PARTIAL DIFFERENTIABILITY OF INVARIANT SPLITTINGS

CHARLES PUGH, MICHAEL SHUB, AND AMIE WILKINSON

Abstract. A key feature of a general nonlinear partially hyperbolic dynamical system is the absence of differentiability of its invariant splitting. In this paper, we show that often partial derivatives of the splitting exist and the splitting depends smoothly on the dynamical system itself.

Dedicated to David Ruelle on his 65th birthday.

October 15, 2002

Shub’s work was partly funded by NSF Grant #DMS-9988809. Wilkinson’s work was partly funded by NSF Grant #DMS-0100314.

1. Introduction

One of the major technical barriers to the understanding of Anosov diffeomorphisms is the fact that unstable bundles are not in general differentiable along stable bundles. This situation persists for partially hyperbolic diffeomorphisms, where there are also center bundles present. Under mild bunching conditions, however, the unstable bundles are differentiable along the center bundles, see Theorem A below. This fact has already been observed and exploited in several special situations. First, for Anosov diffeomorphisms themselves, the unstable bundles are differentiable with respect the diffeomorphism, as long as partial derivatives are taken in certain dynamically defined directions given by conjugating maps [7]. Consequently entropy and SRB states also vary differentiably with parameters for Anosov diffeomorphisms and flows [5, 6, 11]. Differentiability of the unstable bundle along the center was a crucial ingredient in proving stable ergodicity for many partially hyperbolic diffeomorphisms [3, 14, 9, 10]. It was also an ingredient in the construction of nonuniformly hyperbolic diffeomorphisms with pathological foliations [13, 2, 12]. While we have similar applications in mind for Theorem A, we will content ourselves here with some general theorems. We describe the main results of this paper in the following section; the proofs occupy the remaining sections.

2. Statements of Results

Suppose that f : M → M is a partially hyperbolic diffeomorphism. The tangent bundle splits as E^{u}⊕E^{c}⊕E^{s}. In general the splitting is continuous but not C^{1}. Here we show that under some mild pointwise bunching conditions, E^{u} is continuously differentiable in the E^{c} direction, i.e.,

\frac{∂E^{u}(p)}{∂E^{c}}

exists and is a continuous function of p ∈ M. More precisely, we prove:

Theorem A. Suppose f : M → M is C^{2} and partially hyperbolic with splitting T M = E^{u} ⊕ E^{c} ⊕ E^{s}. Then, under the pointwise bunching condition

\begin{aligned}& sup_{p} \frac{∥T_{p}^{c}f∥}{m(T_{p}^{u}f)m(T_{p}^{c}f)} < 1, \\ (1) \end{aligned}

E^{u} is continuously differentiable with respect to E^{c}.

Theorem A is a corollary of a more general result about partial differentiability of dominated splittings – see Theorem 5.1 in Section 5.

Next we show, under the same bunching hypothesis, that in a family t ↦ f_{t} of partially hyperbolic diffeomorphisms, the unstable bundle E^{u}(f_{t}) is always continuously differentiable along “dynamically defined” curves in M. Roughly speaking, a dynamically defined curve t ↦ φ_{p}(t) through p ∈ M is a C^{1} curve along which the hyperbolic component of the dynamics of f_{t} varies as little as possible. For example, if f_{t} is Anosov and h_{t} : M → M is the conjugacy from f_{t} to f_{0}, so that h_{t}f_{0} = f_{t}h_{t}, then t ↦ h_{t}(p) is a dynamically defined curve. In the language of Section 7, a dynamically defined curve is the M-component of an integral curve of the center distribution 𝔼^{c} of the evaluation map Eval : M × I → M × I:

(p, t) ↦ (f_{t}p, t).

We prove that dynamically defined curves always exist, and unstable bundles, subject to a bunching condition, vary in a C^{1} way along them.

Theorem B. Let {f_{t} : M → M}_{t∈(−ϵ,ϵ)} be a C^{2} family of C^{2}, partially hyperbolic diffeomorphisms having, for each t ∈ (−ϵ, ϵ), a T f_{t}-invariant splitting:

T M = E^{u}(f_{t}) ⊕ E^{c}(f_{t}) ⊕ E^{s}(f_{t}).

Then there exists ϵ_{0} > 0 so that, for every p ∈ M there exists a C^{1} path

φ_{p} : (−ϵ_{0}, ϵ_{0}) → M

with φ_{p}(0) = p, and with the following property.

If the pointwise bunching condition

\begin{aligned}& sup_{p} \frac{∥T_{p}^{c}f_{0}∥}{m(T_{p}^{u}f_{0})m(T_{p}^{c}f_{0})} < 1 \\ (2) \end{aligned}

holds, then t ↦ E_{φ_{p}(t)}^{u}(f_{t}) is C^{1}.

Theorem B follows from a more general result, Theorem 7.4, which states that any invariant, dominated subbundle of a partially hyperbolic diffeomorphism is continuously differentiable along dynamically defined paths, subject to a bunching condition on the bundle. In addition, Theorem 7.4 produces, for any v ∈ E_{p}^{c}(f_{0}), a dynamically defined path φ_{p,v} so that φ̇_{p,v}(0) ∈ v + E^{u} ⊕ E^{s}(f_{0}).

The machinery behind the proofs of Theorems A and B is Theorem 3.1, a refinement of the C^{1} Section Theorem from [4] that handles partial derivatives of a section.

In Section 8, we address the question of when t ↦ E_{p}^{u}(f_{t}) is differentiable at t = 0. The issue here is of a slightly different nature than that in Theorems A and B. While t ↦ E^{u}(f_{t}) is always continuously differentiable along dynamically defined paths, the requirement that the constant path t ↦ p be dynamically defined for all p is a stringent one, satisfied only for very special families.

If, instead of requiring that t ↦ E_{p}^{u}(f_{t}) be C^{1} in a given family, we just ask that it be differentiable at t = 0 but for all families through f_{0}, then the actual dynamics of f_{0} becomes irrelevant. It is easy to see that p ↦ E_{p}^{u}(f_{0}) must be C^{1} for this property to hold. What is interesting is that nonsmoothness of p ↦ E_{p}^{u}(f_{0}) is the only obstruction to the differentiability of t ↦ E_{p}^{u}(f_{t}) at t = 0 in every family. Building on Theorem B, one can show:

Theorem C. Let {f_{t} : M → M}_{t∈(−ϵ,ϵ)} be a C^{1} family of C^{2}, partially hyperbolic diffeomorphisms having, for each t ∈ (−ϵ, ϵ), a T f_{t}-invariant splitting:

T M = E^{u}(f_{t}) ⊕ E^{c}(f_{t}) ⊕ E^{s}(f_{t}).

Assume that the pointwise bunching condition

\begin{aligned}& sup_{p} \frac{∥T_{p}^{c}f_{0}∥}{m(T_{p}^{u}f_{0})m(T_{p}^{c}f_{0})} < 1 \\ (3) \end{aligned}

holds. Assume also that E^{u}(f_{0}) is a C^{2−ϵ} subbundle of T M, for all ϵ > 0.

Then for all p ∈ M,

t ↦ E_{p}^{u}(f_{t})

is differentiable at t = 0.

If φ_{p} is any dynamically defined path through p ∈ M given by Theorem B, then:

E_{p}^{u}(f_{t}) − E_{p}^{u}(f_{0}) = (\frac{d}{dt}E_{φ_{p}t}^{u}(f_{t}) |_{t=0} − D_{p}E^{u}(f_{0})(\frac{d}{dt}φ_{p}t |_{t=0})) t + O(t^{1+η}),

for some η > 0.

Subsequent to proving Theorem C, we learned of a more general result, due to Dolgopyat:

Theorem D (Dolgopyat, [2], Theorem 3). Let {f_{t} : M → M}_{t∈(−ϵ,ϵ)} be a C^{1} family of C^{2} diffeomorphisms having, for each t ∈ (−ϵ, ϵ), a T f_{t}-invariant dominated splitting:

T M = R(f_{t}) ⊕ S(f_{t}) ⊕ T (f_{t}).

Suppose that p ↦ S_{p}(f_{0}) is C^{1}. Then, for every p ∈ M, t ↦ S_{p}(f_{t}) is differentiable at t = 0.

