STABLE ERGODICITY AND FRAME FLOWS

Keith Burns and Mark Pollicott

Northwestern University and Manchester University

Abstract. In this note we show that all diffeomorphisms close enough to the timeone map of the frame flow on certain negatively curved manifolds are ergodic. As a simple corollary we deduce that the frame flows are ergodic for all compact manifolds with curvature pinched sufficiently close to −1, thus providing results in the case of manifolds of dimension 7 or 8 which were missing from the results of Brin and Karcher.

0. Introduction

The problem of determining the ergodicity of natural measure preserving transformations is a classical one. Anosov proved ergodicity for uniformly hyperbolic diffeomorphisms of compact manifolds which are C^{2} and preserve a smooth measure. Since uniform hyperbolicity is an open property, these systems are stably ergodic — any C^{2} diffeomorphism that is close enough in the C^{1} topology and preserves the same smooth measure is also ergodic. The situation is more complicated for partially hyperbolic diffeomorphisms because it is difficult to control the behaviour of the map in the neutral direction. Until the work of Grayson, Pugh and Shub [GPS] in the early 1990’s, uniformly hyperbolic diffeomorphisms were the only ones known to be stably ergodic. Let V be a smooth n-dimensional compact manifold with negative sectional curvatures. For simplicity we shall always assume that V is oriented. Results for nonorientable V can be obtained by studying the double cover. The sectional curvature K will be assumed to satisfy the pinching condition −Λ^{2} < K < −λ^{2}.

Let M = SV be the unit tangent bundle for V and ϕ_{t} : M → M be the geodesic flow. The geodesic flow preserves the (normalized) Liouville measure, which we denote by μ. Both the geodesic flow and its time-one map f = ϕ_{1} : M → M are known to be ergodic. Moreover f is partially hyperbolic, the original flow direction giving a one-dimensional neutral direction. It is now known that the map f is always stably ergodic. This was proved by Grayson, Pugh and Shub [GPS] when V is a surface with constant negative curvature; by Wilkinson [W] when V is a surface of variable negative curvature; by Pugh and Shub [PS2] for higher dimensional manifolds of constant, or nearly constant, negative curvature; and for general V by Katok and Kononenko [KK]. In fact Katok and Kononenko showed that the time-one map of any contact Anosov flow is stably ergodic.

Typeset by AMS-TEX

Their result was generalized by Burns, Pugh and Wilkinson [BPW] who showed that the time-one map of a volume preserving Anosov flow is stably ergodic unless the strong stable and strong unstable foliations for the flow are jointly integrable.

In this paper we will be interested in a related class of partially hyperbolic diffeomorphisms. Let M̂ be the space of positively oriented orthonormal n-frames in T V. This gives a fibre bundle where the natural projection ρ : M̂ → M takes a frame to its first vector. The associated structure group SO(n − 1) acts on fibres by rotating the frames, whilst keeping the first vector fixed. In particular, we can identify each fibre of M̂ with SO(n − 1). Let ϕ̂_{t} : M̂ → M̂ denote the frame flow, which acts on frames by moving their first vectors according to the geodesic flow and moving the other vectors by parallel transport along the geodesic defined by the first vector. The map ρ is a semi-conjugacy from ϕ̂_{t} to ϕ_{t}; i.e. ρ ∘ ϕ̂_{t} = ϕ_{t} ∘ ρ for each t ∈ ℝ. In particular, ϕ̂_{t} is an SO(n − 1)-group extension of ϕ_{t}. The frame flow ϕ̂_{t} preserves the measure μ̂ = μ × ν_{SO(n−1)}, where ν_{SO(n−1)} is (normalized) Haar measure on SO(n − 1). It was shown in [BP, §6] that the time-one map of the frame flow is a partially hyperbolic diffeomorphism. The neutral direction has dimension 1 + dim SO(n − 1) and is spanned by the flow direction and the fibre direction.

The ergodicity of the frame flow has been extensively studied. The frame flow is known to be ergodic and to have the K property under the following conditions:

(1) if V has constant negative curvature (Brin [Br2]);

(2) for a set of metrics with negative curvature that is open and dense in the C^{3} topology (Brin [Br1]);

(3) if n is odd, but not equal to 7 (Brin and Gromov [BG]);

(4) if n is even, but not equal to 8, and λ/Λ > 0.93 (Brin and Karcher [BK]). Our first result partially extends (3) and (4) to the cases where n = dim V is 7 or 8.

Theorem 0.1. The frame flow is ergodic and K if (5) n = 7 and λ/Λ > 0.99023 …, or

(6) n = 8 and λ/Λ > 0.99023 . . ..

Our next result addresses the problem of stable ergodicity for the time-one maps of frame flows.

Theorem 0.2. The time-one map of the frame flow f̂ = ϕ̂_{1} : M̂ → M̂ is stably ergodic and stably K in each of the cases (1) – (6).

Theorem 0.2 is a generalization of Theorem 0.1 because a flow is automatically ergodic (resp. K) if its time-one map is stably ergodic (resp. stably K). The frame flow is actually Bernoulli in all of the cases (1) – (6); this follows from the K property of the frame flows, the fact that the geodesic flow is Bernoulli [Ra], and a result of Rudolph about isometric extensions of Bernoulli systems [Ru]. We do not know if the corresponding time-one maps are stably Bernoulli.

Theorem 0.2 shows that the time-one map of the frame flow of a manifold with negative curvature is stably ergodic in all cases where it is known to be ergodic. We expect the time-one map of the frame flow to be stably ergodic whenever it is ergodic.

The frame flow is not always ergodic. Kähler manifolds with negative curvature and real dimension at least 4 — in particular quotients of the complex and quaternionic hyperbolic spaces — have nonergodic frame flows because the complex structure is invariant under parallel translation [BG]. The curvature tensor in these examples is invariant under parallel translation and, if the metric is suitably scaled, every vector lies in both a plane with curvature −1 and a plane with curvature −1/4. These are the only examples with negative curvature and nonergodic frame flow known to the authors. It has been conjectured that if the sectional curvatures are strictly pinched between −1 and −1/4, then the frame flow is ergodic [BK, Br3]. Brin also conjectures in [Br3] that the frame flow should be ergodic unless the holonomy of the manifold is a proper subgroup of SO(n).

It is unknown to us if the time-one frame flow on a negatively curved manifold is ever robustly transitive, i.e. all sufficiently close diffeomorphisms are transitive. Indeed, we do not even know whether the time-one map of the geodesic flow for a surface of constant negative curvature is robustly transitive. On the other hand, it seems possible that the methods of [BD] could be used to produce robustly transitive diffeomorphisms arbitrarily close to the time-τ map of the frame flow for suitable τ.

The first author thanks the University of Manchester for hospitality during the period when this work was begun.

1. A criterion for stable ergodicity

A key ingredient in our analysis is the important program of Pugh and Shub to establish stable ergodicity from a few simply verified hypothesis. In this section we recall their main result. Let N be a C^{∞} compact manifold and, denoting by μ the Riemannian volume on N, we let Diff_{μ}^{2}(N) denote those C^{2} diffeomorphisms on N which preserve μ.

Definition. We call a C^{∞} diffeomorphism f_{0} : N → N which preserves μ stably ergodic if there is an open neighbourhood U ⊂ Diff_{μ}^{2}(N) of f_{0} such that any f_{1} ∈ U is ergodic.

