Stable ergodicity of skew products (Ergodicité stable des produits croisés)

Keith Burns and Amie Wilkinson Northwestern University

February 10, 1999

Abstract

Stable ergodicity is dense among compact Lie group extensions of Anosov diffeomorphisms of compact manifolds. Under the additional assumption that the base map acts on an infranilmanifold, an extension that is not stably ergodic must have a factor that has one of three special forms. A consequence is that stable ergodicity and stable ergodicity within skew products are equivalent in this case.

L’ergodicité stable est dense pour les extensions de difféomorphismes d’Anosov des variétés compactes par groupes de Lie compacts. Si, par ailleurs, le difféomorphisme d’Anosov agit sur une infranilvariété, alors toute extension qui n’est pas stablement ergodique possède un facteur qui prend une de trois formes particulières. Il s’ensuit que les notions de “ergodicité stable” et de “ergodicité stable entre les produits croisés” sont équivalentes.

Introduction

A diffeomorphism F of a compact manifold M is partially hyperbolic if the tangent bundle T M splits as a Whitney sum of T F -invariant subbundles:

T M = E^{u} ⊕ E^{c} ⊕ E^{s},

and there exist a Riemannian (or Finsler) metric on M and constants λ < 1 and μ > 1 such that for every p ∈ M,

m(T_{p}F |_{E^{u}}) > μ > ∥T_{p}F |_{E^{c}}∥ ≥ m(T_{p}F |_{E^{c}}) > λ > ∥T_{p}F |_{E^{s}}∥ > 0.

(The co-norm m(A) of a linear operator A between Banach spaces is defined by m(A) := inf_{∥v∥=1} ∥A(v)∥.) The bundles E^{u}, E^{c} and E^{s} are referred to as the unstable, center and stable bundles of F, respectively. A special case of a partially hyperbolic diffeomorphism is an Anosov diffeomorphism, for which E^{c} = {0}.

It has been known since the 1960’s that volume preserving Anosov diffeomorphisms are stably ergodic. By a volume preserving diffeomorphism, we mean a diffeomorphism that is at least C^{1+α} for some α > 0 and preserves a smooth measure on M. Stable ergodicity of a volume preserving diffeomorphism F means that any volume preserving diffeomorphism which preserves the same smooth measure as F and is close enough to F in the C^{1} topology is ergodic.

Starting with the seminal work of Grayson, Pugh and Shub [GPS], a series of recent papers ([Wi], [PS1], [PS2], [KK], [BPW]) has given many examples of (non Anosov) partially hyperbolic, volume preserving diffeomorphisms that are stably ergodic. Indeed, Pugh and Shub have conjectured [PS1] that stably ergodic diffeomorphisms should form an open dense subset of the volumepreserving, partially hyperbolic diffeomorphisms of a compact manifold. This is really a conjecture that the set is dense, since openness of the set is an immediate consequence of its definition.

The present paper addresses this conjecture in the special case of skew products. Let f : M → M be a diffeomorphism of a compact manifold M and let G be a compact Lie group. A function φ : M → G (assumed to be at least C^{1+α}) defines a skew-product f_{φ} : M × G → M × G by

f_{φ}(x, g) = (f(x), φ(x)g).

A skew product is also called a G-extension of f. If the base diffeomorphism f is Anosov, then f_{φ} is partially hyperbolic. Since Haar measure on G is translationinvariant, f_{φ} will be volume preserving if f is volume preserving (and f_{φ} will preserve the product of Haar measure with the volume preserved by f).

We are able to verify Pugh and Shub’s conjecture in the case of skew products. If f is a volume preserving Anosov diffeomorphism of a compact manifold M and G is a compact Lie group, then f_{φ} is stably ergodic for a dense set of functions φ : M → G (see Theorem A in Section 4). Under the further assumption that M is an infranilmanifold, we can show (see Theorem B in Section 4) that a skew product f_{φ} which fails to be stably ergodic must have a factor that has one of three specific forms:

1. f × Id_{Y}, where Y is a nontrivial quotient of G;

2. f × R_{α}, where R_{α} is a rotation of a circle;

3. f_{ψ}, where ψ : M → T^{d} is homotopic to a constant map and ψ(M) lies in a coset of a lower dimensional Lie subgroup of T^{d}.

A corollary of this result is that in the case when G is semisimple, a skew product is stably ergodic if and only if it is ergodic.

Our results are based on a recent theorem of Pugh and Shub (see Section 1) that provides a comprehensive criterion for a partially hyperbolic system to be stably ergodic. We also use ideas and results from Brin’s work on skew products [B1, B2]. Brin introduced a sequence of Lie subgroups of the fiber of the skew product, and used these subgroups to characterize the ergodic skew products as well as those that are K-systems. He also proved an analogue of our density result: the ergodic skew products form an open and dense subset of the set of all skew products.

It follows from our results that a skew product over an Anosov diffeomorphism of an infranilmanifold is stably ergodic (among all volume preserving diffeomorphisms) if and only if it is stably ergodic within skew products.

Stable ergodicity within skew products has been the focus of a series of recent papers of Adler, Kitchens and Shub [AKS], Parry and Pollicott [PP], Field and Parry [FP], and Walkden [Wa]. The main interest has been in the case of Hölder continuous skew products with a subshift of finite type as the base.

A detailed formulation of the above results will be found in Section 4.

