − k ∘ (g′_{2})^{−1}(x_{0})| = 1. Then

∥D_{3}A(g_{1}, g_{2}, 0) − D_{3}A(g_{1}, g′_{2}, 0)∥
≥ |D(g_{1})_{(g_{2})^{−1}(x_{0})} k ∘ g_{2}^{−1}(x_{0}) − D(g_{1})_{(g′_{2})^{−1}(x_{0})} k ∘ (g′_{2})^{−1}(x_{0})|.

This stays bounded away from zero as g′_{2} goes to g_{2}.

However the following lemma gives a partial result in this direction.

Lemma 7. Let T : D × D × Γ(M) → Γ(M) be defined by

T (g_{1}, g_{2}, h) = D_{3}A(g_{1}, g_{2}, 0)h.

Then T is continuous in all variables.

In fact D_{3}A(g_{1}, g_{2}, 0)h is a continuous function of g_{1}, g_{2}, and h. The point is that it is not necessary to take the supremum over all h of unit length but just those near h_{0}. The proof is left to the reader.

The following lemma is what we need to prove the stability of the fixed point of A. It is based on the last paragraph of page 144 in [4]. If D_{3}A : D × D × Γ(M) → L(Γ(M), Γ(M)) were continuous, then we could use a standard fixed point theorem or the Implicit Function Theorem.

Lemma 8. Let P be a topological space. Let F_{1} ⊕ F_{2} be a Banach space with the norm equal to the maximum of the norms on the two factors. Let T : P ×F_{1} ⊕F_{2} → F_{1}⊕F_{2} be a function (not necessarily continuous) such that for each x ∈ P, T (x, ·) : F_{1} ⊕F_{2} → F_{1} ⊕F_{2} is a continuous linear isomorphism. Assume ∥T_{1}(x, ·, 0)^{−1}∥ ≤ τ, ∥T_{2}(x, 0, ·)∥ ≤ τ, ∥T_{1}(x, 0, ·)∥ ≤ μ, and ∥T_{2}(x, , ·, 0)∥ ≤ μ where T_{i}(x, ·, 0) : F_{1} → F_{i}. We also have ϵ > 0 such that τ + μ + ϵ < 1. Let U_{1} ⊕ U_{2} ⊂ F_{1} ⊕ F_{2} be a ball about the origin of radius R. Assume f : P × U_{1} ⊕ U_{2} → F_{1} ⊕ F_{2} is a function such that for all x ∈ P, (i) the Lipschitz constant L(f(x, ·) − T (x, ·)|U_{1} ⊕ U_{2}) < ϵ and (ii) |f(x, 0, 0)| ≤ (1 − τ − μ − ϵ)R. Then there exists a function u : P → U_{1} ⊕ U_{2} such that f(x, u(x)) = u(x) and |u(x)| ≤ |f(x, 0, 0)|/(1 − τ − μ − ϵ). Further, if f and T are continuous then so is u.

Proof. Define g : P × U_{1} ⊕ U_{2} → F_{1} ⊕ F_{2} by

g(x, y_{1}, y_{2}) = (T_{1}(x, ·, 0)^{−1}(y_{1} + T_{1}(x, y_{1}, 0) − f_{1}(x, y_{1}, y_{2})), f_{2}(x, y_{1}, y_{2})).

Note the fixed points of g(x, ·) are the same as those of f(x, ·).

First we show g(x, ·) is a contraction with contraction constant τ + μ + ϵ. Let y = (y_{1}, y_{2}) and y′ = (y′_{1}, y′_{2}).

|g_{1}(x, y) − g_{1}(x, y′)|
≤ τ(|y_{1} − y′_{1}| + L(T_{1} − f_{1})|y − y′| + |T_{1}(x, 0, y_{2} − y′_{2})|)
≤ τ(1 + ϵ + μ)|y − y′|
≤ (τ + ϵ + μ)|y − y′|
|g_{2}(x, y) − g_{2}x, y′)|
≤ |T_{2}(x, y − y′)| + L(f_{2}(x, ·) − T_{2}(x, ·))|y − y′|
≤ (τ + ϵ + μ)|y − y′|.

By [6, 10.1.1], [4, 1.1], or [5], g has a fixed point, u(x), for each x ∈ P with

|u(x)| ≤ |g(x, 0)|/(1 − τ − μ − ϵ) ≤ |f(x, 0)|/(1 − τ − μ − ϵ).