A C^{1} diffeomorphism f_{0} : N → N is called partially hyperbolic if there are a Df_{0}- invariant splitting T N = E^{s} ⊕E^{c} ⊕E^{u} and constants λ_{s} < μ_{s} < λ_{c} ≤ 1 ≤ μ_{c} < λ_{u} < μ_{u} and C ≥ 1 such that for all n ≥ 0

\begin{aligned}v ∈ E^{s} & ⇒ C^{−1}λ_{s}^{n} ≤ ∥Df_{0}^{n}(v)∥ ≤ Cμ_{s}^{n}, \\ v ∈ E^{c} & ⇒ C^{−1}λ_{c}^{n} ≤ ∥Df_{0}^{n}(v)∥ ≤ Cμ_{c}^{n}, and \\ v ∈ E^{u} & ⇒ C^{−1}λ_{u}^{n} ≤ ∥Df_{0}^{n}(v)∥ ≤ Cμ_{u}^{n}. \end{aligned}

The bundles E^{s}, E^{c}, and E^{u} are called stable, central (or neutral), and unstable respectively.

We say that f_{0} : N → N is centre bunched if μ_{c}/λ_{c} is sufficiently close to 1. The precise meaning of “sufficiently close” is specified in Pugh and Shub’s paper [PS2]. Centre bunching is C^{1} open. It certainly holds when the derivative of the map acts as an isometry on E^{c}, because then we can take λ_{c} = μ_{c} = 1.

We can associate to the bundles E^{s} and E^{u} the C^{1} stable and unstable foliations of N defined by

\begin{aligned}W_{f_{0}}^{s}(x) & = \{y ∈ N : d(f_{0}^{n}x, f_{0}^{n}y) → 0, as n → +∞\} and \\ W_{f_{0}}^{u}(x) & = \{y ∈ N : d(f_{0}^{−n}x, f_{0}^{−n}y) → 0, as n → +∞\}. \end{aligned}

These foliations have C^{1} leaves which are tangent to the bundles E^{s} and E^{u} respectively. The foliations are absolutely continuous, but the leaves may not vary smoothly.

A partially hyperbolic diffeomorphism f_{0} may not have continuous foliations W_{f_{0}}^{cs}, W_{f_{0}}^{c} and W_{f_{0}}^{cu} whose leaves are tangent to the bundles E^{c} ⊕ E^{s}, E^{c} and E^{c} ⊕ E^{u} respectively; f_{0} is called dynamically coherent if:

• W_{f_{0}}^{cs}, W_{f_{0}}^{c} and W_{f_{0}}^{cu} exist;

• W_{f_{0}}^{c} and W_{f_{0}}^{s} both subfoliate W_{f_{0}}^{cs}; and also
• W_{f_{0}}^{c} and W_{f_{0}}^{u} both subfoliate W_{f_{0}}^{cu}.

These foliations are called the centre-stable, central (or neutral) and centre-unstable foliations respectively. They again have C^{1} leaves but they may not be absolutely continuous (see [SW] for an explicit example). This lack of absolute continuity is the root of many of the technical difficulties in [GPS, PS1, PS2]. Dynamical coherence is stable under small C^{1} perturbations of f_{0}, provided the central foliation for f_{0} is C^{1}. This follows from the corollary to Proposition 2.3 in [PS1].

Definition. We call a piecewise C^{1} path ψ : [0, 1] → N from x to y a us-path if it is contained in the union of a finite number of stable and unstable manifolds. That is there exist a finite sequence of times 0 = t_{0} ≤ t_{1} < ⋯ < t_{n−1} ≤ t_{n} = 1 with ψ(t_{2i}) ∈ W_{f_{0}}^{s}(ψ(t_{2i+1})) and ψ(t_{2i+1}) ∈ W_{f_{0}}^{u}(x_{2i+2}) such that ψ([t_{2i}, t_{2i+1}]) ⊂ W_{f_{0}}^{s}(ψ(t_{2i+1})) and ψ([t_{2i+1}, t_{2i+2}]) ⊂ W_{f_{0}}^{u}(ψ(t_{2i+2})).

Figure 1. A us-path from x to y

We say that f_{0} : N → N has the accessibility property if for every x, y ∈ N there exists a us-path from x to y. We say that f_{0} : N → N has the stable accessibility property if there is a neighbourhood U ⊂ Diff_{μ}^{2}(M) such that any f_{1} ∈ U also has the accessibility property. The following fundamental result of Pugh and Shub [PS2] will be crucial in our subsequent analysis.

Theorem 1.1 (Pugh-Shub Theorem). Assume that a partially hyperbolic diffeomorphism f_{0} : N → N is centre bunched, dynamically coherent with C^{1} central foliation, and stably accessible. Then f_{0} is stably ergodic and stably K.

Although there are earlier versions of this result [PS1] which would serve for our purposes, the statement in [PS2] is the most convenient for our use. The fact that Pugh and Shub’s hypotheses imply that f_{0} is stably K as well as stably ergodic was pointed out in Corollary 1.2 of [BW]; it is not known whether their theorem can be further extended to conclude that f_{0} is stably Bernoulli. In practice, it is the third of the three hypotheses in the theorem that is difficult to establish.

The time-one maps of geodesic flows and frame flows for manifolds with negative curvatures are partially hyperbolic; see §6 of [BP1]. In the case of the time-one map of the geodesic flow (indeed for the time-one map of any Anosov flow), the stable and unstable foliations are the strong stable and strong unstable foliations for the flow. For v ∈ M, let H^{s}(v) and H^{u}(v) be the stable and unstable horospheres that are orthogonal to v. Then the stable and unstable manifolds W^{s}(v) and W^{u}(v) consist of the unit normals to H^{s}(v) and H^{u}(v) that point to the same side as v. If v̂ ∈ M̂ is a frame with first vector v, W^{s}(v̂) and W^{u}(v̂) are lifts to M̂ of W^{s}(v) and W^{u}(v). The element of W^{s}(v̂) that projects to a vector w ∈ W^{s}(v) is the unique frame ŵ with first component w such that ϕ̂_{t}(v̂) and ϕ̂_{t}(ŵ) converge exponentially as t → ∞. One can think of ŵ as being obtained from v̂ by parallel translating v̂ to infinity along the geodesic determined by v and then parallel translating back again along the geodesic determined by w. We shall call this operation on frames horospherical translation. We point out that even when V is a surface of constant negative curvature, horospherical translation is not defined by parallel translation along an arc joining the footpoints of v and w in H^{s}(v). Similar remarks apply to the unstable manifold. Moving a frame along a us-path amounts to performing a sequence of horospherical translations.

It is easily seen that the time-one maps of geodesic flows and frame flows are dynamically coherent with smooth central foliations. In the case of the time-one map of the geodesic flow for a manifold with negative curvature (as for the time-one map of any Anosov flow), the central foliation is the orbit foliation. For the time-one map of the frame flow, the central foliation is the complete lift of the orbit foliation by the projection ρ : M̂ → M. In other words, a leaf of the central foliation consists of all frames whose first components lie on a single orbit of the geodesic flow.

For both the geodesic flows and frame flows, the derivative of the time-one map acts as an isometry on E^{c}, which automatically implies that the time-one map is centre bunched. Proofs that the time-one map of the geodesic flow for a manifold with negative curvature is stably accessible can be found in [KK] and [BPW]. Proving that the timeone map of the frame flow is stably accessible under the conditions (1) – (6) in the Introduction is our main task in the present paper.

2. The Brin groups and ergodicity of frame flows

We now want to recall a useful tool for studying ergodicity of frame flows, originally introduced by Brin. Suppose that we have a closed us-path for the time-one map of the geodesic flow that begins and ends at a vector v. There is an associated sequence of horospherical translations that moves any frame with first component v around the path and brings it back to a new frame with first component v. This process defines an element of SO(n − 1) that acts on the orthogonal complement of v; it depends only on the path and not which frame with first component v we consider. The set of all elements of SO(n − 1) that arise in this way forms a subgroup of SO(n − 1) that we shall denote by H_{v}. Since the groups H_{v} and H_{w} are conjugate for any unit vectors v and w, and the properties that we are interested in depend only on the conjugacy class of the group, we shall henceforth omit the subscript v.