We would like to thank Mike Field, John Franks and Charles Pugh for several helpful discussions and Viorel Niţică for introducing us to Brin’s papers. The first author was partially supported by NSF Grant DMS 9504760.

1 Pugh and Shub’s theorem

A function is C^{k−} if its partial derivatives of order j exist and satisfy a Hölder condition with exponent k − j, where j is the greatest integer less than k. By convention, we let C^{∞−} = C^{∞}. Unless otherwise stated, we assume throughout this paper that k ∈ (1, ∞]. For these k the composition of two C^{k−} functions is again C^{k−}, and so the set Diff^{k−}(M) of C^{k−} diffeomorphisms of a manifold M is a group. We endow this group with the C^{j} topology, where j is again the greatest integer less than k; Diff^{k−}(M) is then a closed subgroup of Diff^{j}(M).

Let μ be a smooth volume element of M. The μ-preserving C^{k−} diffeomorphisms also form a group, denoted by Diff_{μ}^{k−}(M). We again use the C^{j} topology, where j is the greatest integer less than k. A diffeomorphism F ∈ Diff_{μ}^{k−}(M) is stably ergodic if every F′ ∈ Diff_{μ}^{k−}(M) that is close enough to F in the C^{1} topology is ergodic (with respect to μ). We similarly define stably weak mixing, stably mixing, and stably Kolmogorov (stably K).

If F is C^{k−} and partially hyperbolic, then its stable and unstable bundles are uniquely integrable and are tangent to foliations W_{F}^{u} and W_{F}^{s}, whose leaves are C^{k−}. A partially-hyperbolic diffeomorphism is said to have the accessibility property if, for every pair of points p, q ∈ M, there is a piecewise C^{1} path γ : [0, 1] → M such that:

• γ(0) = p and γ(1) = q;

• there exist 0 = t_{0} < t_{1} < ⋯ < t_{n} = 1 such that γ([t_{i}, t_{i+1}]) ⊆ W_{F}^{a_{i}}(γ(t_{i})), where a_{i} = u or s, for i = 0, …, n − 1.

The path γ is called a W^{u,s}(F)-path with n legs.

Partial hyperbolicity is an open property in the C^{1} topology on diffeomorphisms of M, and so any diffeomorphism F′ of M that is sufficiently C^{1}-close to the partially hyperbolic diffeomorphism F has stable and unstable foliations W_{F′}^{u} and W_{F′}^{s}. We say that F has the stable accessibility property if every F′ sufficiently C^{1}-close to F has the accessibility property.

Pugh and Shub have shown that partial hyperbolicity and stable accessibility imply stable ergodicity under some relatively mild technical hypotheses. This theorem is the cornerstone of many of our arguments.

Theorem 1.1 [PS2] Let F ∈ Diff_{μ}^{k−}(M) where M is compact and k > 1. If F is center-bunched, partially hyperbolic, stably dynamically coherent, and stably accessible, then F is stably ergodic.

Before proceeding, we discuss the technical conditions “center-bunched” and “stably dynamically coherent”.

A partially hyperbolic diffeomorphism F is center-bunched if, for every p ∈ M, the quantity

μ_{c} = ∥T_{p}F |_{E_{F}^{c}} ∥/m(T_{p}F |_{E_{F}^{c}})

is close to one. The details can be found in section 4 of [PS2]. This property is C^{1}-open, and immediately satisfied when μ_{c} = 1.

A partially hyperbolic diffeomorphism F is dynamically coherent if the distributions E_{F}^{c}, E_{F}^{c} ⊕E_{F}^{u}, and E_{F}^{c} ⊕E_{F}^{s} are integrable, and everywhere tangent to foliations W_{F}^{c}, W_{F}^{cu}, and W_{F}^{cs}, called the center, center-unstable, and center-stable foliations, respectively. Stably dynamically coherent means that all diffeomorphisms close enough to F in the C^{1} topology are dynamically coherent. If F is dynamically coherent and W_{F}^{c} is a C^{1} foliation, then F is stably dynamically coherent. For in this case W_{F}^{c} is normally hyperbolic and plaque-expansive. Then the theory developed in [HPS] can be applied to show that dynamical coherence is C^{1}-open.

Pugh and Shub’s theorem can be strengthened slightly.

Corollary 1.2 Let F satisfy the hypotheses of Theorem 1.1. Then F is stably K.

Recall that a volume-preserving diffeomorphism F : M → M has the Kolmogorov property if there is a sub-σ-algebra A of the Borel σ-algebra B such that F^{−1}A ⊆ A, ⋃_{n=−∞}^{∞} F^{−n}A generates B, and ⋂_{n=0}^{∞} F^{−n}A is the trivial σ-algebra. A diffeomorphism F that has the Kolmogorov property is called a K-system. K-systems are ergodic and have no nontrivial factors of zero entropy (see [P]).

Proof of Corollary 1.2: A set A ⊂ M is called essentially u, s-saturated if there are sets A_{u} and A_{s} that are respectively a union of entire unstable leaves and a union of entire stable leaves respectively and satisfy μ(A △ A_{u}) = 0 = μ(A△A_{s}). The argument in section 9 of [PS2] shows that, under the hypotheses of Theorem 1.1, every μ-preserving diffeomorphism that is close enough to F in the C^{1}-topology has the property that all essentially u, s-saturated sets have measure 0 or full measure. Proposition 5.2 of [BP] says that this latter property implies the K-property. □

2 Skew products

Let f ∈ Diff_{ν}^{k−}(M), let G be a Lie group and let μ be the product of the volume ν on M and Haar measure on G. A C^{k−} function φ : M → G defines a skewproduct over f, f_{φ} : M × G → M × G, by

f_{φ}(x, g) = (f(x), φ(x)g).