Dominated splittings are defined in Section 5. In particular, Theorem D applies when S is E^{u}, E^{c}, E^{s}, E^{cu}, or E^{cs}. We present an exposition of Dolgopyat’s proof of Theorem D in Section 8.

3. Partial Derivatives of an Invariant Section

Let

V \underrightarrow{F} V
π↓  ↓π
M \underrightarrow{f} M

be a C^{1} fiber preserving map, where V is a smooth, finite dimensional fiber bundle over the compact manifold M, and f is a diffeomorphism. In addition assume that there is a section σ : M → V, invariant under F in the sense that

F (σ(p)) = σ(f(p))

for all p ∈ M.

In general there is no reason that σ is smooth, or even continuous. For example, if F is the identity map, every section of V is F -invariant. In [4], we showed that if V is a Banach bundle and F is a fiber contraction then σ is unique and continuous, and furthermore, if the fiber contraction dominates the base contraction sufficiently then the σ is of class C^{r}.

Since F preserves fibers, T F preserves the “vertical” subbundle, Vert ⊂ T V whose fiber at v ∈ V is kernel T_{v}π. We write T_{v}^{Vert}F for the restriction of T_{v}F to Vert_{v},

T_{v}^{Vert}F : Vert_{v} → Vert_{F v} .

We assume that T V carries a Finsler structure and that k_{p} = ∥T_{σp}^{Vert}F ∥ has

sup_{p∈M} k_{p} < 1,

which means that F is a fiber contraction in the neighborhood of σM.

Theorem 3.1. Suppose that E ⊂ T M is a continuous T f-invariant subbundle such that

sup_{p∈M} k_{p}∥(T_{p}^{E}f)^{−1}∥ < 1

where T^{E}f is the restriction of T f to E. Then σ is continuously differentiable in the E-direction in the sense that there is a continuous map H : E → T V such that

(a) For each p ∈ M, H : E_{p} → T_{σp}V is linear.

(b) T π ∘ H = Id : E → E.

(c) If γ is a C^{1} arc in M that is everywhere tangent to E then

(σ ∘ γ)′(t) = H(γ′(t)).

In particular, if E is integrable then the restriction of σ to each E-leaf is C^{1}.

We refer to H as the partial derivative of σ in the E-direction

H = \frac{∂σ}{∂E} .

Remark. If, in addition, there exist C^{r} submanifolds everywhere tangent to E, for some r ∈ (0, ∞), then C^{r} smoothness of σ along E (i.e., along these manifolds) can be assured by assuming that

sup_{p} k_{p}∥(T_{p}^{E}f)^{−1}∥^{r} < 1.

Remark. When E is integrable, the proof of Theorem 3.1 is a fairly simple application of the Invariant Section Theorem of [4]. It is the non-integrable case that requires some new ideas.

Remark. There is a uniformity about ∂σ/∂E. (In the integrable case, this uniformity is automatic.) Fix p ∈ M and extend each w ∈ E_{p} with |w| ≤ 1 to a continuous vector field X_{w} everywhere subordinate to E, and do so in a way that depends continuously on w. Let γ_{w} be an integral curve of X_{w} through p. Since E is only continuous, the integral curve γ_{w} need not be uniquely determined by X_{w}. Nevertheless, for all p in any fixed C^{1} chart, as t → 0 we have

\frac{σ ∘ γ_{w}(t) − σp}{t} → H(w)

uniformly.

Remark. Since M is finite dimensional, Peano’s Existence Theorem implies that there exist C^{1} arcs everywhere tangent to a continuous plane field, and thus the hypothesis of assertion (c) in Theorem 3.1 is satisfied. In the infinite dimensional case, however, Peano’s Theorem fails and (c) could become vacuous.

Proof of Theorem 3.1. We proceed by the graph transform techniques in [4]. Choose a continuous subbundle Hor ⊂ T V, complementary to Vert,

Hor ⊕ Vert = T V.

For example, we could introduce a Riemann structure on T V and take Hor_{v} as the orthogonal complement to Vert_{v}. Note that T π sends each subspace Hor_{v} isomorphically onto T_{p}M, p = πv. With respect to the horizontal / vertical splitting we write

T_{v}F = [\begin{matrix}A_{v} & 0 \\ C_{v} & K_{v} \end{matrix}] = [\begin{matrix}A_{v} : Hor_{v} → Hor_{F v} & 0 \\ C_{v} : Hor_{v} → Vert_{F v} & K_{v} : Vert_{v} → Vert_{F v} \end{matrix}] .

Let \overline{E} ⊂ Hor be the lift of E. That is, T π sends the plane \overline{E}_{v} isomorphically to E_{p}, p = πv. Since E is T f-invariant and F covers f, \overline{E} is A-invariant in the sense that

\overline{E}_{v} \underrightarrow{A_{v}} \overline{E}_{F v}
T π↓  ↓T π
E_{p} \underrightarrow{T f} E_{fp}

commutes.

Let L be the bundle over M whose fiber at p is

L_{p} = L(\overline{E}_{σp}, Vert_{σp}).

An element in L_{p} is a linear transformation P : \overline{E}_{σp} → Vert_{σp}. Let LF be the graph transform on L that sends P ∈ L_{p} to

P′ = (C_{σp} + K_{σp}P)(A_{σp}|_{\overline{E}_{σp}})^{−1} ∈ L_{σp}.

Then T F sends the graph of P to the graph of P′ and LF is an affine fiber contraction

\begin{matrix}L & \underrightarrow{LF} & L \\ π↓ & & ↓π \\ M & \underrightarrow{f} & M. \end{matrix}.

By [4], L has a unique LF -invariant section Λ : M → L, and Λ is continuous. Define H_{p} : E_{p} → T_{σp}V by commutativity of

\overline{E}_{σp} \underrightarrow{Id_{p} ⊕Λ_{p}} \overline{E}_{σp} ⊕ Vert_{σp}
T π↓  ↓Inclusion
E_{p}  \underrightarrow{H_{p}}  T_{σp}V

where Id_{p} is the identity map \overline{E}_{σp} → \overline{E}_{σp}. Then H : E → T V is the unique bundle map such that HE is a T F -invariant subbundle of T_{σM}V,

HE \underrightarrow{T F} HE
T π↓  ↓T π
E \underrightarrow{T f} E

commutes, and T π ∘ H = Id_{E}. We claim that H is the partial derivative of σ in the E-direction.

Let γ : (a, b) → M be a C^{1} arc such that γ is everywhere tangent to E. To complete the proof of the theorem, we must show that

(σ ∘ γ)′(t) = H(γ′(t)).

For n ∈ ℤ, set γ_{n} = f^{n} ∘ γ and

Γ = ⨆_{n∈ℤ} γ_{n}.

This means that we consider the disjoint union of the arcs γ_{n}, so if two of them cross in M, we ignore the crossing in Γ. The one dimensional manifold Γ is noncompact; it has countably many components γ_{n}. In the same way, we discretize V as

V_{Γ} = ⨆_{n} V |_{γ_{n}}.

We equip V_{Γ} and T Γ with the Finslers they inherit from V and M. Then F_{Γ} = F |_{V_{Γ}} is a fiber contraction

V_{Γ} \underrightarrow{F_{Γ}} V_{Γ}
π↓  ↓π
Γ \underrightarrow{f} Γ,

and the fiber contraction dominates the base contraction since

sup_{p} k_{p}∥T_{p}^{E}f∥ < 1

and T Γ ⊂ E. Furthermore, F_{Γ} is uniformly C^{1} bounded since M is compact. The Invariant Section Theorem of [4] then implies that V_{Γ} has a unique F_{Γ}-invariant section σ_{Γ}, and σ_{Γ} is of class C^{1}. Furthermore the tangent bundle of σ_{Γ}(Γ) is the unique nowhere vertical T F_{Γ}-invariant line field in T V_{Γ}.

The restriction of σ to Γ = ⨆_{n} γ_{n} is F_{Γ}-invariant, so by uniqueness

σ_{Γ} = ⨆_{n} σ|_{γ_{n}}.

We claim that

H(T Γ) = T (σ_{Γ}Γ).

Again the reason is uniqueness. We know that T (σ_{Γ}Γ) is the unique T F_{Γ}-invariant, nowhere vertical line field defined over σ_{Γ}Γ. But commutativity of

HE \underrightarrow{T F_{Γ}} HE
H↑  ↑H
E \underrightarrow{T f} E
Inclusion↑  ↑Inclusion
T Γ \underrightarrow{T f} T Γ

implies that H(T Γ) is a second such line field. By uniqueness they are equal.