Let H^{0} be the subgroup of H consisting of the elements defined by us-paths that are null homotopic. It is not difficult to show that H^{0} is path connected and hence a Lie subgroup by the theorem of Kuranishi and Yamabe. One then sees that H is also a Lie subgroup and H^{0} is the connected component of the identity in H; see Proposition 8.1 in [BW] for a more detailed exposition in a closely related situation.

The groups H^{0} ⊂ H ⊂ \overline{H} ⊂ SO(n − 1) will be called the Brin groups. They are intimately connected with the ergodic properties of the time-one map f̂ of the frame flow. Brin proved the following elegant result (Proposition 3 in [Br2]):

Proposition 2.1. f̂ has the K-property if and only if \overline{H} = SO(n − 1).

An immediate consequence is

Corollary 2.2. If \overline{H} = SO(n − 1), then the frame flow ϕ̂_{t} is ergodic.

In sections 5–8 we shall prove a partial generalization of Brin’s result:

Proposition 2.3. f̂ is stably K if H = SO(n − 1).

Proposition 2.3 is analogous to Theorem 9.1 of [BW], where the condition is expressed in terms of H^{0}. Since SO(n − 1) is connected, H = SO(n − 1) is equivalent to H^{0} = SO(n − 1).

Theorem 0.2 follows immediately from Proposition 2.3 and the next result.

Proposition 2.4. H = SO(n − 1) in each of the cases (1) – (6) listed in the Introduction.

Proof. Cases (1), (5) and (6) will be considered in the next two sections. Case (2). Brin’s proof that \overline{H} = SO(n−1) for an open and dense set of metrics actually gives H^{0} = SO(n − 1). See Proposition 1 in [Br2]. Cases (3) and (4). It was shown in [BG] and [BK] that \overline{H} = SO(n − 1) in these cases. In fact they obtain H = SO(n−1). We outline the argument. In addition to the bundle M̂ over M one needs to consider the intermediate bundle M^{⊥} whose fibre over a unit vector v ∈ M consists of the unit vectors orthogonal to v. Thus M^{⊥} has fibre which can be identified with S^{n−2} and there is a natural fibre preserving projection from M̂ to M^{⊥} that maps a frame to its second component. By restricting the action of the group H on the fibre of M̂ to the second component of the frames, one obtains an action of H on the sphere S^{n−2}. It is shown in [BK] that this action is transitive whenever the pinching condition λ/Λ > 0.93 … is satisfied. On the other hand, Brin and Gromov observe in [BG] that, by results of Oniščik [O], if n is even and n ≠ 8, then the only Lie subgroup of SO(n − 1) that can act transitively on S^{n−2} is SO(n − 1) itself. Hence H = SO(n − 1) in case (4).

Brin and Gromov also observe that the structure groups of the bundles M̂ → M and M^{⊥} → M can be reduced to H and that, for odd n, the structure group of the latter bundle cannot be reduced to a subgroup which is not transitive on the fibre S^{n−2}. It follows that H and hence H^{0} must act transitively on S^{n−2} when n is odd. Finally they prove, using Oniščik’s results, that if n is odd and n ≠ 7, then the structure group of M̂ → M cannot be reduced to a proper subgroup of SO(n − 1) which acts transitively on S^{n−2}. Hence H = SO(n − 1) in case (3).

3. Parallel translation around ideal triangles in constant curvature

The case of constant curvature manifolds is amenable to a particularly straightforward geometric analysis described in [BK]. Let V be an n-dimensional compact manifold with all sectional curvatures equal to −1. We can identify the universal cover Ṽ of V with the n-dimensional disk 𝔻^{n} = {x = (x_{1}, …, x_{n}) ∈ ℝ^{n} : ∥x∥ < 1}. The Riemannian metric on V lifts to the usual Poincaré metric on 𝔻^{n} given by

ds^{2} = 4\frac{dx_{1}^{2} + ⋯ + dx_{n}^{2}}{(1 − ∥x∥^{2})^{2}}

with ∥x∥^{2} = x_{1}^{2} + ⋯ + x_{n}^{2}. Horospheres in V lift to n − 1 dimensional euclidean spheres H in 𝔻^{n} which are tangent to the unit sphere S^{n−1} at a single point, i.e.

H = {x ∈ ℝ^{n} : ∥x − αv∥ = 1 − α}

for some v ∈ S^{n−1} and some α ∈ (0, 1). The leaves of the foliations W^{s} and W^{u} lift to the unit tangent vectors orthogonal to these horocycles (inwardly pointing for W^{s}, and outwardly pointing for W^{u}). It is easily seen by considering the upper half plane model of hyperbolic geometry that the horospheres (with the metric induced by the Poincaré metric) are isometric to ℝ^{n−1} and horospherical translation is the same as translation in this Euclidean structure.

We denote by e_{1}, …, e_{n} the basis vectors in ℝ^{n}. We can then consider the natural reference frame F_{0} based at the centre \underline{0} = (0, …, 0) of the Poincaré disk, with the first vector in the frame being e_{1}. Consider the strong stable and unstable manifolds (on 𝔻^{n}) associated respectively to the horospheres

H_{1} = \{x ∈ ℝ^{n} : ∥x − \frac{1}{2}e_{1}∥_{2} = \frac{1}{2}\} and H_{2} = \{x ∈ ℝ^{n} : ∥x + \frac{1}{2}e_{1}∥_{2} = \frac{1}{2}\} .

Figure 2. Horospherical transport of frames in constant curvature

For any u that lies in the intersection of S^{n−1} with the hyperplane orthogonal to e_{1}, we can introduce a third horosphere

H_{3} = \{x ∈ ℝ^{n} : ∥x − \frac{2}{3}u∥ = \frac{1}{3}\}

which has the property of being tangent to both H_{1} and H_{2}.

Following Brin and Karcher, one can first horospherically translate the reference frame F_{0} along the horocycle H_{1} (corresponding to an unstable manifold) until it meets the point where H_{1} touchs H_{3}. Next one horospherically translates the resulting frame along the horosphere H_{3} (corresponding to a stable manifold) until it meets the point where H_{3} touchs H_{2}. Finally one horospherically translates the resulting frame along the horosphere H_{2} (corresponding to a stable manifold) until it returns to the original base point \underline{0} ∈ 𝔻^{n}. The composition of these three horospherical translations can be viewed as parallel translation of F_{0} around the ideal triangle whose vertices are e_{1}(∞), e_{1}(−∞) and u(∞). The frame F_{1} that is obtained by parallel translating F_{0} around this ideal triangle is simply the frame based at \underline{0} that is obtained by reflecting F_{0} in the plane spanned by e_{1} and u. We can repeat the procedure with u replaced by another unit vector v that is orthogonal to e_{1}. If F_{1} is parallel translated around the ideal triangle with vertices e_{1}(∞), e_{1}(−∞) and v(∞) the resulting frame F_{2} will be based at \underline{0} and will differ from F_{0} only by a rotation through angle 2∢(u, v) in the two dimensional plane spanned by u and v; the orthogonal complement of this plane will be fixed.

Since parallel translation along an edge of an ideal triangle is a horospherical translation within either a stable or an unstable leaf, it follows that any rotation of the type considered above must belong to the group H. In particular the rotation R_{θ}^{i,j} through angle θ in the plane spanned by e_{i} and e_{j} belongs to H for 2 ≤ i < j ≤ n. Since the elements R_{θ}^{i,j} generate SO(n − 1), we see that H = SO(n − 1).

It is not difficult to show that the set of metrics for which H = SO(n − 1) is open, and thus there is a neighbourhood of each constant curvature metric in which H = SO(n − 1). But this does not imply that there is a pinching assumption which implies that H = SO(n − 1), because we do not know a priori that there is a uniform estimate on the size of the neighbourhood of a metric of constant negative curvature in which H = SO(n − 1). We show in the next section that for each dimension there is a pinching constant l_{n} such that if λ/Λ > l_{n}, then the group H contains an open subset of SO(n − 1) and hence H = SO(n − 1).