Since Haar measure is translation-invariant, f_{φ} ∈ Diff_{μ}^{k−}(M ×G). Skew products over f are also called G-extensions of f. The set of all C^{k−} skew products over f, denoted

Ext^{k−}(f, G) = {f_{φ} | φ ∈ C^{k−}(M, G)},

is a closed subset of Diff_{μ}^{k−}(M × G); its connected components correspond to homotopy classes of φ.

In this paper, we consider C^{k−} skew products f_{φ} for which f is an Anosov diffeomorphism of a compact manifold and φ maps into a connected Lie group with a Riemannian metric that is bi-invariant, i.e. invariant under both left and right translations. The bi-invariance of the metric means that both left and right translations are isometries. When we refer to “skew products”, we implicitly make these assumptions.

We think of M × G as a bundle over M, so that M is the base and the fibers are the sets {x} × G for x ∈ M. Let π_{M} : M × G → M and π_{G} : M × G → G be the projections of M × G onto M and G, respectively. We equip M × G with a product metric. On M we use a Riemannian metric adapted to the Anosov diffeomorphism f. On G we use the chosen bi-invariant metric.

We denote left and right translations by h ∈ G, by L_{h} and R_{h}, respectively. By abuse of notation, we shall also denote by L_{h} and R_{h} the maps on M × G that send (x, g) to (x, hg) and to (x, gh) respectively.

Brin and Pesin showed that skew products of the type we are considering are partially hyperbolic; see Theorem 2.2 in [BP]. The next proposition describes the partially hyperbolic splitting for f_{φ} and the associated foliations. Let T M = E_{f}^{u} ⊕ E_{f}^{s} be the Anosov splitting for f, and let W_{f}^{u} and W_{f}^{s} be the stable and unstable foliations for f.

Proposition 2.1 [BP] The skew product f_{φ} is partially hyperbolic, dynamically coherent, and center-bunched. The leaves of the center foliation W_{f_{φ}}^{c} are the fibers of M × G. For a = u or s, each leaf of W_{f_{φ}}^{ac} is the product of G with a leaf of W_{f}^{a}. Each W_{f_{φ}}^{a} leaf is the graph of a C^{k−} function from a leaf of W_{f}^{a} to G. For any g ∈ G, the right translation R_{g} carries W_{f_{φ}}^{a} leaves to W_{f_{φ}}^{a} leaves.

An immediate consequence of Theorem 1.1, Corollary 1.2 and the previous proposition is:

Corollary 2.2 Let G be compact. Then f_{φ} : M × G → M × G is stably ergodic and stably K if it has the stable accessibility property.

3 Algebraic factors of skew products

Skew products commute with the projection π_{M} to the base manifold and with all right translations R_{g} of M ×G. It is natural to consider factors of skew products that share the same properties. This motivates the following definition.

We shall say that the C^{k−} dynamical system h : M × Y → M × Y is an algebraic factor of the C^{k−} skew product f_{φ} if Y = H\G, where H is a closed subgroup of G, and there exists a C^{k−} function Φ : M → G/H for which we have a commutative diagram

[figure: M × G; M × G; M × H\G; M × H\G; f_{φ}; h; π_{Φ}; π_{Φ}]

in which π_{Φ}(x, g) = (x, Φ(x)^{−1}g). By Φ(x)^{−1} we mean {g^{−1} | g ∈ Φ(x)}, which is an element of H\G.

Any function Φ : M → G/H induces a partition P of M ×G that is invariant under any right translation R_{g}, g ∈ G, in the sense that R_{g} permutes elements of the partition. The elements of P are the set P = ⋃_{x∈M}{x} × Φ(x) and its translates by the maps R_{g}, g ∈ G. The map π_{Φ} sends elements of P to constant sections of H\G. The existence of the quotient map h is equivalent to the partition P being f_{φ}-invariant. Conversely, a partition of M × G each of whose elements meets every fiber of M × G induces an algebraic factor if it is invariant under f_{φ} and right translations; cf. Section 6.

Note that an algebraic factor preserves the product of volume on M with the projection of Haar measure to H\G. Thus an algebraic factor is a factor in the ergodic-theory sense. All ergodic properties, such as ergodicity, weak mixing, mixing, and the Kolmogorov property are inherited by factors. Hence if a skew product has an algebraic factor that does not possess one of these properties, then the skew product itself does not have the property.

If we restrict further to the class of algebraic factors that are themselves skew products, then skew-stable ergodic properties are also preserved:

Lemma 3.1 Let N be a normal subgroup of G, and let h : M × N\G → M × N\G be an algebraic factor of the C^{k−} skew product f_{φ} : M × G → M × G.

Then h is an N\G-skew product, and if f_{φ} is stably ergodic/mixing/K among G-skew products then h is stably ergodic/mixing/K among N\G-skew products.

Proof: Let p : G → N\G. Let h be an algebraic factor of f_{φ} induced by Φ : M → G/N. It follows from the definition of algebraic factor that h(x, y) = (f(x), w(x, y)), where, for all g ∈ G,

w(x, Φ(x)^{−1}g) = Φ(f(x))^{−1}φ(x)g = Φ(f(x))^{−1} · p(φ(x)) · p(g).