To complete the proof, we show that the line field equality implies the vector equality

\frac{d}{dt} σ ∘ γ(t) = H(γ′(t)),

as the theorem asserts. Differentiating γ(t) = π ∘ σ_{Γ} ∘ γ(t) gives

γ′(t) = T π ∘ T σ_{Γ}(γ′(t)).

The vector T σ_{Γ}(γ′(t)) lies in the span of H(γ′(t)), so there is a real number c(t) such that T σ_{Γ}(γ′(t)) = H(c(t)γ′(t)). This gives

γ′(t) = T π ∘ H(c(t)γ′(t)).

Since T π ∘ H = Id_{E} we have

γ′(t) = c(t)γ′(t)

and c(t) = 1. Thus

\frac{d}{dt} σ_{Γ} ∘ γ(t) = T σ_{Γ}(γ′(t)) = H(c(t)γ′(t)) = H(γ′(t)).

□

Remark. Above, it is assumed that γ is everywhere tangent to E. One might expect that tangency of γ to E at p = γ(0) suffices to prove that (σ ∘ γ)′(0) = H(γ′(0)). This is not so. For example E can be the flow direction of an Anosov flow. The bundle E^{u} can be Hölder, but not C^{1}. Say its Hölder exponent is θ < 1. One can construct a C^{1} curve γ(t) which is tangent to E at p = γ(0), but which diverges from E at a rate t^{1+ϵ}. The difference between E_{γ(t)}^{u} and E_{p}^{u} is then on the order of t^{θ+ϵθ}. If ϵ is small this exponent is < 1, and the map t ↦ E_{γ(t)}^{u} fails to be differentiable at t = 0.

4. A Series Expression for ∂σ/∂E

As above σ is the unique F -invariant section and H = Id_{E} ⊕Λ is its partial derivative in the E-direction. Naturally, ∂σ/∂E depends on the choice of horizontal subbundle Hor ⊂ T V. We use the isomorphism T π : Hor_{σp} → T_{p}M to identify the linear map A_{σp} : Hor_{σp} → Hor_{σ(fp)} with its T π-conjugate T_{p}f. Then, using the

canonical isomorphism Vert_{σp} ≈ V_{p}, we can express T F = [\begin{matrix}A & 0 \\ C & K \end{matrix}] as

T_{σp}F = [\begin{matrix}T_{p}f : T_{p}M → T_{fp}M & 0 \\ C_{p} : T_{p}M → V_{fp} & K_{p} : V_{p} → V_{fp} \end{matrix}] .

Thus, the bundle map LF : L → L becomes

P ↦ (C_{p} + K_{p}P) ∘ (T_{fp}^{E}f^{−1}).

Denote by Λ_{0} the zero section of L, and call its N^{th} iterate in L,

Λ_{N} = (LF)^{N}(Λ_{0}).

We know that Λ_{N} → Λ uniformly as N → ∞. Also, we claim that

Λ_{N}(p) = ∑_{n=0}^{N−1} K_{p}^{n} ∘ C_{f^{−n−1}(p)} ∘ (T_{p}^{E}f^{−n−1})

where K^{0} = Id and for n ≥ 1,

K_{p}^{n} = K_{f^{−1}(p)} ∘ ⋯ ∘ K_{f^{−n}(p)} : V_{f^{−n}(p)} → V_{p}.

If N = 1 we have

Λ_{1}(p) = (C_{f^{−1}(p)} + K_{f^{−1}(p)}P_{0})(T_{p}^{E}f^{−1}) = C_{f^{−1}(p)}T_{p}^{E}f^{−1}

because Λ_{0} = 0 implies that P_{0} = 0. Thus, the assertion holds with N = 1; the proof is completed by induction.

Since the partial sums of the infinite series ∑_{n=0}^{∞} K_{p}^{n}C_{f^{−n−1}(p)}T_{p}^{E}f^{−n−1} converge uniformly to Λ, we are justified in writing

\frac{∂σ}{∂E} = H(p) = Id_{E} ⊕ ∑_{n=0}^{∞} K_{p}^{n}C_{f^{−n−1}(p)}T_{p}^{E}f^{−n−1}.

5. The Dominated and Partially Hyperbolic Cases: Proof of Theorem A

Let f : M → M be a diffeomorphism of a compact Riemannian (or Finslered) manifold. In this section, we consider T f-invariant dominated and partially hyperbolic splittings of the tangent bundle T M. We recall the definitions. The conorm of a linear transformation T : X → Y is

m(T) = inf_{x} \frac{|T x|}{|x|}.

Suppose that T M splits as a T f-invariant sum of two bundles:

T M = R ⊕ S.

This splitting is dominated if the following condition holds:

inf_{p} \frac{m(T_{p}^{R}f)}{∥T_{p}^{S}f∥} > 1

where the notation T^{X}f is used for the restriction of T f to the bundle X. We say that f has a dominated decomposition if there is a T f-invariant dominated splitting of T M.

Then f is partially hyperbolic if it has a T f-invariant splitting T M = E^{u}⊕E^{c}⊕E^{s} such that:

(1) E^{u} ⊕ (E^{c} ⊕ E^{s}) and (E^{u} ⊕ E^{c}) ⊕ E^{s} are both dominated splittings of T M, (2)

inf_{p} m(T_{p}^{u}f) > 1  sup_{p}∥(T_{p}^{s}f)∥ < 1

In other words, T f expands vectors in E^{u}, contracts vectors in E^{s}, and is relatively neutral on vectors in E^{c}.

Theorem 3.1 can be applied in the dominated decomposition and partially hyperbolic contexts, as follows.

Theorem 5.1. Suppose that the C^{2} diffeomorphism f : M → M has a dominated decomposition:

T M = R ⊕ S,

And let E ⊂ T M be a T f-invariant subbundle. Then, under the pointwise bunching condition

\begin{aligned}& sup_{p} \frac{∥T_{p}^{S}f∥}{m(T_{p}^{R}f)m(T_{p}^{E}f)} < 1, \\ (4) \end{aligned}

R is continuously differentiable with respect to E.

Theorem A is an immediate corollary of Theorem 5.1 , where we set R = E^{u}, S = E^{c} ⊕ E^{s}, and E = E^{c}.

In [4] we defined the “bolicity” of a linear transformation to be the ratio of its norm to its conorm, so the poinwise bunching condition in Theorem A can be re-stated as

sup_{p} \frac{bol(T_{p}^{c}f)}{m(T_{p}^{u}f)} < 1.

Similarly, to show that E^{s} is differentiable along E^{c}, we assume

\begin{aligned}& sup_{p}∥T_{p}^{s}f∥ bol(T_{p}^{c}f) < 1. \\ (5) \end{aligned}

Proof of Theorem 5.1. Let d be the fiber dimension of R in the dominated decomposition

T M = R ⊕ S.

T f acts naturally on the Grassmann G = G(d, T M) of all d-planes in T M,

Gf : G → G.

If Π is a d-plane in T_{p}M then Gf(Π) = T_{p}f(Π). Since f is C^{2}, Gf is a C^{1} fiber preserving map,

\begin{matrix}G & \underrightarrow{Gf} & G \\ π↓ & & ↓π \\ M & \underrightarrow{f} & M \end{matrix},

where π sends Π ⊂ T_{p}M to p. Besides, p ↦ R_{p} is a Gf-invariant section of G. We show that, at the invariant section R, the fiber contraction rate dominates the contraction rate along E as follows.

A compact neighborhood N_{p} of R_{p} in G_{p} consists of d-planes Π such that Π = graph P where P : R_{p} → S_{p} is a linear transformation with ∥P ∥ ≤ 1. Give G a Finsler which is the operator norm on each N_{p} and any other Finsler on the rest of G. Then Gf is a fiber preserving map whose fiber contraction rate at R_{p} is

k_{p} = \frac{∥T_{p}^{S}f∥}{m(T_{p}^{R}f)} < 1.

Since this dominates the contraction rate m(T_{p}^{E}f) along E, Theorem 3.1 applies and p ↦ R_{p} is seen to be a continuously differentiable function of p in the E direction. □

6. A series formula for ∂E^{u}/∂E^{c}

From Section 4 we know that there is a series that expresses ∂E^{u}/∂E^{c}. We write this formula out after making a convenient choice of the horizontal bundle.