4. Comparison of constant and variable curvature

We begin with some simple remarks about SO(n − 1); see e.g. [Ga, Ch. IX, §13]. If g_{1}, g_{2} ∈ SO(n − 1), let

∢(g_{1}, g_{2}) = sup_{∥u∥=1} ∢(g_{1}(u), g_{2}(u)).

Up to scaling ∢(g_{1}, g_{2}) is the distance between g_{1} and g_{2} in the metric induced by the Killing form on 𝔰𝔬(n − 1). For each g ∈ SO(n − 1), there is an orthogonal direct sum decomposition

ℝ^{n−1} = E_{1}(g) ⊕ E_{−1}(g) ⊕ F_{1}(g) ⊕ ⋯ ⊕ F_{k(g)}(g),

in which E_{1}(g) and E_{−1}(g) are the eigenspaces for 1 and −1 respectively, and each F_{i}(g) is a g-invariant plane on which g acts as rotation through an angle α_{i}(g) ∈ (0, π) with respect to one of the two possible orientations of F_{i}(g). This decomposition is unique, up to the order of the F_{i}(g). The dimension of E_{−1}(g) is even and E_{−1}(g) can be expressed as a direct sum of invariant planes with g acting as rotation by π in each of them. We see that every element of SO(n − 1) can be expressed as the product of at most

k_{n−1} =^{def} ⌊\frac{n − 1}{2}⌋

rotations, each of which fixes all elements of the (n − 3)-dimensional complement of the plane in which it acts, i.e., k(g) ≤ k_{n−1} for all g ∈ SO(n − 1). Since a rotation in the plane is the product of two reflections, we see that every element of SO(n − 1) is the product of at most 2k_{n−1} reflections, each of which reverses the sign of a unit vector and fixes all unit vectors in its orthogonal complement.

If dim E_{−1} ≥ 2, then ∢(g, id) = π; otherwise

∢(g, id) = max_{1≤i≤k(g)} α_{i}(g) < π.

If ∢(g, id) < π, we can define g_{t} for 0 ≤ t ≤ 1 so that all the g_{t} have the same splitting and α_{i}(g_{t}) = (1−t)α_{i}(g). The path α_{t} is the shortest geodesic from g to id in the metric defined by the Killing form. More generally any g′, g′′ ∈ SO(n−1) with ∢(g, g′) < π are joined by a unique shortest geodesic γ(t). Indeed if g = g′(g′′)^{−1}, then γ(t) = g_{t}g′′ for 0 ≤ t ≤ 1, where g_{t} is the path described above. We see that {g ∈ SO(n − 1) : ∢(g, id) < θ} is contractible for any θ ≤ π, in particular θ = π/2. It follows that {g ∈ SO(n − 1) : ∢(g, γ) < π/2} is contractible for any γ ∈ SO(n − 1).

We now compare parallel translation around ideal triangles in manifolds with variable and constant negative curvature. Let Ṽ be the universal cover of a compact manifold V whose sectional curvature K satisfies the pinching condition −Λ^{2} < K < −λ^{2}. Suppose F = (e_{1}, …, e_{n}) is a frame at a point p ∈ Ṽ and u ∈ T_{p}Ṽ is a unit vector orthogonal to e_{1}. Just as in the hyperbolic case, we can define parallel translation of F around the ideal triangle e_{1}(∞), e_{1}(−∞), u(∞) as the composition of horospherical translations for three horospheres that are pairwise tangent and orthogonal to the sides of the triangle. This parallel translation will map e_{1} to −e_{1} and acts on the orthogonal complement e_{1}^{⊥} of e_{1} as a linear isometry, which we shall denote by P_{u}. Let R_{u} be the reflection of e_{1}^{⊥} that reverses the sign of u and fixes all vectors in e_{1}^{⊥} that are orthogonal to u. We saw above that P_{u} = R_{u} when the curvature is constant. Brin and Karcher showed in [BK] how one can use pinching of the curvature to control the difference between P_{u} and R_{u}. Using their results we can prove the following:

Lemma 4.1. There is a constant l_{n} that depends only on the dimension (and not on V or F) such that if λ/Λ > l_{n}, then

∢(P_{u}(v), R_{u}(v)) < \frac{π}{4k_{n−1}}

for all unit vectors u, v ∈ e_{1}^{⊥}.

Before proving the lemma, we use it to show that H = SO(n − 1) when λ/Λ > l_{n}. The idea is to show that H contains an open subset of SO(n − 1). We fix a frame F = (e_{1}, …, e_{n}) and think of SO(n − 1) acting on e_{1}^{⊥}. Recall that every element of SO(n − 1) is the product of at most 2k_{n−1} reflections. Let U_{n−1} denote the set of all (2k_{n−1})-tuples u = (u_{1}, …, u_{2k_{n−1}}) of unit vectors in e_{1}^{⊥}. For u ∈ U_{n−1}, let R_{u} = R_{u_{2k_{n−1}}} ∘ ⋯ ∘ R_{u_{1}} and P_{u} = P_{u_{2k_{n−1}}} ∘ ⋯ ∘ P_{u_{1}}.

Corollary 4.2. There is ϵ > 0 such that for any u ∈ U_{n−1} we have

∢(R_{u}(v), P_{u}(v)) < \frac{π}{2} − ϵ

for all unit vectors v ∈ e_{1}^{⊥}.

Proof. By Lemma 4.1, ∢(R_{u}(v), P_{u}(v)) < 2k_{n−1} · π/4k_{n−1} = π for all u and all v. Now use compactness.

Choose γ ∈ SO(n − 1) such that k(γ) = k_{n−1} and α_{i}(γ) = π/2 for 1 ≤ i ≤ k_{n−1}. Our goal is to show that H contains a neighbourhood of γ from which it follows immediately that H = SO(n − 1).

Let

Δ = { g ∈ SO(n − 1) : ∢(g, γ) < π/2 }.

We know from the remarks at the beginning of the section that Δ is contractible. We now show that k(g) = k_{n−1} for any g ∈ Δ. This is equivalent to showing that the eigenspce E_{−1}(g) is trivial and that E_{1} has dimension one if n − 1 is odd and is trivial if n − 1 is even. Observe that

∢(γ(u), u) = π/2

for all unit vectors u ∈ F (γ) =^{def} F_{1}(γ) ⊕ ⋯ ⊕ F_{k_{n−1}}(γ). If g ∈ Δ and u is a unit vector in F (γ), then g(u) ≠ ±u, since ∢(u, γ(u)) = π/2 and ∢(γ(u), g(u)) < π/2. If n − 1 is even, F (γ) = ℝ^{n−1} and hence E_{1} and E_{−1} are both trivial if n − 1 is even. If n − 1 is odd, F (γ) has codimension one and hence E_{1} and E_{−1} both have dimension at most one in this case. Since E_{1} has dimension at least one when n − 1 is odd and E_{−1} is always even dimensional, it follows that E_{1} is one dimensional and E_{−1} is trivial when n − 1 is odd. Hence k(g) = k_{n−1} for all g ∈ Δ.

For each g ∈ Δ, let

U(g) = { u ∈ U_{n−1} : u_{2i−1}, u_{2i} ∈ F_{i}(g) and 2∢(u_{2i−1}, u_{2i}) = α_{i} }.

Then R_{u} = g for all u ∈ U(g). Let U(Δ) = ⋃_{g∈Δ} U(g). Observe that U(Δ) has a natural fibre bundle structure. The fibre consists of k_{n−1}! copies of the torus T^{k_{n−1}}; there is one copy for each possible ordering of the k_{n−1} spaces F_{i}(g). Since the base of U(Δ) is contractible, there is a continuous section Γ : Δ → U(Δ).