Now assume f and T are continuous, so g is continuous. For x_{0} ∈ P, by [6,10.1.1], [4, 1.1], or [5],

|u(x_{0}) − u(x)| ≤ |g(x, u(x_{0})) − u(x_{0})|/(1 − τ − μ − ϵ).

This shows that u is continuous.

T (f, f, ·) = D_{3}A(f, f, 0) = f_{∗} : Γ(M) → Γ(M) is hyperbolic and has a splitting Γ(M) = Γ(E^{u}) ⊕ Γ(E^{s}). We can approximate E^{u} and E^{s} by smooth bundles F^{u} and F^{s}. Let F_{1} = Γ(F^{u}) and F_{2} = Γ(F^{s}). For a small neighborhood V of f we can insure that T (g_{1}, g_{2}, h) = D_{3}A(g_{2}, g_{2}, 0)h satisfies Lemma 8. This follows from the continuity of the norms of the coordinate functions and Lemmas 6 and 7. On F_{1} ⊕ F_{2} we take the norm |(y_{1}, y_{2}) = max{|y_{1}|, |y_{2}|}.

Thus there exists neighborhoods V_{1} of f in D and U′_{1} of 0 in Γ(M) such that for g_{1}, g_{2} ∈ V_{1} there exists a unique k = u′(g_{1}, g_{2}) ∈ U′_{1} such that A(g_{1}, g_{2}, k) = k. Let U_{1} = ϕ(U′_{1}) = exp(U′_{1}) ⊂ C, u = ϕ ∘ u′, and h = u(g_{1}, g_{2}). Then g_{1} ∘ h = h ∘ g_{2}. Also u is a continuous function of g_{1} and g_{2}. Let U_{2} be a smaller neighborhood of id in C such that for all h_{1}, h_{2} ∈ U_{2}, h_{1} ∘ h_{2} ∈ U_{1}. This exists since composition is continuous. By continuity of u or by continuity of A and the estimate

|u′(g_{1}, g_{2}) − id| ≤ |A(g_{1}, g_{2}, id) − id|/(1 − τ − μ − ϵ),

there exists a smaller neighborhood V_{2} of f in D such that for g_{1}, g_{2} ∈ V_{2}, u(g_{1}, g_{2}) ∈ U_{2}. If g ∈ V_{2} let h = u(g, f) and h′ = u(f, g). Then g ∘ h = h ∘ f and f ∘ h′ = h′ ∘ g. Thus h ∘ h′ ∘ g = h ∘ f ∘ h′ = g ∘ h ∘ h′. Also h′ ∘ h ∘ f = f ∘ h′ ∘ h. h′ ∘ h, h ∘ h′ ∈ V_{1} so by uniqueness we get h′ ∘ h = h ∘ h′ = id. Thus h is a homeomorphism.

Remark 5. The proof given above applies directly to prove the local stability of basic sets. See [4, Theorem 7.3]

Remark 6. Using the Implicit Function Theorem instead of Lemma 8, we can solve for an h such that g ∘ h = h ∘ f. We do this by always keeping f fixed. h has to be onto by a degree argument. Using the stable manifold theorem of [4] or [5], we can show f is expansive, i.e., there exists an r > 0 such that for any two points x, y ∈ M there exists an integer n such that the distance from f^{n}(x) to f^{n}(y) is greater than r. From this property, it can be shown that h has to be one to one.

Remark 7. The proof indicated in the Remark 6 does not apply in the general setting of a basic set of Remark 5.

References

[1] J. Moser, On a theorem of Anosov, Differential Equations 5 (1969), pp. 411–440.

[2] S. Smale, Differential Dynamical Systems, Bull. Amer. Math. Soc. 73 (1967), pp. 747–817.

[3] J. Palis, On the structure of hyperbolic points in Banach spaces, Anais Acad. Bras. Ciêcias 40 (1968).

[4] M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Proc. of Symposium in Pure Math. 14 (1970), Amer. Math. Soc., pp. 133–165.

[5] G. Tavares dos Santos, Variedade estável de elemento crítico, in Seminario de Sistemas Dinamicos, ed. J. Palis, Monografias de Matematica 4 (1971), Instituto de Matematica Pura e Applicada, Rio de Janeiro, Brazil, Chapter 3.

[6] J. Dieudonné, Foundations of Modern Analysis, Academic Press, 1960.

[7] R. Abraham and J. Robbin, Transversal Mappings and Flows, Benjamin, 1967.

Department of Mathematics, Northwestern University, Evanston IL 60208

   E-mail address: clark@math.nwu.edu, alberto@matcuer.unam.mx