To do so, we coordinatize G near E_{p}^{u} as follows. Fix a smooth Riemann structure on M that exhibits the partial hyperbolicity of f, and let exp be its exponential map. Abusing notation, we denote by ℝ^{u} and ℝ^{cs} the planes ℝ^{u} × 0 and 0 × ℝ^{cs} in ℝ^{m}. For each p ∈ M, define a linear map I_{p} : ℝ^{m} → T_{p}M that carries ℝ^{u} and ℝ^{cs} isometrically to E_{p}^{u} and E_{p}^{cs}. The restriction of exp_{p} ∘I_{p} to a small neighborhood U = U_{p} of 0 in ℝ^{m} is a diffeomorphism of U to a neighborhood Q = Q_{p} of p in M,

φ_{p} : U → Q

and (a) φ_{p}(0) = p

(b) T_{0}φ_{p} carries ℝ^{u} and ℝ^{cs} isometrically to E_{p}^{u} and E_{p}^{cs}.

(c) If we denote by E_{pq}^{u} and E_{pq}^{cs} the planes T_{x}φ_{p}(ℝ^{u}) and T_{x}φ_{p}(ℝ^{cs}), where

q = φ_{p}(x), then q ↦ E_{pq}^{u} ⊕ E_{pq}^{cs} is a smooth splitting of T Q that reduces to
E_{p}^{u} ⊕ E_{p}^{cs} when p = q.

We now coordinatize G near E_{p}^{u}. Let M be the space of (u × cs)-matrices, thought of as linear transformations X : ℝ^{u} → ℝ^{cs}. Given (x, X) ∈ U × M, let q = φ_{p}(x) and consider the linear transformation S : E_{pq}^{u} → E_{pq}^{cs}, defined by commutativity of

ℝ^{u} \underrightarrow{X} ℝ^{cs}
T_{x}φ_{p}↓  ↓T_{x}φ_{p}
E_{pq}^{u} \underrightarrow{S} E_{pq}^{cs}.

The graph of S is a plane Π ∈ G near E_{p}^{u}, and thus

Φ_{p} : (x, X) ↦ Π

is a local trivialization of G at E_{p}^{u}.

Because U × M is a product, T (U × M) carries a natural horizontal structure, the horizontal space at (x, X) being

ℝ^{m} × 0 ⊂ ℝ^{m} × M = T_{(x,X)}(U × M).

We define the horizontal space at Π = Φ_{p}(0, X) ∈ G_{p} to be

Hor_{Π} = T_{(0,X)}Φ_{p}(ℝ^{m} × 0).

Writing T (Gf) : T G → T G with respect to the horizontal / vertical splitting of T G gives

T_{Π}(Gf) = [\begin{matrix}A_{Π} : Hor_{Π} → Hor_{Gf(Π)} & 0 \\ C_{Π} : Hor_{Π} → Vert_{Gf(Π)} & K_{Π} : Vert_{Π} → Vert_{Gf(Π)} \end{matrix}]

Take Π = E_{p}^{u} and identify

T_{E_{p}^{u}}G_{p} = Vert_{E_{p}^{u}} ≈ L(E_{p}^{u}, E_{p}^{cs}).

Fix v ∈ E_{p}^{c}. Then C_{f^{−n−1}p} ∘T^{c}f^{−n−1}(v) is a linear transformation Y_{n}(v) : E_{f^{−n}p}^{u} → E_{f^{−n}p}^{cs}, and Y_{n}(v) is susceptibe to the n^{th} power of the linear graph transform, which converts it to a linear transformation E_{p}^{u} → E_{p}^{cs} defined by

T_{f^{−n}p}^{cs}f^{n} ∘ (Y_{n}(v)) ∘ T_{p}^{u}f^{−n}.

This is the same as the repeated action of K. (That is, the graph transform of T f^{n} is the same as the n^{th} power of the graph transform of T f.) Thus, by the formula in Section 4,

\frac{∂E^{u}}{∂E^{c}} (v) = ∑_{n=0}^{∞} T_{f^{−n}p}^{cs}f^{n} ∘ (C_{f^{−n−1}p} ∘ T^{c}f^{−n−1}(v)) ∘ T_{p}^{u}f^{−n}.

We also can express this in charts as follows. Writing f in the φ-charts gives

f_{p} = φ_{fp}^{−1} ∘ f ∘ φ_{p}

and

(Df_{p})_{x} = [\begin{matrix}D_{x}^{u,u}f_{p} & D_{x}^{cs,u}f_{p} \\ D_{x}^{u,cs}f_{p} & D_{x}^{cs,cs}f_{p} \end{matrix}]

where the D_{x}^{u,u}f_{p} block consists of the partial derivatives of the u-components of f_{p} with respect to the u-variables, evaluated at the point x, etc. At x = 0, the off-diagonal blocks are zero, while the diagonal blocks are T φ-conjugate to T_{p}^{u}f and T_{p}^{cs}f. Thus, the coordinate expression of Gf becomes

(Gf)_{p} : (x, X) ↦ (f_{p}x, (D_{x}^{u,cs}f_{p} + (D_{x}^{cs,cs}f_{p})X)(D_{x}^{u,u}f_{p} + (D_{x}^{cs,u}f_{p})X)^{−1}.

Differentiating this with respect to x and X at the origin (0, 0) ∈ ℝ^{m} × M yields

(D((Gf)_{p}))_{(0,0)} = [\begin{matrix}A_{p} : ℝ^{m} → ℝ^{m} & 0 \\ C_{p} : ℝ^{m} → M & K_{p} : M → M \end{matrix}] ,

where A_{p} is T φ-conjugate to T_{p}f,

A_{p} = (T_{0}φ_{fp})^{−1} ∘ T_{p}f ∘ T_{0}φ_{p},

C_{p} represents the second derivatives of f in the φ-charts,

C_{p} = \frac{∂}{∂x}|_{x=0}(D_{x}^{u,cs}f_{p})(D_{x}^{u,u}f_{p})^{−1}
= (D_{x}(D_{x}^{u,cs}f_{p}))(D_{x}^{u,u}f_{p})^{−1}
− (D_{x}^{u,cs}f_{p})(D_{x}^{u,u}f_{p})^{−1}(D_{x}(D_{x}^{u,u}f_{p}))(D_{x}^{u,u}f_{p})^{−1},

and, because the off-diagonal blocks vanish at the origin, K_{p} is T Φ-conjugate to the graph transform of T f,

P ↦ T_{p}^{cs}f ∘ P ∘ (T_{p}^{u}f)^{−1}.

It is worth noting that the norm of C is uniformly bounded on a neighborhood of E^{u} in G because f is C^{2} and M is compact. Also, this is clear from the formula expressing C in the φ-charts.

7. Dependence of E^{u}, E^{s} on f: Proof of Theorem B

As has been highlighted in the Katok-Milnor examples [8], the conjugacy between an Anosov diffeomorphism and its perturbations is a smooth function of the perturbation, even though the conjugacies themselves are only continuous. For example, consider a 1-parameter family of Anosov diffeomorphisms g_{t} : M → M. The map g_{0} is conjugate to g_{t} by a homeomorphism h_{t} : M → M, and h_{t} is uniquely determined by the requirements that h_{0} = Id and t ↦ h_{t} is continuous. The map

\begin{matrix}(−ϵ, ϵ) × M & → (−ϵ, ϵ) × M \\ (t, p) & ↦ (t, g_{t}p) \end{matrix}

is smooth, partially hyperbolic, and supports a center foliation W^{c} whose leaf through (0, p) is {(t, h_{t}(p)) : t ∈ (−ϵ, ϵ)}. As a foliation W^{c} is only continuous, but its leaves are smooth. Theorem 3.1 applies perfectly well to this situation, and we conclude that E_{h_{t}p}^{u}, E_{h_{t}p}^{s} are C^{1} functions of t. In this section we replace the Anosov condition by partial hyperbolicity, and derive an analogous result.

We assume that f_{0} : M → M is a C^{2}, partially hyperbolic diffeomorphism with splitting

T M = E^{u} ⊕ E^{c} ⊕ E^{s}

and that F is a small neighborhood of f_{0} in Diffeo^{2} M. By the usual linear graph transform techniques, all f ∈ F are partially hyperbolic and their splittings

T M = E^{u}(f) ⊕ E^{c}(f) ⊕ E^{s}(f)

depend continuously on f.