Since N is normal and Φ(x) ∈ G/N, we have Φ(x) ∈ N\G and p(g) = Φ(x) · Φ(x)^{−1}g for all g ∈ G. Hence

w(x, y) = Φ(f(x))^{−1} · p(φ(x)) · Φ(x) · y,

for all y ∈ N\G and h = f_{ψ}, where

ψ = (Φ ∘ f)^{−1} · (p ∘ φ) · Φ.

With respect to the bi-invariant metrics on G and N\G, the canonical projection p : G → N\G is a Riemannian submersion. Riemannian submersions have an isometric path-lifting property: given any geodesic arc γ : [0, 1] → N\G and any point q ∈ p^{−1}γ(0), there is a unique geodesic arc γ̃ : [0, 1] → G such that p∘γ̃ = γ, γ̃(0) = q, and such that all vectors tangent to γ̃ are horizontal, i.e. orthogonal to the fibers of the projection p (see e.g. Prop 3.31 in [CE]). Using this property, we can lift a function ψ_{1} : M → N\G, whenever d_{C^{0}}(ψ_{1}, p ∘ φ) is sufficiently small, to a function ψ̃_{1} : M → G, as follows. Suppose that the C^{0} distance between ψ_{1} and p∘φ is smaller than the injectivity radius of N\G. Then for every x ∈ M, there is a unique horizontal geodesic γ with γ(0) = p ∘ φ(x) and γ(1) = ψ_{1}(x). Let γ̃ be the geodesic lift of γ such that γ̃(0) = φ(x), and let

ψ̃_{1}(x) = γ̃(1).

It is immediate that ψ̃_{1} has the following properties:

1. p ∘ ψ̃_{1} = ψ_{1};

2. if ψ_{1} is C^{k−} then so is ψ̃_{1};

3. d_{C^{k−}}(ψ̃_{1}, φ) = d_{C^{k−}}(ψ_{1}, p ∘ φ).

Now suppose that h is not stably ergodic among skew products. Then there exists a C^{k−} function ψ_{0} : M → N\G such that f_{ψ_{0}} is not ergodic, with ψ_{0} arbitrarily C^{k−}-close to ψ. For ψ_{0} sufficiently close to ψ, the function ψ_{1} = (Φ∘f)·ψ_{0}·Φ^{−1} lifts to a C^{k−} function ψ̃_{1} that is C^{k−}-close to φ. Notice that f_{ψ_{1}} is conjugate to f_{ψ_{0}} by the measure-preserving diffeomorphism (x, y) ↦ (x, Φ(x)y), and so f_{ψ_{1}} is not ergodic.

Since it has f_{ψ_{1}} as a non-ergodic factor, fψ̃_{1} is not ergodic. As such a ψ̃_{1} can be found arbitrarily C^{k−}-close to φ, this implies that f_{φ} is not stably ergodic.

The same proof works when “ergodic” is replaced by “mixing” or “K”. □

4 Statement of results

Our first result is that the conjecture of Pugh and Shub holds for skew products. Recall that we have given C^{k−}(M, G) the C^{j} topology, where j is the greatest integer less than k.

Theorem A: [Density of stable ergodicity] Let f : M → M be a C^{k−} volume-preserving Anosov diffeomorphism of a compact manifold, and let G be a compact, connected Lie group. There is a subset E of C^{k−}(M, G) such that E is dense in the C^{k−} topology and the skew product f_{φ} is stably ergodic, and in fact, stably a K-system for every φ ∈ E.

Our second theorem characterizes the skew products that are not stably ergodic. In this result and its corollaries we assume that M is an infranilmanifold, although we actually use only the (possibly weaker) conditions that π_{1}(M) is virtually nilpotent and the action of f^{∗} − I on H^{1}(M, R) is invertible. These conditions are also satisfied, for example, if M has finite fundamental group, but since there are no known Anosov diffeomorphisms other than those on infranilmanifolds, we will not dwell on these conditions.

Theorem B: [Characterization of stable ergodicity] Let f : M → M be a C^{k−} volume-preserving Anosov diffeomorphism of an infranilmanifold, let G be a compact, connected Lie group and let φ : M → G be C^{k−}.

If f_{φ} is not stably ergodic, then it has an algebraic factor h : M × H\G → M × H\G, where one of the following holds:

1. H ≠ G, and h is the product of f with Id_{H\G};

2. H is normal, H\G is a circle, and h is the product of f with a rotation;

3. H is normal, H\G is a d-torus, and h = f_{ψ}, where ψ is homotopic to a constant and maps M into a coset of a lower dimensional Lie subgroup of T^{d}.

If f_{φ} has an algebraic factor of type (1), it is not ergodic; if f_{φ} has an algebraic factor of type (2), but none of type (1), then it is ergodic, but not weak mixing; otherwise f_{φ} is Bernoulli.

The classification of skew products in Theorem B is based on a sequence of Lie subgroups of G that are determined by the behavior of W^{u,s}(f_{φ})-paths. This sequence was first studied by Brin in [B2] and is described in detail in Section 8. In our notation,^{1} the sequence is:

H^{0} ⊂ H ⊂ \overline{H} ⊂ K ⊂ G.