We have R_{Γ(g)} = g for each g ∈ Δ. Let P (g) = P_{Γ(g)} for g ∈ Δ. Set Δ̂ = {g ∈ Δ : ∢(g, γ) > π/2 − ϵ/2} and ŜO(n − 1) = {g ∈ SO(n − 1) : ∢(g, γ) > ϵ/2}, where ϵ is the constant from Corollary 4.2. For each g ∈ Δ we have ∢(P (g), g) < π/2 − ϵ. If g ∈ Δ̂, then the short geodesic in SO(n − 1) from g to P (g) lies in ŜO(n − 1). We see that

P : (Δ, Δ̂) → (SO(n − 1), ŜO(n − 1))

and there is a homotopy from P to the identity through mappings of (Δ, Δ̂) into (SO(n − 1), ŜO(n−1)). It follows that P (Δ) contains the open set {g ∈ SO(n−1) : ∢(g, γ) < ϵ/2}. Since P (Δ) ⊂ H, we must have H = SO(n − 1).

We now turn to the proof of Lemma 4.1. For manifolds with sectional curvatures close to −1, Brin and Karcher used a perturbation argument to estimate ∢P_{u}(v), R_{u}(v). For convenience, set G = log((\sqrt{5} + 1)/2) and given l > 0 define the following technical quantities:

a(l) = 6(\sqrt{1 + l^{2}}/\sqrt{2} − l) log (cosh[(1/l)G]) ,
b(l) = 6 sin^{−1} (( cosh[(2/l)G] − exp [(3/2 − 1/l) log 2
+ \frac{1}{l} log[cosh[(l − 1/2) log 2]]]) / sinh[(2/l)G]) ,
c(l) = \frac{3}{5}π((1/l)^{2} − 1),
e(l) = 12(\sqrt{1 + (1/l)^{2}}/\sqrt{2} − l)G, and
f(l) = 12(\sqrt{1 + (1/l)^{2}}/\sqrt{2} − 1)(2\sqrt{1 + (1/l)^{2}}/\sqrt{2} − 1)
× exp [(\sqrt{1 + (1/l)^{2}}/\sqrt{2}) log[cosh[G/l]]]G^{2}.

The following gives explicit upper bounds on the difference between changes in the frame F by parallel transporting around the an ideal triangle in the constant curvature case (as above) and in the case of variable curvature.

Proposition 4.3 (Brin-Karcher). Comparing parallel transport around the horocycle triangle for the cases of K = −1 and −Λ^{2} ≤ K ≤ −λ^{2}, we set l = λ/Λ and then:

(1) ∢P_{u}(v), R_{u}(v) ≤ d(l) := a(l) + b(l) + c(l) if v ∈ span{e_{1}, e_{i}}; and

(2) ∢P_{u}(v), R_{u}(v) ≤ g(l) := b(l) + c(l) + e(l) + f(l) if v ⊥ span{e_{1}, e_{i}}.

Thus the conclusion of Lemma 4.1 will hold if l is chosen so that

d(l) < \frac{π}{4k_{n−1}}  and  g(l) < \frac{π}{4k_{n−1}} .

Since d and g are decreasing functions of l, it suffices to take the maximum of the solutions to d(l′_{n}) = π/4k_{n−1} and g(l′′_{n}) = π/4k_{n−1}. Numerical results obtained using Mathematica can be found in Table 1.

n  \frac{π}{4k_{n−1}}  l′_{n}  l′′_{n}  max\{l′_{n}, l′′_{n}\}
3 or 4  0.78539  0.95369  0.97090  0.97090
5 or 6  0.39269  0.97675  0.98537  0.98537
7 or 8  0.26179  0.98448  0.99023  0.99023
9 or 10  0.19635  0.98835  0.99267  0.99267
11 or 12  0.15708  0.99068  0.99413  0.99413

Table 1. Dimension and the pinching bound

Combing the results for n = 7, n = 8 with the estimates of Brin and Karcher (cf. Introduction) yields the following.

Proposition 4.4. If the sectional curvatures of V are pinched between −1 and −0.99023, then the Brin transitivity group H satisfies H = SO(n − 1).

5. Centre engulfing and stable accessibility

As we have observed in Section 2, the hypotheses of centre bunching and dynamical coherence in Theorem 1.1 are automatically valid for the time-one map of the frame flow. In order to apply Theorem 1.1 it remains to establish the third property of stable accessibility. A practical approach to showing stable accessibility for a dynamically coherent partially hyperbolic diffeomorphism f_{0} : N → N involves showing that the us-paths can be arranged to vary continuously (at least locally in N) as f is changed to a nearby diffeomorphism. To formalise this, it is convenient to use the notion of centre engulfing, as introduced by Burns, Pugh and Wilkinson [BPW, BW].

Given a partially hyperbolic diffeomorphism f_{0} : N → N, let us denote by d the dimension of the centre bundle E^{c}. The diffeomorphism f_{0} : N → N is centre engulfing at a point x ∈ N if:

(1) there is a continuous family of us-paths ψ_{z} : [0, 1] → N indexed by points z ∈ Z, where Z is a compact d-dimensional manifold with boundary;

(2) the number of legs in ψ_{z} is uniformly bounded;

(3) each path ψ_{z} begins at x and ends in the centre manifold W^{c}(x) through x (i.e. ψ_{z}(0) = x and ψ_{z}(1) ∈ W^{c}(x) for all z ∈ Z);

(4) the paths ψ_{z} corresponding to z ∈ ∂Z do not end at x; and finally

(5) the map (Z, ∂Z) → (W^{c}(x), W^{c}(x) − {x}) defined by z ↦ ψ_{z}(1) wraps around x non-trivially (i.e. the map has non-zero degree in that the induced map on homology H_{d}(Z, ∂Z, ℤ) = ℤ to H_{d}(W^{c}(x), W^{c}(x) − {x}, ℤ) = ℤ is nonzero).

Observe that these five conditions are open under C^{1}-small perturbations of f_{0}.

Figure 3. Centre engulfing

In particular, these five conditions imply that the set of endpoints of the paths contains a neighbourhood of x in W^{c}(x), i.e. that x lies in the interior (as a subset of W^{c}(x)) of U = {ψ_{z}(1) : z ∈ Z}. Moreover, assumptions (3) and (4) imply that the ends of the paths genuinely “wrap around” a neighbourhood of x in the centre manifold.

The following criterion of Burns and Wilkinson is useful in establishing stable accessibility.

Proposition 5.1. [BW, Corollary 5.3] If f_{0} : N → N is centrally engulfing at one point and it is possible to join any point of N to x by a us-path, then f_{0} is stably accessible.

By virtue of Proposition 5.1, it suffices for the proof of Proposition 2.3 to show centre engulfing under the condition H = SO(n − 1). In order to simplify matters, we introduce in the next section an intermediate step of first showing the analogous statement for the fibres of SO(n − 1).

6. Fibre engulfing and centre engulfing

It is convenient of introduce a restricted version of centre engulfing as the first step in the proof of Proposition 2.3. Let d = dim(SO(n − 1)) + 1, so d is the dimension of the centre bundle for the the time-one map f : M̂ → M̂. We shall say that f : M̂ → M̂ is fibre-engulfing at a frame v̂ ∈ M̂ with first element v ∈ M if

(i) there is a continuous family of us-paths ψ_{z} : [0, 1] → M̂ indexed by z ∈ Z, where Z is a compact (d − 1)-dimensional manifold with boundary;

(ii) the number of legs in ψ_{z} is uniformly bounded; (iii) each path begins at v̂ and ends in the fibre M̂_{v} of M̂ containing v̂ (i.e. ψ_{z}(0) = v̂

and ψ_{z}(1) ∈ M̂_{v} for all z ∈ Z); (iv) the paths corresponding to z ∈ ∂Z do not end at v̂;

(v) the map (Z, ∂Z) → (M̂_{v}, M̂_{v} − {v̂}) defined by z ↦ ψ_{z}(1) wraps around v̂ nontrivially (i.e. the map has non-zero degree in that the induced map on homology H_{d−1}(Z, ∂Z, ℤ) = ℤ to H_{d−1}(M̂_{v}, M̂_{v} − {v̂}, ℤ) = ℤ is nonzero).