The space Diff^{2}(M) is a Banach manifold, see [1] for details. For f ∈ Diff^{2}(M), the tangent space to Diff^{2}(M) at f has a natural description. Let X_{f} be the Banach space of C^{2} sections of the pullback bundle f^{∗}T M, that is, the bundle whose fiber over p ∈ M is T_{fp}M. We write f + g to indicate the diffeomorphism exp_{f} ∘g. That is, if g is a small vector field in X_{f}, then

(f + g)(p) = exp_{f}(g(p))

is in Diff^{2}(M) and is close to f. So a small disk in X_{f} is a chart for a small neighborhood of f, and X_{f} is thereby identified with the tangent space T_{f}Diff^{2}(M).

Define the map

\begin{matrix}Eval : F × M & → F × M \\ (f, p) & ↦ (f, fp). \end{matrix}

Eval is C^{2} because left-composition is a smooth operation on functions.

Lemma 7.1. If the diameter of F is sufficiently small, then Eval has a partially hyperbolic splitting:

T (F × M) = 𝔼^{u} ⊕ 𝔼^{c} ⊕ 𝔼^{s},

where

𝔼_{f,p}^{u} = 0 × E_{p}^{u}(f) ⊂ 0 × T M
𝔼_{f,p}^{s} = 0 × E_{p}^{s}(f) ⊂ 0 × T M,

and 𝔼_{f,p}^{c} is the graph of a linear map:

P_{f,p} : X_{f} ⊕ E_{p}^{c}(f) → E_{p}^{u}(f) ⊕ E_{p}^{s}(f).

Proof. The tangent to Eval at (f, p) acts on a vector [\begin{matrix}g \\ v \end{matrix}] ∈ T_{f,p}(F × M) as

T_{f,p} Eval [\begin{matrix}g \\ v \end{matrix}] = [\begin{matrix}g \\ g(p) + T_{p}f(v) \end{matrix}] = [\begin{matrix}Id_{F} & 0 \\ ev_{p} & T_{p}f \end{matrix}] [\begin{matrix}g \\ v \end{matrix}] ,

where ev_{p} evaluates the section of f^{∗}T M at p. In particular, this implies that the subbundles 𝔼^{u} = 0×E^{u}, 𝔼^{s} = 0×E^{s} cited above are T Eval-invariant. (The bundle 0 × E^{c} is also T Eval-invariant, but it is too small to be the 𝔼^{c} we want.)

Note that the subbundle T F ⊕ 0 is not T Eval-invariant, nor is the subbundle T F ⊕E^{c} whose fiber at (f, p) is X_{f} ⊕E_{p}^{c}(f). For if v ∈ E_{p}^{c}(f) then the T Eval-image

of [\begin{matrix}g \\ v \end{matrix}] is [\begin{matrix}g \\ g(p) + T^{c}f_{p}(v) \end{matrix}], and this vector need not lie in T F ⊕E^{c}. Nevertheless, by

the domination hypotheses, the T Eval graph transform defines a fiber contraction of the bundle whose fiber at (f, p) is

L(X_{f} ⊕ (E^{u} ⊕ E^{c})_{p}(f), E^{s}(f)).

The resulting invariant section is the unique T Eval-invariant subbundle 𝔼^{c} ⊕ 𝔼^{s} ⊂ T (F × M) whose fiber at (f, p) projects isomorphically onto X_{f} ⊕ (E^{c} ⊕ E^{s})_{p}(f).

Similarly, we find the unique T Eval^{−1}-invariant subbundle 𝔼^{c} ⊕𝔼^{u} ⊂ T (F ×M) whose fiber at (f, p) projects isomorphically onto X_{f} ⊕ (E^{c} ⊕ E^{u})_{p}(f). Intersecting these bundles, we obtain the T Eval^{−1}-invariant subbundle 𝔼^{c}. □

Remark. At the end of this section, we give a series expression for 𝔼^{cu}.

Corollary 7.2. Suppose that f : M → M is C^{2} and partially hyperbolic, with splitting:

T M = E^{u} ⊕ E^{c} ⊕ E^{s}.

If f_{0} satisfies the pointwise bunching condition:

\begin{aligned}& sup_{p} \frac{∥T_{p}^{c}f_{0}∥}{m(T_{p}^{u}f_{0})m(T_{p}^{c}f_{0})} < 1, \\ (6) \end{aligned}

then, for all p ∈ M, E^{u} is continuously differentiable at (f_{0}, p) with respect to 𝔼^{c}, where 𝔼^{c} is given by Lemma 7.1.

In fact this corollary can be stated in a more general form that can be useful in applications. Not only is it possible to differentiate E^{u} along 𝔼^{c}, but in fact bundles in dominated decompositions can be differentiated along 𝔼^{c} as well. If f_{0} has a dominated decomposition

T M = R ⊕ S,

then standard graph-transform arguments apply to show that for f sufficiently C^{1}-close to f_{0}, this decomposition has a unique continuation

T M = R(f) ⊕ S(f)

that is dominated for T f. Under appropriate bunching hypotheses, we can differentiate R(f) in the 𝔼^{c} direction:

Corollary 7.3. Suppose that f : M → M is C^{2} and partially hyperbolic, with splitting:

T M = E^{u} ⊕ E^{c} ⊕ E^{s}.

Suppose also that

T M = R ⊕ S

is a dominated decomposition for f_{0}. If f_{0} satisfies the pointwise bunching condition:

\begin{aligned}& sup_{p} \frac{∥T_{p}^{S}f_{0}∥}{m(T_{p}^{R}f_{0})m(T_{p}^{c}f_{0})} < 1, \\ (7) \end{aligned}

then, for all p ∈ M, R is continuously differentiable at (f_{0}, p) with respect to 𝔼^{c}, where 𝔼^{c} is given by Lemma 7.1.

Proof of Corollary 7.3. We first construct the bundle over F × M whose fiber over (f, p) is the space of linear maps L(R_{p}(f), S_{p}(f)). Since Eval preserves the factors {f} × M, its tangent map T Eval induces a graph transform map on this bundle, covering Eval, which is a fiber contraction, with:

k_{f,p} = \frac{∥T_{p}^{S}f∥}{m(T_{p}^{R}f)}.

The unique invariant section of this graph transform is ℝ = 0 ⊕ R ⊂ T F × T M. (note that the bundle ℝ is not to be confused with the real numbers R).

Now suppose γ is any curve tangent to 𝔼^{c}. As in the proof of Theorem 3.1, we obtain differentiability of ℝ (and hence, of R) along γ when k_{f,p} dominates the contraction along γ at (f, p) The contraction along γ at (f, p) is bounded below by the conorm of T_{f,p}^{𝔼^{c}} Eval, which in turn is approximately given by

m(T_{f_{0},p}^{𝔼^{c}} Eval) = min{1, m(T^{c}f_{0})}.

Hence, ℝ = 0 ⊕ R is differentiable along γ if

\begin{aligned}& sup_{f,p} \frac{k_{f,p}}{min\{m(T_{p}^{c}f_{0}), 1\}} < 1. \\ (8) \end{aligned}

Since k_{f,p} < 1 for all p, and by the bunching hypothesis (7), k_{f,p} < m(T_{p}^{c}f_{0}), the condition in (8) is satisfied.

Now we apply Theorem 3.1 and conclude that ℝ is continuously differentiable along 𝔼^{c}.

Note that Theorem 3.1 needs to be re-proved in this more general context, but because its original proof relied on uniform estimates (this was the only necessity for the compactness assumption on M), it is not hard to do.

□

We are now ready to prove Theorem B. As mentioned in the introduction, Theorem B is a corollary of the following more general result.

Theorem 7.4. Let {f_{t} : M → M}_{t∈(−ϵ,ϵ)} be a C^{2} family of C^{2}, partially hyperbolic diffeomorphisms having, for each t ∈ (−ϵ, ϵ), a T f_{t}-invariant splitting:

T M = E^{u}(f_{t}) ⊕ E^{c}(f_{t}) ⊕ E^{s}(f_{t}).