Brin proved the following (Propositions 1, 2, and 3 in [B2]):

(i) there is a C^{k−}-dense set of φ for which H^{0} = G (see Theorem 9.8);

(ii) f_{φ} is ergodic if and only if K = G;

(iii) f_{φ} is a K-system if and only if \overline{H} = G.

It follows from (iii) and work of Rudolph [R] that:

(iv) f_{φ} is Bernoulli if and only if \overline{H} = G.

We show (Theorem 9.1 and Propositions 12.2, 12.3, and 12.7):

(A) if H^{0} = G, then f_{φ} is stably accessible;

(B1) if K ≠ G, then f_{φ} has an algebraic factor of type (1);

(B2) if \overline{H} ≠ G and K = G, then f_{φ} has an algebraic factor of type (2);

(B3) if H^{0} ≠ G and \overline{H} = G, then f_{φ} has an algebraic factor of type (3).

Case (B3) requires the assumption that the base M is an infranilmanifold; the other cases do not.

Theorem A is an immediate consequence of (A) and Corollary 2.2. It is through this corollary that we apply the theorem of Pugh and Shub.

Since the cases (A), (B1), (B2), and (B3) are mutually exclusive, we see that Theorem B will follow from the above statements, provided one can show that f_{φ} is not stably ergodic if it has an algebraic factor of types (1), (2), or (3). A factor of type (1) is nonergodic, so f_{φ} itself is nonergodic if it has a factor of type (1). We shall see that factors of types (2) and (3) are not stably ergodic; since the group H is normal in these cases, it follows from Lemma 3.1 that f_{φ} is not stably ergodic if it has a factor of types (2) or (3). Case (2) can be approximated by the product of f with a rational rotation, which is not ergodic. In case (3), since ψ is homotopic to a constant, we can make a small change to ψ to give us a function ψ_{0} : M → T^{d} that takes values in a coset of a closed subgroup H_{0} of T^{d}. The skew product f_{ψ_{0}} belongs to case (1); it has f × Id_{H_{0}\T^{d}} as a factor, and so is not ergodic.

1In Remark 2 of [B2], the sequence appears as

H_{loc} ⊂ H_{C} ⊂ H\overline{C} ⊂ H ⊂ G.

Brin’s H_{loc} is a subgroup of our H^{0}; the other terms are the same except for notation.

Figure 1: Stable ergodicity and stable ergodicity within skew products are the same.

Furthermore, we see that if f_{φ} has a factor of type (1), (2), or (3), then either f_{φ} is itself not ergodic or it can be perturbed to a nonergodic skew product. Thus f_{φ} is not stably ergodic within skew products if it has a factor of types (1), (2) or (3). On the other hand, if f_{φ} has no factors of these types, then f_{φ} is stably ergodic and even stably K (with perturbations allowed among all volume preserving diffeomorphisms).

We obtain

Corollary B1: Let f_{φ} be a compact group extension of a volume preserving Anosov diffeomorphism f of a compact infranilmanifold. The following are equivalent:

1. f_{φ} is stably ergodic within skew products;

2. f_{φ} is stably ergodic;

3. f_{φ} is stably a K-system.

The picture is summarized in Figure 1.

Stable ergodicity within skew products was first studied by Brin ([B1], [B2]), who proved the analogue of Theorem A: the set of C^{k−} skew products that are stably ergodic within skew products is C^{k−} dense and C^{1} open (see Remark 2.1 and Proposition 2.3 in [B1]).

Since a semisimple Lie group does not have abelian quotients, semisimple skew products cannot have algebraic factors of type (2) or (3). This gives

Corollary B2: Let f_{φ} be as in Theorem B and suppose in addition that the group G is semisimple. Then f_{φ} is stably ergodic if and only if it is ergodic.

For circle extensions, we have the following:

Corollary B3: Let f_{φ} be as in Theorem B and suppose in addition that the group G is the circle T. Let m be the index of (f^{∗} − Id)H^{1}(M, Z) in H^{1}(M, Z). Then f_{φ} is stably ergodic if and only if there are no functions Φ ∈ C^{k−}(M, T) that satisfy an equation of the form

mφ = Φ ∘ f − Φ + c,

where c ∈ T is a constant.

This corollary is proved at the end of Section 12.

5 Stable accessibility

As far as we know, this property was first studied systematically in the context of partially hyperbolic systems by Brin and Pesin [BP] in 1974. They used accessibility to show ergodicity of partially hyperbolic systems where the center foliation is Lipschitz.

The following discussion is based on [BPW], although it differs in the details. Let F_{1} and F_{2} be a pair of topological foliations on a compact connected manifold M with dimension n.

Definition: An F_{1}, F_{2}-path is a path ψ : [0, 1] → M consisting of a finite number of consecutive arcs — called legs — each of which is a curve that lies in a single leaf of one the two foliations^{2}. The pair F_{1}, F_{2} is transitive if any two points in M are joined by an F_{1}, F_{2}-path and is stably transitive if any pair of foliations sufficiently close to F_{1}, F_{2} is transitive.

Recall that if Z is a compact connected orientable n-dimensional manifold with boundary and q_{0} ∈ M, we can define the degree of a continuous map ψ : (Z, ∂Z) → (M, M \ {q_{0}}) to be the unique integer l ≥ 0 such that there are generators ζ_{Z} for H_{n}(Z, ∂Z) and ζ_{M} for H_{n}(M, M \ {q_{0}}) with ψ_{∗}(ζ_{Z}) = lζ_{M}, where ψ_{∗} is the map induced by ψ on n-dimensional homology. Two properties of degree are important in the following.