Compare with [BW, §9]. This is almost exactly the same as the definition of centre engulfing in the previous section, except that the ends of the paths are additionally required to lie in the fibre of M̂ containing v̂ (which is codimension one in the leaf of the central foliation W^{c} that contains v̂).

If f is fibre engulfing at v̂, then f is also fibre engulfing at every frame in W^{c}(v̂). Indeed if ŵ ∈ W^{c}(v̂), there are τ ∈ ℝ and g ∈ SO(n − 1) such that ŵ = ϕ̂_{τ} (g · v̂); and if ψ_{z} is the family of us-paths considered above, the family of us-paths indexed by z,

t ↦ ϕ̂_{τ} (g · ψ_{z}(t)),

defines a central engulfing at ŵ.

The choice of τ and g is unique unless the first vector v of v̂ is periodic under the geodesic flow ϕ_{t}. In all cases, the correspondence between ŵ and (τ, g) defines a diffeomorphism between W_{0}^{c}(v̂) and (−τ_{0}, τ_{0}) × SO(n − 1), where τ_{0} is half the least period of v under the geodesic flow and W_{0}^{c}(v̂) is the set of all frames whose first vector is ϕ_{τ} (v) with |τ| < τ_{0}.

For ŵ ∈ W_{0}^{c}(v̂), we define the family of us-paths Ψ_{ŵ,z} : [0, 1] → M̂ by

Ψ_{ŵ,z}(t) = ϕ̂_{τ} (g · ψ_{z}(t)),

where |τ| < τ_{0} and ŵ = ϕ̂_{τ} (g · v̂). This family depends continuously on both z and ŵ.

Proposition 6.1. If the time one map of the frame flow is fibre engulfing at v̂ ∈ M̂, then it is also centre engulfing at v̂.

Proof. Let v be the first element of v̂. The arguments in [KK] and [BPW] show that the time one map ϕ_{1} of the geodesic flow on M = SV is centrally engulfing at v. Since the centre leaf of v is its orbit under the geodesic flow, this means that we can choose a continuous family of us-paths α_{ζ} : [0, 1] → M, indexed by ζ ∈ [−1, 1], and a small δ > 0 such that

• α_{ζ}(0) = v and α_{ζ}(1) ∈ ϕ_{[−δ,δ]}(v) for each ζ;

• α_{−1}(1) and α_{1}(1) are not equal to v and lie on opposite sides of v in the orbit

segment ϕ_{[−δ,δ]}(v). The idea in these arguments is to first choose a short us-path with four legs, whose types are usus, that starts at v and ends in the orbit of v under the geodesic flow, close to but not at v. Such a path must exist unless the stable and unstable folations for ϕ_{1} are integrable in a neighbourhood of v; and these foliations are nowhere integrable because the geodesic flow of manifold with negative curature is a contact Anosov flow. There must also be a short us-path with four legs, whose types are susu, that starts at v and ends in the orbit of v close to but not at v. The endpoints of the two paths will be on opposite sides of v. One can then continuously shorten the legs of the two paths, keeping their endpoints in the orbit of v, until the paths become the constant path at v.

Figure 4. The paths α_{ζ}.

For each ζ ∈ [−1, 1], the us-path α_{ζ} for ϕ_{1} lifts to a us-path α̂_{ζ} : [0, 1] → M̂ for the time one map f = ϕ̂_{1} of the frame flow. The path α̂_{ζ} begins at v̂ and ends in the fibre of M̂ containing α_{ζ}(1). The frame α_{ζ}(1) may not lie on the frame flow orbit of v̂, but it does lie in the fibre of M̂ that passes through a point of the orbit close to v̂; and the frames in the orbit corresponding to α̂(−1) and α̂(1) lie on opposite sides of v̂ in the orbit.

By hypothesis there is a fibre engulfing at v̂ defined by a family ψ_{z} of us-paths indexed by z ∈ Z, where Z is a (d − 1)-dimensional manifold with boundary. Let Ψ_{ŵ,z} be the family of us-paths defined in the remarks earlier in this section.

In order to define a centre-engulfing at v̂ we want to concatenate each path α̂_{ζ} with the fibre engulfing at the frame α̂_{ζ}(1) ∈ W^{c}(v̂). This is achieved by defining the family of us-paths Φ_{(ζ,z)} : [0, 1] → M̂ indexed by (ζ, z) ∈ [−1, 1] × Z as follows:

Φ_{(ζ,z)}(t) = \{ \begin{matrix}α̂_{ζ}(t) & if 0 ≤ t ≤ 1/2, \\ Ψ_{α̂_{ζ}(1),z}(2t − 1) & if 1/2 ≤ t ≤ 1. \end{matrix}

It is easy to see that this family defines a center engulfing at v̂.

7. Fibre engulfing in the constant curvature case

In Section 3 we saw that we were fortunate enough to have in the case of manifolds of constant negative curvature a very explicit understanding of the stable and unstable manifolds, and the nature of the effect of parallel transporting frames. In this section we observe that fibre engulfing is easily established for frame flows on such manifolds. It then follows immediately from Proposition 6.1, Proposition 5.1 and Theorem 1.1 that time-one frame flows on manifolds of constant negative sectional curvature are stably ergodic. This is weaker than the statement of Theorem 1, in that it does not have quantifiable pinching estimates, and the proof of the complete statement requires results from the next section. However, this does provide a rather transparent example where fibre engulfing (and centre engulfing) can be easily seen.

Proposition 7.1. Let ϕ̂: M̂ → M̂ be the frame flow on a manifold with constant negative sectional curvatures. Then the time-one map is fibre engulfing.

Proof. The group SO(n−1) corresponds to orientation preserving isometries on the sphere S^{n−1}. This d = n(n − 1)/2 dimensional group is generated by rotations R_{θ}^{i.j} in each of the planes span(e_{i}, e_{j}) (2 ≤ i < j ≤ n) where e_{1}, …, e_{n} are the standard basis vectors for ℝ^{n}.

As recalled in section 3, we can realize R_{θ}^{i,j} by parallel transporting the reference frame around a us-path consisting of 6 components. We can define a map [−δ, δ]^{d} → SO(n − 1) by

(θ_{1}, θ_{2}, …, θ_{d−1}, θ_{d}) ↦ \underbrace{R_{θ_{1}}^{2,3}R_{θ_{2}}^{2,4} ⋯ R_{θ_{1}}^{n−2,n}R_{θ_{1}}^{n−1,n}}_{Group multiplication in SO(n−1)}

and for δ > 0 sufficiently small this is a fibre engulfing, as required. This completes the proof of Proposition 7.1.

8. Fibre engulfing when H = SO(n − 1)

In this section we show fibre engulfing under the more general hypothesis that the Brin transitivity group H equals all of SO(n − 1). We need the following modified version of a result of Burns and Wilkinson (Theorem 9.1 in [BW]).

Proposition 8.1. If H = SO(n−1), then the time-one map of the frame flow ϕ̂: M̂ → M̂ is fibre engulfing.

Proof. The proof of this result is very similar to that of Theorem 9.1 of [BW, Theorem 9.1], which is the analogous result in the context of skew products. For completeness we outline the main steps in the context of the frame flow.