Then there exists ϵ_{0} > 0 so that, for every p ∈ M and every v ∈ E^{c}(p), there exists a C^{1} path

φ_{p} : (−ϵ_{0}, ϵ_{0}) → M

with the following properties:

(1) φ_{p,v}(0) = p

(2) φ̇_{p,v}(0) ∈ v + E^{u} ⊕ E^{s},

(3) If

T M = R(f_{0}) ⊕ S(f_{0})

is any dominated decomposition for f_{0} satisfying the pointwise bunching condition

\begin{aligned}& sup_{p} \frac{∥T_{p}^{S}f_{0}∥}{m(T_{p}^{R}f_{0})m(T_{p}^{c}f_{0})} < 1, \\ (9) \end{aligned}

then t ↦ R_{φ_{p,v}(t)}(f_{t}) is C^{1}.

Remark. E^{c} is allowed to be the trivial bundle in Theorem 7.4, in which case f_{0} is Anosov. If f_{0} is Anosov, then 𝔼^{c} is uniquely integrable, φ_{p,0} is unique, and p ↦ φ_{p,0}(t) is the homeomorphism conjugating f_{0} to f_{t}.

Remark. Similarly, if E^{c} is integrable and tangent to a plaque-expansive foliation W^{c}, then 𝔼^{c} is also tangent to a foliation 𝕎^{c}. The maps p ↦ φ_{p,0}(t) can be shown to be leaf conjugacies between f_{0} and f_{t}.

If f_{0} is r-normally-hyperbolic:

\begin{aligned}& ∥T^{c}f∥^{r} < m(T^{u}f) & ∥T^{s}f∥ < m(T^{c}f)^{r}, \\ (10) \end{aligned}

then the leaves of 𝕎^{c} are C^{r}. In this case, t ↦ φ_{p,v}(t) can also be chosen to be C^{r}. If, in addition, the stronger center bunching condition:

\begin{aligned}& sup_{p} \frac{∥T_{p}^{c}f_{0}∥}{m(T_{p}^{u}f_{0})m(T_{p}^{c}f_{0})^{r}} < 1 \\ (11) \end{aligned}

holds, then the C^{r} Section Theorem implies that 𝔼^{u} is C^{r} along the leaves of 𝕎^{c} (and so t ↦ E_{φ_{p,v}(t)}^{u}(f_{t}) is also C^{r}).

Remark. A simple refinement of the proof shows that both t ↦ R_{φ_{p,v}(t)}(f_{t}) and t ↦ φ_{p,v}(t) are C^{1+α}, where there is a bound on the α-Hölder norm of the tderivative that is uniform in p, v. The exponent α is determined by several bunching conditions.

Proof of Theorem 7.4. Let F and, for (f, q) ∈ F × M, the linear map P_{f,q} : X_{f} ⊕ E_{q}^{c}(f) → E_{q}^{u}(f) ⊕ E_{q}^{s}(f) be given by Lemma 7.1, so that

𝔼_{f,q}^{c} = graph(P_{f,q}).

Choose ϵ_{0} > 0 so that f_{t} ∈ F, for all t ∈ (−ϵ_{0}, ϵ_{0}).

We identify the submanifold

{f_{t} × M | t ∈ (−ϵ_{0}, ϵ_{0}) ⊂ F × M

with (−ϵ_{0}, ϵ_{0}) × M in the obvious way. In the tangent bundle R × T M to this manifold, P_{f_{t},q} and 𝔼_{f_{t},q}^{c} have their counterparts P_{t,q} : R⊕E_{f_{t}}^{c}(q) → (E^{u}⊕E^{s})_{f_{t}}(q) and 𝔼_{t,q}^{c}, where

𝔼_{t,q}^{c} = graph(P_{t,q}).

Let p and v be given, and let V be any continuous vector field on M with the properties: V (p) = v and V (q) ∈ E^{c}(q), for all q ∈ M. Define a vector field Ω on (−ϵ_{0}, ϵ_{0}) × M by:

Ω(t, q) = \frac{∂}{∂t} + V (q) + P_{t,q}(\frac{∂}{∂t} + V (q)).

Notice that Ω(t, q) ∈ 𝔼_{t,q}^{c}, for all (t, q) ∈ (−ϵ_{0}, ϵ_{0})×M. It follows from the bunching hypothesis and Corollary 7.3 that R is differentiable along the integral curves of Ω.

Let φ̂_{p,v} be any integral curve of Ω through (p, 0). Now φ_{p,v} is defined to be the M coordinate of φ̂_{p,v}:

φ̂_{p,v} = (t, φ_{p,v}).

It is straightforward to check that φ_{p,v} satisfies (1)-(3). □

7.1. A series expansion for 𝔼^{cu}. We give a series expression for 𝔼^{cu} as follows. Define the linear map P_{f,p}^{cu} : X ⊕ E_{p}^{cu}(f) → E_{p}^{s}(f) as the series

P_{f,p}^{cu}(g, v) = ∑_{k=0}^{∞} T_{f^{−k}p}^{s}f^{k}(g^{s}(f^{−k}p)).

Note that the series does not depend on v. The domination conditions imply that the series converges. Under T Eval, the graph of P_{f,p}^{cu} is sent to the graph of P_{f,fp}^{cu}. Hence, by uniqueness,

𝔼_{f,p}^{cu} = graph(P_{f,p}^{cu}).

In the same way we get a unique T Eval-invariant subbundle 𝔼^{cs} ⊂ T (F ×M) whose fiber at (f, p) projects isomorphically onto X ⊕ E_{p}^{cs}(f), and 𝔼_{f,p}^{cs} = graph(P_{f,p}^{cs}) where

P_{f,p}^{cs}(g, v) = ∑_{k=0}^{∞} T_{f^{k}p}^{u}f^{−k}(g^{u}(f^{k}p)).

The intersection of these two subbundles is the center bundle 𝔼^{c}. Namely, at (f, p), the fiber of 𝔼^{c} is the graph of the map P_{f,p}^{c} : X ⊕ E_{p}^{c}(f) → (E_{p}^{u}(f) ⊕ E_{p}^{s}(f)), where

P_{f,p}^{c}(g, v) = ∑_{k=0}^{∞} T_{f^{k}p}^{u}f^{−k}(g^{u}(f^{k}p)) + ∑_{k=0}^{∞} T_{f^{−k}p}^{s}f^{k}(g^{s}(f^{−k}p)).

8. When f ↦ E_{p}^{u}(f) actually is differentiable

We described in the previous section how f ↦ E_{p}^{u}(f) is generally not differentiable, even if f is Anosov. In fact, if p ↦ E_{p}^{u}(f_{0}) fails to be differentiable in even one direction at p_{0}, then f ↦ E_{p_{0}}^{u}(f) is not differentiable at f_{0}. For in that case, it is easy to construct a smooth 1-parameter family of diffeomorphisms φ_{t} : M → M such that t ↦ E_{φ_{t}p_{0}}^{u}(f_{0}) is not differentiable at t = 0; but then E_{p_{0}}^{u}(φ_{t}f_{0}φ_{t}^{−1}) = T φ_{t}(E_{φ_{t}p_{0}}^{u}(f_{0})) is not differentiable at t = 0.

It turns out that, under the usual center bunching hypothesis, nonsmoothness of p ↦ E_{p}^{u}(f_{0}) is the only obstruction to differentiability of f ↦ E_{p}^{u}(f) at f_{0} in all directions.

The results that follow apply to 1-parameter families of C^{2} diffeomorphisms {f_{t}}_{t∈I} such that t ↦ f_{t} is a C^{1} map from I into Diff^{2}(M) – a C^{1} family of C^{2} diffeomorphisms. Since the original proof of Theorem C is somewhat lengthy and the result is subsumed by Theorem D, we omit the proof of Theorem C and present instead a proof of Theorem D, following closely the approach of Dolgopyat in [2].

Assume that for each t ∈ I, f_{t} is partially hyperbolic with splitting

T M = E_{t}^{u} ⊕ E_{t}^{c} ⊕ E_{t}^{s}.

Write

E_{t} = E_{t}^{c} H_{t} = E_{t}^{u} ⊕ E_{t}^{s}.

Theorem 8.1 (Theorem D). If E_{0} is a C^{1} bundle then the curve of subbundles t ↦ E_{t} is differentiable with respect to t at t = 0, and the derivative (dE_{t}(x)/dt)_{t=0} depends continuously on x ∈ M.