• If ψ̃ : (Z, ∂Z) → (M, M \ {q_{0}}) is close enough to ψ in the C^{0} topology, then ψ̃ and ψ have the same degree.

• If deg ψ ≠ 0, then ψ^{−1}(q_{0}) ≠ 0.

Definition: A point q_{0} can be F_{1}, F_{2}-engulfed from a point p_{0} if there is a continuous map Ψ : Z × [0, 1] → M such that:

2We make the convention that consecutive legs must be of opposite types, although it is permissible to have a leg of length 0. The initial leg may be of either type.

1. Z is a compact, connected, orientable, n-dimensional manifold with boundary;

2. for each z ∈ Z, the curve ψ_{z}(·) = Ψ(z, ·) is an F_{1}, F_{2}-path with ψ_{z}(0) = p_{0};

3. there is a constant C such that every path ψ_{z} has at most C legs;

4. ψ_{z}(1) ≠ q_{0} for all z ∈ ∂Z.

5. the map (Z, ∂Z) → (M, M \ {q_{0}}) defined by z ↦ Ψ(z, 1) has positive degree.

It is evident that q_{0} can be reached from p_{0} along an F_{1}, F_{2}-path if q_{0} can be engulfed from p_{0}. Engulfing is stable under small perturbations of p_{0}, q_{0} and the foliations F_{1} and F_{2}.

Proposition 5.1 Suppose that q_{0} can be F_{1}, F_{2}-engulfed from point p_{0} and there is an F_{1}, F_{2}-path from q_{0} to q_{1}. Then q_{1} can be F_{1}, F_{2}-engulfed from p_{0}.

Proof: The proposition follows easily from the special case in which q_{0} and q_{1} are joined by an F_{1}, F_{2}-path with one leg. We now consider that case and assume, after possibly renaming the foliations, that q_{0} and q_{1} lie in same leaf of F_{1}. Let d be the dimension of the leaves of F_{1}. It is possible to find an open set Q containing q_{0} and q_{1} and a homeomorphism ρ : R^{n−d} × R^{d} → Q such that:

1. ρ sends each set {const} × R^{d} into a single leaf of F_{1};

2. q_{0} = ρ(0, 0) and q_{1} ∈ ρ({0} × R^{d}).

For q ∈ Q, let γ_{q} be the path that is the image under ρ of the line segment from ρ^{−1}(q) to ρ^{−1}(q) + ρ^{−1}(q_{1}). The map that sends q to the other end of γ_{q} is a homeomorphism of Q which maps a neighborhood of q_{0} to a neighborhood of q_{1}.

Suppose that we could choose the map Ψ as in the definition of engulfing and with the additional property that Ψ(Z × {1}) ⊂ Q. Then we could define a new map \overline{Ψ} : Z × [0, 1] → N by setting \overline{Ψ}(z, t) = \overline{ψ}_{z}(t) where \overline{ψ}_{z} is the path formed by concatenating ψ_{z} and γ_{Ψ(z,1)}. It is easily seen that \overline{Ψ} is an engulfing of q_{1} from p_{0}.

It remains to show that Ψ can be chosen so that Ψ(Z × {1}) ⊂ Q. Let Ψ_{0} : Z_{0} × [0, 1] → M be any engulfing of q_{0} from p_{0}. Set ψ_{0}(z) = Ψ_{0}(z, 1). Choose δ > 0 so that \overline{B(q_{0}, 3δ)} ⊂ Q and ψ_{0}(∂Z_{0}) ∩ \overline{B(q_{0}, δ)} = ∅. Choose a smooth map ψ̃_{0} : Z_{0} → M such that dist(ψ̃_{0}(z), ψ(z)) < δ/2 for all z ∈ Z_{0}. Then ψ̃(∂Z_{0}) ⊂ M \ {q_{0}} and ψ_{0} and ψ̃_{0} have the same degree as maps of (Z_{0}, ∂Z_{0}) into (M, M \ {q_{0}}).

Now choose ρ ∈ (δ, 2δ) such that ψ̃_{0} is transverse to the geodesic sphere of radius ρ and center q_{0}, i.e. ρ is a regular value of dist^{2}(q_{0}, ψ̃_{0}(·)). Then ψ̃_{0}^{−1}(\overline{B(q_{0}, ρ)}) is a compact smooth manifold with boundary, which must have a component Z such that ψ̃ = ψ̃_{0}|_{Z} has nonzero degree. Let ψ = ψ_{0}|_{Z}. Then

ψ : (Z, ∂Z) → (\overline{B(q_{0}, 3δ)}, \overline{B(q_{0}, 3δ)} \ B(q_{0}, δ/2)) ⊂ (Q, Q \ \{q_{0}\})

and ψ has the same degree as ψ̃ as a map of (Z, ∂Z) into (Q, Q \ {q_{0}}).

Finally we choose Ψ to be the restriction of Ψ_{0} to Z × [0, 1]. It is clear from the above Ψ has the desired properties. □

In the situation where F_{1} and F_{2} are the foliations W^{u} and W^{s} for a dynamically coherent partially hyperbolic diffeomorphism F : M → M, there is a simpler condition which implies that one point can be engulfed from another. Let c be the dimension of the leaves of W^{c}.