To show that the time-one map is fibre engulfing we need to construct for each x ∈ M̂ a family of us-paths with the properties (i)–(v) described at the beginning of §5. We identify the fibre of M̂ containing x with SO(n − 1) in such a way that the identity e ∈ SO(n − 1) corresponds to x. The strategy of the proof is to choose a basis ξ_{1}, …, ξ_{d−1} in the Lie algebra 𝔰𝔬(n − 1) and to consider the map ψ^{(1)} : [−1, 1] → SO(n − 1) defined by

ψ^{(1)} : z = (z_{1}, …, z_{d−1}) ↦ exp(z_{1}ξ_{1}) ⋯ exp(z_{d−1}ξ_{d−1}).

We would like ψ^{(1)} to be the “endpoint” map z ↦ ψ_{z}(1) corresponding to a family of us-paths indexed by z ∈ [−1, 1]^{d−1} that satisfies properties (i)–(iii) from the definition of fiber engulfing. We call ψ^{(1)} achievable if this is the case.

If ψ^{(1)} is achievable, we are done, since it is obvious that the remaining properties (iv) and (v) that are required for the family of us-paths ψ_{z} to be a fibre engulfing will also hold if we restrict z to [−δ, δ]^{−1} for a small enough δ. Instead of ψ^{(1)} being achievable, it is enough if ψ^{(1)} is approximable, i.e. ψ^{(1)} can be approximated arbitrarily closely in the uniform topology by achievable maps.

Clearly, the notion of approximability or achievability can be introduced for maps into SO(n − 1) of any parameter space, in particular [−1, 1]. Observe that the product of a family of maps θ_{i} : [−1, 1] → SO(n − 1) for i = 1, …, d − 1 defined by

(z_{1}, …, z_{d−1}) ↦ θ_{1}(z_{1}) ⋯ θ_{d−1}(z_{d−1})

is again achievable or approximable if each of the maps θ_{i} has the relevant property.

It thus suffices to show that each of the maps z ↦ exp(zξ_{i}) is approximable. In order to do this, we call a vector ξ ∈ 𝔰𝔬(n − 1) approximable if the map from [−1, 1] to SO(n − 1) defined by t ↦ exp tξ is approximable. Observe that for any approximable vector ξ and any λ ∈ ℝ the vector λξ is again approximable. Moreover, one can show that if ξ and η are approximable vectors, then so are ξ + η and [ξ, η] (cf. [BW, Lemma 9.7]). Let 𝔥_{app} ⊂ SO(n − 1) denote the Lie subalgebra of approximable vectors, and let H_{app} = exp 𝔥_{app} ⊂ SO(n − 1) be the associated Lie subgroup.

It remains to show that H_{app} = SO(n−1). Assume for a contradiction that there exists g ∈ SO(n − 1) − H_{app}. Our assumption that H = SO(n − 1) means that there exists a us-path in M̂ that begins at the point x ∈ M̂ around which we are trying to construct a fibre engulfing and ends at the point y in the fibre of M̂ through x which corresponds to g under the identification of the fibre with SO(n − 1) and x with e which was introduced above.

Since SO(n−1) is connected, H = SO(n−1) entails H^{0} = SO(n−1). This means that we can choose the us-path from x to y so that its projection to M is contractible. Call this us-path c_{1} and let c_{0} denote the constant path from x to itself. Then c_{0} and c_{1} are homotopic. As in [BW] it is not difficult to show that this homotppy can be performed through us-paths; cf. Proposition 7.2 of [BW]. Let c_{s} be a homotopy, indexed by s ∈ [0, 1], from c_{0} to c_{1} through us-paths in M̂ which begin at x and end in the fibre of x. The endpoints of this one parameter family of us-paths form a curve from x to y in the fibre of x in M̂, which corresponds under our identification to a curve α : [0, 1] → SO(n − 1).

Since α(0) = e ∈ H_{app} and α(1) ∉ H_{app}, we can find an interval [s_{0}, s_{1}] ⊂ [0, 1] such that α(s_{0}) ∈ H_{app} and α(s) ∉ H_{app} for s_{0} < s ≤ s_{1}. For s ∈ (s_{0}, s_{1}], let β be the point in H_{app} that is closest to α(s) in the canonical binvariant metric induced on SO(n − 1) by the Cartan-Killing form. Let ξ(s) be the unit vector at β(s) that points towards α(s) and η(s) the unit vector at β(s) that points towards α(s_{0}). Then ξ(s) is orthogonal to 𝔥_{app}. Let ξ be a unit vector at α(s_{0}) that is the limit of ξ(s_{k}) for a sequence s_{k} ↘ s_{0}. Then ξ is orthogonal to 𝔥_{app}.

We now want to show that ξ is approximable, this will be the desired contradiction. Let us define an approximable curve in the obvious way as a continuous map from a closed interval into SO(n − 1) that is approximable. Concatenations of approximable curves are approximable and so is any curve obtained by restricting an approximable curve to a subinterval in its domain. A curve obtained by translating an approximable curve by multiplying it from the right by a fixed element of SO(n − 1) is approximable. Any curve in H_{app} is approximable, as is any achievable curve.

We see that for each s ∈ (s_{0}, s_{1}] there is an approximable curve σ_{s} from β(s) to α(s), which consists of the shortest geodesic from β(s) to α(s_{0}) followed by the segment of α from α(s_{0}) to β(s). The first half of σ_{s} lies in H_{app} and the second half is part of an approximable curve.

Now observe that we can approximate the geodesic t ↦ exp tξ(s) in the C^{0} topology with a concatenation of right translates of σ_{s}. Since the length of σ_{s} approaches 0 as s ↘ s_{0}, this approximation becomes better and better as s ↘ s_{0}. We then see that the geodesic t ↦ exp tξ is apporximated arbitrarily closely in the C^{0} topology by approximable curves, which themselves are approximated arbitrarily well by achievable curves. It follows that t ↦ exp tξ(s) is approximable and hence ξ ∈ 𝔥_{app}, which is absurd since we saw above that ξ is orthogonal to 𝔥_{app}.

This completes the outline proof of Proposition 8.1.

To complete the proof of Proposition 2.3, we observe that for frame flows satisfying the hypotheses of the theorem we have that G = SO(n − 1). By Proposition 8.1 above, we see that the time one maps of such frame flows are fibre engulfing, and thus by Proposition 6.1 also centre engulfing. By Proposition 5.1 we deduce that the time one maps of these frame flows are stably accessible and, finally, by Theorem 1.1 that they are stably ergodic.

9. k-frame flows

In this section we briefly recall the study of Brin and his coauthors on k-frame flows. We begin by defining the k-frame flows for 1 ≤ k ≤ n = dim V. A k-frame on V is an ordered k-tuple of orthonormal vectors in a single tangent space to V. We denote the space of all kframes by M^{(k)}. The k-frame flow ϕ_{t}^{(k)} on M^{(k)} acts as the geodesic flow on the first vector of a frame and parallel translates the remaining vectors along the geodesic determined by the first vector. Thus M^{(1)} = M and ϕ_{t}^{(1)} is the geodesic flow ϕ_{t}, while M^{(n)} = M̂ and ϕ_{t}^{(n)} = ϕ̂_{t}. For k ≤ l, the natural projection ρ_{kl} : M^{(l)} → M^{(k)}, which ignores the last l −k components of a frame, defines a semiconjugacy from ϕ_{t}^{(l)}onto ϕ_{t}^{(k)}. The frame flow ϕ̂_{t}^{(k)} can be viewed as a SO(k − 1) extension of the geodesic flow ϕ_{t} : M → M. The flow ϕ̂^{(k)} preserves the measure μ^{(k)} = μ × ν_{SO(n−1)}, where ν_{SO(k−1)} is (normalized) Haar measure on SO(n − 1), and furthermore ρ_{nk}^{∗}μ^{(n)} = μ^{(k)}. Since the Bernoulli property is preserved under such factor maps we can deduce the following.