Remark. Theorem D remains valid, and the proof is the same, if the partially hyperbolic splitting is replaced by a dominated triple splitting R_{t}⊕S_{t}⊕T_{t}. Namely, the middle bundle S_{t} is differentiable with respect to t at t = 0, provided that S_{x,0} is C^{1}. Similarly, there is nothing special about the one-dimensionality of the parameter t.

The following facts about weak continuity will be used. We assume that W is a Banach space, but that W also carries a weak topology. Of course, if W has finite dimension, the weak and strong topologies coincide. We have in mind the case that W is a space of operators on the the infinite dimensional Banach space of continuous sections of a vector bundle and Λ = ℝ.

Definition 8.2. A function h : Λ → W is weakly continuous at μ ∈ Λ if h(λ) tends weakly to h(μ) and ∥h(λ)∥ stays bounded as λ → μ.

Proposition 8.3 (Weak Inversion Rule). If a curve of invertible operators t ↦ A_{t} is weakly continuous at t = 0 and if the operators’ conorms are uniformly positive then the curve of inverse operators is also weakly continuous at t = 0.

Proof. Let t ↦ A_{t} be the curve of operators, and let V be the Banach space on which they operate. Then, as t → 0, A_{t} converges weakly to A_{0} and ∥A_{t} − A_{0}∥ stays bounded. The conorm assumption means that for all small t, ∥A_{t}^{−1}∥ ≤ M.

For each v ∈ V,

\begin{aligned}|A_{t}^{−1}(v) − A_{0}^{−1}(v)| & = |A_{t}^{−1} ∘ (A_{0} − A_{t}) ∘ A_{0}^{−1}(v)| \\ & ≤ M|v − A_{t}(A_{0}^{−1}(v))|. \end{aligned}

Since A_{0}^{−1}(v) is fixed, and A_{t} converges weakly to A_{0}, A_{t}(A_{0}^{−1}(v)) → v as t → 0, which completes the proof that A_{t}^{−1} converges weakly to A_{0}^{−1} as t → 0. But also,

\begin{aligned}|A_{t}^{−1}(v) − A_{0}^{−1}(v)| & = |A_{t}^{−1} ∘ (A_{0} − A_{t}) ∘ A_{0}^{−1}(v)| \\ & ≤ M∥A_{0} − A_{t}∥M|v| \end{aligned}

implies that ∥A_{t}^{−1} − A_{0}^{−1}∥ stays bounded as t → 0, and completes the proof that the inverse curve is weakly continuous. □

Now we return to the splitting T M = E_{t} ⊕H_{t}, where H_{t} is the hyperbolic part of the partially hyperbolic splitting for f_{t}, and E_{t} is the center part. We are assuming that E = E_{0} is a C^{1} bundle.

Let H̃ be a smooth approximation to H_{0}, and express T f_{t} with respect to the splitting T M = E ⊕ H̃ as

T_{x}f_{t} = [\begin{matrix}A_{x,t} & B_{x,t} \\ C_{x,t} & K_{x,t} \end{matrix}] .

Since f_{t} is a C^{1} curve of C^{2} diffeomorphisms, A, B, C, K are C^{1} functions of x, t. At t = 0 we have

C_{x,0} = 0  and  A_{x,0} = T_{x}f_{0}|_{E}

for all x. Furthermore, when H̃ closely approximates H, ∥B∥ is small. Consequently, if P : E → H̃ has norm ≤ 1 then A + BP is invertible and the norm of its inverse is uniformly bounded. Uniformity refers to P, x, t.

Let L be the vector bundle over M whose fiber at x is L_{x} = L(E_{x}, H̃_{x}). Equipping L_{x} with the operator norm gives L a Finsler; let L(1) be its unit ball bundle. Denote by Sec(L) the Banach space of continuous sections X : M → L, equipped with the sup norm ∥ ∥. Its unit ball is Sec(L(1)).

T f_{t} defines a graph transform

L(1) \underrightarrow{(T f_{t})_{#}} L
π↓  ↓π
M  \underrightarrow{f_{t}} M

according to the condition T_{x}f_{t}(graph P) = graph((T_{x}f_{t})_{#}(P)). That is,

(T_{x}f_{t})_{#}(P) = (C_{x,t} + K_{x,t}P) ∘ (A_{x,t} + B_{x,t}P)^{−1},

which is a linear map E_{f_{t}x} → H̃_{f_{t}x}. The graph transform naturally induces a nonlinear map on the space of sections,

G_{t} : Sec(L(1)) → Sec(L)

such that

G_{t}(X) = (T f_{t})_{#} ∘ X ∘ f_{t}^{−1}.

Proposition 8.4. G_{t} is uniformly analytic.

Remark. (T f_{t})_{#} is not analytic, it is only C^{1}. Nevertheless, for each fixed t, its action on the space of continuous sections is analytic. The uniformity refers to t.

We prove Proposition 8.4 by factoring G_{t} into a product of several analytic maps. Let E, E_{t} and E_{t}^{−1} denote the bundles whose fibers at x ∈ M are E_{x} = L(E_{x}, E_{x}), E_{x,t} = L(E_{x}, E_{f_{t}x}), and E_{x,t}^{−1} = L(E_{f_{t}x}, E_{x}). Let A, A_{t}, and A_{t}^{−1} denote the invertible elements in E, E_{t} and E_{t}^{−1}, and denote sectional inversion as Inv : Sec(A) → Sec(A), Inv_{t} : Sec(A_{t}) → Sec(A_{t}^{−1}).

Lemma 8.5. Sectional inversion is uniformly analytic.

Proof. Consider the identity section Id of A. Any section near Id is inverted by the power series

A^{−1} = ∑_{k=0}^{∞}(Id −A)^{k},

and hence sectional inversion is analytic in a neighborhood of the identity section. For A in a neighborhood of the general section A_{0} : M → A, sectional inversion factors according to the commutative diagram

A \underrightarrow{Inv near A_{0}} A^{−1}
L_{A_{0}^{−1}} ↓  ↑R_{A_{0}}
A_{0}^{−1}A \underrightarrow{Inv near Id} A^{−1}A_{0}

where L_{A_{0}^{−1}} and R_{A_{0}} are left and right multiplication by the sections A_{0}^{−1} and A_{0}. Since L_{A_{0}^{−1}} and R_{A_{0}} are continuous linear transformations of the section spaces, they are analytic, which completes the proof of the lemma for sections in a neighborhood of the identity section. The corresponding diagram

Sec(A_{t}) \underrightarrow{Inv_{t} near A_{0}} Sec(A_{t}^{−1})
L_{A_{0}^{−1}} ↓  ↑R_{A_{0}}
Sec(A) \underrightarrow{Inv near Id} Sec(A)

applies to sectional inversion in the neighborhood of a section A_{0} : M → A_{t}, and shows that Inv_{t} is analytic.

Uniform analyticity means that for any r, the r^{th} derivative of Inv_{t} is uniformly bounded on sets of sections such that ∥A∥ and ∥A^{−1}∥ are uniformly bounded; this is clear from the higher order chain rule and the factorization of sectional inversion given above. □

Proof of Proposition 8.4. We have G_{t}(X) = (T f_{t})_{#} ∘ X ∘ f_{t}^{−1}, and must show that G_{t} is a uniformly analytic function of X ∈ Sec(L). We factor G_{t} as the Cartesian product of two affine maps on section spaces, followed by inversion in one of the two spaces, followed by sectional linear composition, all of which is expressed by commutativity of

Sec(L)  \underrightarrow{G_{t}}  Sec(L)
Aff_{1} × Aff_{2}↓  ↑composition
Sec(L_{t}) × Sec(A_{t}) \underrightarrow{Id × Inv_{t}} Sec(L_{t}) × Sec(A_{t}^{−1})

where L_{t} is the bundle over M whose fiber at x is L(E_{x}, H̃_{f_{t}x}), and

Aff_{1}(X) = C_{t} + K_{t}X Aff_{2}(X) = A_{t} + B_{t}X.

Uniform analyticity of G_{t} then follows from Lemma 8.5. □

The r^{th}-order Taylor expansion of G_{t} at the zero section is

G_{t}(X) = Z_{t} + Q_{t}(X) + ⋯ + \frac{1}{r!}(D^{r}G_{t})_{0}(X^{r}) + R_{t}(X),

where Z_{t} = G_{t}(0), Q_{t} = (DG_{t})_{0}.