Definition: A point q_{0} can be centrally engulfed from a point p_{0} if there is a continuous map Ψ : Z × [0, 1] → M such that:

1. Z is a compact, connected, orientable, c-dimensional manifold with boundary;

2. for each z ∈ Z, the curve ψ_{z}(·) = Ψ(z, ·) is a W^{u,s}(F)-path with ψ_{z}(0) = p_{0} and ψ_{z}(1) ∈ W^{c}(q_{0});

3. there is a constant C such that every path ψ_{z} has at most C legs;

4. ψ_{z}(1) ≠ q_{0} for all z ∈ ∂Z;

5. the map (Z, ∂Z) → (W^{c}(q_{0}), W^{c}(q_{0}) \ {q_{0}}) defined by z ↦ Ψ(z, 1) has positive degree.

Lemma 5.2 Suppose q_{0} can be centrally engulfed from p_{0}. Then p_{0} can be engulfed from p_{0}.

Proof: Let D^{a} denote the closed unit disc with the same dimension as the leaves of W^{a}. Since W^{c} is integrable, there is a homeomorphism ρ : D^{u} × D^{s} × D^{c} onto a neighborhood V of q_{0} such that:

1. ρ(0, 0, 0) = q_{0};

2. ρ({0} × {0} × D^{c}) ⊂ W^{c}(q_{0});

3. ρ(D^{u} × {0} × {x_{c}}) ⊂ W^{u}(ρ(0, 0, x_{c})) for all x_{c} ∈ D^{c};

4. ρ({x_{u}} × D^{s} × {x_{c}}) ⊂ W^{s}(ρ(x_{u}, 0, x_{c})) for all (x_{u}, x_{c}) ∈ D^{u} × D^{s}.

By the same argument as in the proof of Proposition 5.1, we may assume that Ψ(z, 1) ∈ ρ({0} × {0} × D^{c}) for all z ∈ Z. For z ∈ Z, let (0, 0, x_{c}(z)) = ρ^{−1}(Ψ(z, 1)). Now for any (x_{u}, x_{s}, z) ∈ D^{u}×D^{s}×Z, we can construct a W^{u,s}(F)- path \overline{ψ}_{(x_{u},x_{s},z)} by concatentating ψ_{z} with the images under ρ of the line segments from (0, 0, x_{c}) to (x_{u}, 0, x_{c}) and from (x_{u}, 0, x_{c}) to (x_{u}, x_{s}, x_{c}).

It is easy to see that the map \overline{Ψ} : D^{u} × D^{s} × Z × [0, 1] → N defined by \overline{Ψ}(x_{u}, x_{s}, z, t) = \overline{ψ}_{(x_{u},x_{s},z)}(t) is an engulfing of q_{0} from p_{0}. □

An immediate consequence of the above results is

Corollary 5.3 Let F : M → M be a dynamically coherent, partially hyperbolic diffeomorphism. Suppose that there is a point p_{0} such that any point of M can be reached from p_{0} along a W^{u,s}(F)-path and p_{0} can be centrally engulfed from p_{0}. Then F is stably accessible.

6 Right invariant equivalence relations on M×G

In this section ∼ is an equivalence relation on M × G that is invariant under the right action of G, in the sense that

(x_{1}, g_{1}) ∼ (x_{2}, g_{2}) ⇒ (x_{1}, g_{1}g) ∼ (x_{2}, g_{2}g)

for all g ∈ G and all (x_{1}, g_{1}), (x_{2}, g_{2}) ∈ M ×G. Write g_{1} ∼_{x} g_{2} if (x, g_{1}) ∼ (x, g_{2}). Observe that the following properties hold.

• H_{x} = {g ∈ G : g ∼_{x} e} is a group for each x ∈ M and the ∼_{x} equivalence classes are closets that belong to H_{x}\G.

• If ∼ is invariant under f_{φ}, in the sense that f_{φ}(x_{1}, g_{1}) ∼ f_{φ}(x_{2}, g_{2}) whenever (x_{1}, g_{1}) ∼ (x_{2}, g_{2}), then f_{φ} induces a map on ⋃_{x∈M} H_{x}\G.

• If ∼ is closed (i.e. the ∼ equivalence classes are closed), then each H_{x} is a closed Lie subgroup of G.

• If (x_{1}, g_{1}) ∼ (x_{2}, g_{2}), then H_{x_{2}} = g_{0}H_{x_{1}}g_{0}^{−1}, where g_{0} = g_{2}g_{1}^{−1}. The ∼_{x_{2}} equivalence class of g_{0} is g_{0}H_{x_{1}} = H_{x_{2}}g_{0}.

Suppose now that the equivalence class of (x_{0}, g_{0}) meets {x}×G for every x ∈ M. Then all the equivalence classes have this property and all the subgroups H_{x} belong to a single conjugacy class. The following properties hold.

• The map Φ : x ↦ {g ∈ G : (x, g) ∼ (x_{0}, g_{0})} takes M to G/H_{x_{0}}. The map π_{Φ} : (x, g) ↦ (x, Φ(x)^{−1}g) takes ∼ equivalence classes to constant sections of M × H_{x_{0}}\G.

• If ∼ is closed and invariant under f_{φ}, then Φ induces an algebraic factor of f.

• If f_{φ} maps each ∼ equivalence class into itself, then the map induced on M × H_{x_{0}}\G is f × Id.