Proposition 9.1. If any of the hypotheses (1) – (6) (from the introduction) hold for the orientable manifold V, then each of the flows ϕ_{t}^{(k)} on M^{(k)}, for 1 ≤ k ≤ n, is Bernoulli and thus, in particular, ergodic.

Brin’s papers naturally focus mainly on the flow ϕ_{t}^{(n−1)}. This is equivalent to studying ϕ̂_{t}. The reason for this is that the first n−1 components of a positively oriented orthonormal frame uniquely determine the last component, so that ϕ_{t}^{(n−1)} is really the same flow as ϕ̂_{t} when, as we are assuming, V is orientable. In the case when V is not orientable, there is an obvious conjugacy between ϕ_{t}^{(n−1)} and the n-frame flow on the double cover of V.

Theorem 9.2. If any of the hypotheses (1) – (6) (from the introduction) hold for the orientable manifold V, then each of the time-one maps ϕ_{1}^{(k)} on M^{(k)}, for 1 ≤ k ≤ n, is stably K and thus, in particular, stably ergodic.

Proof. We begin by describing a procedure for rotating a frame v̂ = (v_{1}, …, v_{n}) to a nearby frame ŵ = (w_{1}, …, w_{n}) based at the same point.

First rotate v_{1} to w_{1} in the plane spanned by v_{1} and w_{1}, fixing all vectors orthogonal to the plane. If v_{1} = w_{1} this rotation is not needed and the fact that the plane is undefined doesn’t matter. Let v′_{i} be the new position of v_{i}. Observe that v′_{2}, …, v′_{n} and w_{2}, …, w_{n} are all orthogonal to v′_{1} = w_{1}. Now rotate v′_{2} to w_{2} in the plane that they span, fixing all vectors orthogonal to this plane (in particular v′_{1} = w_{1}) to obtain a new frame v′′_{1} = w_{1}, v′′_{2} = w_{2}, v′′_{3}, …, v′′_{n}. Again there is no need to rotate if v′_{2} = w_{2} and the plane is undefined. We continue in the obvious way.

Let v_{i}^{(k)} denote the position of v_{i} after k steps. Then v_{i}^{(0)} = v_{i}, v_{i}^{(1)} = v′_{i} and so on. The procedure ensures that v_{k}^{(k)} = w_{k}. Since the last term of a positively oriented orthonormal frame is uniquely determined by the preceding terms, and v_{i}^{(n−1)} = w_{i} for 1 ≤ i ≤ n − 1, we must also have v_{n}^{(n−1)} = w_{n}. Thus we can rotate v̂ to ŵ in n − 1 steps.

The procedure is well defined unless we obtain v_{k+1}^{(k)} = −w_{k} after k steps for some k. This cannot occur if all of the angles between v_{i} and the matching element w_{i} of the other basis are small enough. Indeed if all of these angles are less than α, the angles between v_{i}^{(k)} and w_{i} are all less than 2^{k}α. Thus if the initial angles are all less than π/2^{n−1}, then the condition v_{k+1}^{(k)} = −w_{k} will never arise.

Now let ψ : M^{(k)} → M^{(k)} be a C^{1} small perturbation of ϕ_{1}^{(k)} that preserves the measure μ^{(k)}. We shall construct a map ψ̂: M̂ → M̂ that preserves the measure μ̂ and has ψ as a factor. Since ϕ̂_{1} = ϕ^{(n)} is stably K and the K property is inherited by factors, it follows that ψ is K.

The lift ψ̂ is constructed by adapting the procedure for rotating frames described above. Given a k-frame v^{(k)} = (v_{1}, …, v_{k}) based at a point p, let w^{(k)} = (w_{1}, …, w_{k}) based at q denote the k-frame ψ̂(v^{(k)}). Let u^{(k)} = (u_{1}, …, u_{k}) be the k-frame at q obtained by parallel translating v^{(k)} along the shortest geodesic from p to q.

Observe that the first k steps of the procedure for rotating n-frames depend only on the first k elements of the frames. Thus if start with any n-frame at p whose first k elements are v^{(k)}, we can parallel translate this frame to a frame at q whose first k elements are u^{(k)}, and then apply the first k steps of the procedure to obtain a frame whose first k elements are w^{(k)}. This defines the lift ψ̂.

Since ψ̂ acts on all n-frames whose first k elements are v^{(k)} by a common isometry from T_{p}V to T_{q}V, and and ψ preserves the measure μ^{(k)}, it is clear that ψ̂ preserves the measure μ̂ on M̂.

References

[BD] C. Bonatti and L. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math. 143 (1996), 357–396.

[Br1] M. Brin, Topological transitivity of one class of dynamical systems and flows of frames on manifolds of negative curvature, Funktsional. Anal. i Prilozhen. 9 (1975), 9–19, English translation: Functional Anal. Appl. 9 (1975), 8–16.

[Br2] M. Brin, The topology of group extensions of Anosov systems, Mat. Zametki 18 (1975), 453–465, English translation: Math. Notes 18 (1976), 858–864.

[Br3] M. Brin, Ergodic theory of frame flows, Ergodic theory and dynamical systems II, Proc. spec. Year, Maryland 1979–80, Prog. Math. 21,, Birkhäuser, Boston, 1982, pp. 163–183.

[BG] M. Brin and M. Gromov, On the ergodicity of frame flows, Invent. Math. 60 (1980), 1–7.

[BK] M. Brin and H. Karcher, Frame flows on manifolds with pinched negative curvature, Compos. Math. 52 (1984), 275–297.

[BP1] M. Brin and Ya. Pesin, Flows of frames on manifolds of negative curvature, Uspehi Mat. Nauk. 28 (1973), 169–170.

[BP2] M. Brin and Ya. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk. SSSR 38 (1974), 170–212, English translation: Math. USSR Izvestija 8 (1974), 177–218.

[BW] K. Burns and A. Wilkinson, Stable ergodicity of skew products, Ann. Sci. Ecole Norm. Sup. 32 (1999), 859–889.

[BPW] K. Burns, C. Pugh and A. Wilkinson, Stable ergodicity and Anosov flows, Topology 39 (2000), 149–159.

[Ga] F. Gantmacher, The theory of matrices, Chelsea, New York, 1959.

[GPS] M. Grayson, C. Pugh and M. Shub, Stably ergodic diffeomorphisms, Annals of Math. 140 (1994), 295–329.

[KK] A. Katok and A. Kononenko, Cocycle’s stability for partially hyperbolic systems, Math. Res. Lett. 3 (1996), 191–210.

[On] A. Oniščik, Lie groups which are transitive on compact manifolds. III, Mat. Sbornik 75 (117) (1968), 255–263, English translation: Math. of USSR—Sbornik, 4 (1968), 233–240.

[PS1] C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity, J. of Complexity 13 (1997), 125–179.

[PS2] C. Pugh and M. Shub, Stable ergodicity and julienne quasiconformality, J. Eur. Math. Soc. 2 (2000), 1–52.

[Ra] M. Ratner, Anosov flows with Gibbs measures are also Bernoullian, Israel J. Math. 17 (1974), 380–391.

[Ru] D. Rudolph, Classifying the isometric extensions of a Bernoulli shift, J. d’Analyse Math. 34 (1978), 36–60.

[SW] M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents, Invent. Math. 139 (2000), 495–508.

[Wi] A. Wilkinson, Stable ergodicity of the time-one map of a geodesicflow, Ergod. Th. and Dynam. Sys. 18 (1998), 1545-1587.

Keith Burns, Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, U.S.A

Mark Pollicott, Department of Mathematics, Manchester University, Oxford Road, Manchester, M13 9PL, U.K.