Proposition 8.6. For small t, (a) t ↦ Z_{t} is C^{1}.

(b) t ↦ (I − Q_{t})^{−1} is weakly continuous.
(c) ∥R_{t}(X)∥/∥X∥^{2} is uniformly bounded for all small X ∈ Sec(L).

Proof. At the zero section, the 0^{th} and first derivatives of

G_{t}(X) = (C_{t} + K_{t}X)(A_{t} + B_{t}X)^{−1} ∘ f_{t}^{−1},

with respect to X are computed at once as

\begin{aligned}Z_{t} & = (C_{t}A_{t}^{−1}) ∘ f_{t}^{−1} \\ Q_{t}(X) & = (K_{t}XA_{t}^{−1} + C_{t}A_{t}^{−1}B_{t}XA_{t}^{−1}) ∘ f_{t}^{−1} \end{aligned}

Since f_{t} is a C^{1} curve of C^{2} diffeomorphisms, and since the splitting E ⊕ H̃ is C^{1}, the curves t ↦ A_{t}, t ↦ B_{t}, t ↦ C_{t}, t ↦ K_{t} in the appropriate bundles are C^{1}. This makes (a) immediate, and also shows that the curve t ↦ Q_{t} in Sec(L) is weakly continuous.

By inspection, at t = 0, Q_{t} becomes the hyperbolic operator

Q_{0}(X) = (K_{0}XA_{0}^{−1}) ∘ f_{0}^{−1},

because C_{t=0} = 0. Thus, for all small t, I − Q_{t} is uniformly invertible, and Proposition 8.3 implies that t ↦ (I − Q_{t})^{−1} is weakly continuous.

Assertion (c) follows from the Mean Value Theorem and the fact that the second derivative of G_{t} is uniformly bounded near the zero section. □

Proof of Theorem D. Proposition 8.6 implies that

G_{t}(X) = Z_{t} + Q_{t}(X) + R_{t}(X)

and ∥R_{t}(X)∥ = O(1)∥X∥^{2} as ∥X∥ → 0. Let P_{t} : x ↦ P_{x,t} be the unique G_{t}- invariant section of L with norm ≤ 1. Thus P_{x,t} : E_{x} → H̃_{x} and

E_{x,t} = graph P_{x,t} = {v + P_{x,t}(v) ∈ T_{x}M : v ∈ E_{x}}.

Theorem D asserts that E_{t} is differentiable at t = 0. That is,

\frac{dP_{x,t}}{dt} |_{t=0}

exists and is continuous with respect to x.

Plugging X = P_{t} into the Taylor expansion of G_{t} gives

P_{t} = Z_{t} + Q_{t}(P_{t}) + R_{t}(P_{t}),

and since I − Q_{t} is invertible, we get

P_{t} = (I − Q_{t})^{−1}(Z_{t} + R_{t}(P_{t})).

Thus

\begin{aligned}& ∥P_{t}∥ ≤ ∥(I − Q_{t})^{−1}∥(∥Z_{t}∥ + M∥P_{t}∥^{2}). \\ (12) \end{aligned}

(These norms refer to section sup-norms or to operator norms, as appropriate.)

Now we estimate Z_{t} = (C_{t}∘A_{t}^{−1})∘f_{t}^{−1} as follows. It is differentiable with respect to t, and since C_{t=0} = 0, we have Z_{t=0} = 0. Thus ∥Z_{t}∥ = O(1)t as t → 0. Since P_{t} is continuous in t, and P_{0} = 0, we get ∥P_{t}∥_{0}^{2} ≪ ∥P_{t}∥_{0} when t is small, which lets us absorb the squared term into the l.h.s. of the inequality (12), so

∥P_{t}∥ = O(1)t

as t → 0. Consequently, we get a bootstrap effect on the remainder:

∥R_{t}(P_{t})∥ = O(1)t^{2}

as t → 0. Combined with the more exact estimate on Z_{t},

Z_{t} = tZ′_{0} + o(1)t

where Z′_{0} = (d/dt)_{t=0}(Z_{t}), this gives

\frac{P_{t}}{t} = (I − Q_{t})^{−1}Z′_{0} + (I − Q_{t})^{−1}(o(1) + O(1)t).

Proposition 8.6 implies that (I − Q_{t})^{−1} converges weakly to (I − Q_{0})^{−1} as t → 0, so

lim_{t→0}(I − Q_{t})^{−1}Z′_{0} = (I − Q_{0})^{−1}Z′_{0},

while uniform boundedness of ∥(I − Q_{t})^{−1}∥ implies that

lim_{t→0}(I − Q_{t})^{−1}(o(1)) + O(1)t) = 0.

Thus, as t → 0,

\frac{P_{x,t} − P_{x,0}}{t} → (I − Q_{0})^{−1}Z′_{0},

uniformly in x ∈ M, which completes the proof that t ↦ E_{t} is differentiable at t = 0, and that its derivative there, (I − Q_{0})^{−1}Z′_{0}, depends continuously on x ∈ M. □

Remark. Suppose that E_{0} and Df_{t} are C^{r}, r ≥ 2. We tried to show that E_{t} is r^{th}-order differentiable at t = 0 in the sense that there is an r^{th} order Taylor expansion for E_{t} at t = 0. Many ingredients of the preceding proof of the r = 1 case above generalize very nicely to r ≥ 2. There is a natural notion of weak r^{th}-order differentiability, and it behaves well with respect to operator inversion and operator products. However, we would also need affirmative answers to the following two questions:

(a) Is the curve t ↦ (I − Q_{t})^{−1} in Sec(L) weakly differentiable at t = 0?

(b) Does the operator (I − Q_{0})^{−1} send C^{1} sections of L to C^{1} sections?

At first, it would be acceptable to assume analyticity of E_{0} and f_{t}.

References

[1] Abraham, Ralph, and Robbins, Joel Transversal Mappings and Flows, Benjamin, New York, 1967.

[2] Dolgopyat, Dmitry, On differentiability of SRB states, preprint.

[3] Grayson, Matthew; Pugh, Charles; Shub, Michael, Stably ergodic diffeomorphisms. Ann. of Math. (2) 140 (1994), no. 2, 295–329.

[4] Hirsch, M. W.; Pugh, C. C.; Shub, M. Invariant manifolds. Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977.

[5] Katok, A.; Knieper, G.; Pollicott, M.; Weiss, H. Differentiability and analyticity of topological entropy for Anosov and geodesic flows. Invent. Math. 98 (1989), no. 3, 581–597.

[6] de la Llave, R.; Marco, J. M.; Moriyon, R. Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation. Ann. of Math. (2) 123 (1986), no. 3, 537–611.

[7] Mañé, Ricardo, On the dimension of the compact invariant sets of certain nonlinear maps. Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), pp. 230–242, Lecture Notes in Math., 898, Springer, Berlin-New York, 1981.

[8] Milnor, John, Fubini foiled: Katok’s paradoxical example in measure theory. Math. Intelligencer 19 (1997), no. 2, 30–32.

[9] Pugh, Charles; Shub, Michael, Stably ergodic dynamical systems and partial hyperbolicity. J. Complexity 13 (1997), no. 1, 125–179.

[10] Pugh, Charles; Shub, Michael, Stable ergodicity and julienne quasi-conformality. J. Eur. Math. Soc. (JEMS) 2 (2000), no. 1, 1–52.

[11] Ruelle, David, Differentiation of SRB states. Comm. Math. Phys. 187 (1997), no. 1, 227–241.

[12] Ruelle, David, “Extensions of a result by Shub and Wilkinson...” preprint

[13] Shub, Michael; Wilkinson, Amie, Pathological foliations and removable zero exponents. Invent. Math. 139 (2000), no. 3, 495–508.

[14] Wilkinson, Amie Stable ergodicity of the time-one map of a geodesic flow. Ergodic Theory Dynam. Systems 18 (1998), no. 6, 1545–1587.

Mathematics Department, University of California, Berkeley California, 94720, U.S.A.

   E-mail address: pugh@math.berkeley.edu

Watson Research Center, IBM, Yorktown Heights NY, 10598, USA

   E-mail address: shub@us.ibm.com

Mathematics Department, Northwestern University, Evanston Illinois, 60208, U.S.A.

   E-mail address: wilkinso@math.northwestern.edu