• If there is one ∼ equivalence class that meets {x} × G for every x ∈ M, then all the equivalence classes have this property and all the subgroups H_{x} belong to a single conjugacy class.

We define the leaf-wise closure \overline{∼} of ∼ by saying that (x, g) \overline{∼} (x′, g′) if there are sequences g_{n} → g and g′_{n} → g′ such that (x, g_{n}) ∼ (x′, g′_{n}) for each n.

The closure of ∼ is the relation \overline{\overline{∼}} defined by setting (x, g) \overline{\overline{∼}} (y, h) if there are sequences (x_{n}, g_{n}) → (x, g) and (y_{n}, h_{n}) → (y, h) such that (x_{n}, g_{n}) ∼ (y_{n}, h_{n}) for each n.

Lemma 6.1 \overline{∼} is an equivalence relation. The group \{g ∈ G : g \overline{∼}_{x} e\} is the closure of H_{x}. Every \overline{∼} equivalence class is a union of ∼ equivalence classes.

Proof: Observe that if (x, g) \overline{∼} (y, h), then there are sequences g′_{n} → g and h′_{n} → h such that (x, g′_{n}) ∼ (y, h) and (x, g) ∼ (y, h_{n}) for each n. Indeed, if g_{n} and h_{n} are as above, we can take g′_{n} = g_{n}h_{n}^{−1}h and h′_{n} = h_{n}g_{n}^{−1}g. Hence if (x, g) \overline{∼} (y, h) and (y, h) \overline{∼} (z, k), there are sequences g′_{n} → g and k′_{n} → k such that (x, g′_{n}) ∼ (y, h) ∼ (z, k′_{n}), which implies that (x, g) \overline{∼} (z, k). Thus \overline{∼} is transitive. Since \overline{∼} is obviously symmetric and reflexive, it is an equivalence relation.

By the above, g \overline{∼}_{x} e if and only if there is a sequence g′_{n} → g such that g′_{n} ∼_{x} e for each n. This proves the second claim.

Since (x, g) ∼ (x′, g′) ⇒ (x, g) \overline{∼} (x′, g′), the \overline{∼} equivalence class of (x, g) contains the ∼ equivalence class of (x, g). It follows that every \overline{∼} equivalence class is a union of ∼ equivalence classes. □

There is a natural condition under which the relations \overline{∼} and \overline{\overline{∼}} coincide.

Definition: We say that ∼ is continuous if each x ∈ M has a neighborhood U(x) on which there is a continuous function Γ_{x} : U(x) → G such that (u, Γ_{x}(u)) ∼ (x, e) for all u ∈ U(x).

Lemma 6.2 Let ∼ be continuous. Then the relations \overline{∼} and \overline{\overline{∼}} coincide.

Proof: It suffices to show that if (x_{n}, g_{n}) → (x, g), then there is a sequence g′_{n} → g such that (x_{n}, g_{n}) ∼ (x, g′_{n}) for all large enough n. But g′_{n} = Γ_{x}(x_{n})^{−1}g_{n} has the desired properties. □

7 W^{u,s}(f_{φ})-paths

In this section, we develop some properties of W^{u,s}(f_{φ})-paths that will be needed in the following sections. The first result is a corollary of Proposition 2.1. We will use it repeatedly. For j ≥ 1 and x ∈ M, let

φ_{j}(x) = φ(f^{j−1}(x)) ⋯ φ(x).

Note that (f_{φ})^{j} = (f^{j})_{φ_{j}}.

Proposition 7.1 For any g ∈ G and any W^{u,s}(f)-path γ : [0, 1] → M, there is a unique W^{u,s}(f_{φ})-path γ_{g} : [0, 1] → M × G such that γ_{g}(0) = (γ(0), g) and π_{M} ∘ γ_{g} = γ. Moreover, the following properties hold:

1. if γ′ : [0, 1] → M is fixed-end-point homotopic to γ, then γ′_{g}(0) = γ_{g}(0);

2. for any g′ ∈ G, γ_{g′} = R_{g^{−1}g′} ∘ γ_{g}.

3. if γ(0) = x, then (f^{j} ∘ γ)_{φ_{j}(x)} = f_{φ}^{j}(γ_{e}).

The W^{u,s}(f_{φ})-paths can be viewed as being horizontal with respect to a “connection” whose horizontal distribution is E_{f_{φ}}^{s} ⊕ E_{f_{φ}}^{u}. We do not have a connection in the usual sense, since this distribution may not be differentiable, and we are only able to lift W^{u,s}(f)-paths rather than arbitrary piecewise smooth paths. The following proposition shows, however, that the W^{u,s}(f)-paths will be sufficient for our purposes.

Proposition 7.2 Let f : M → M be an Anosov diffeomorphism.

1. Any path can be approximated arbitrarily closely in the C^{0} topology by a W^{u,s}(f)-path with the same endpoints.

2. Every homotopy class of loops based at x_{0} is represented by a W^{u,s}(f)-path.

3. Homotopic W^{u,s}(f)-paths are homotopic through W^{u,s}(f)-paths.

Proof: The first two assertions follow from the fact that W_{f}^{u} and W_{f}^{s} are uniformly transverse.

To prove the third assertion, let γ_{1} and γ_{2} be W^{u,s}(f_{φ})-paths based at x_{0}, and let A(s, t) be a basepoint-fixing homotopy from γ_{1}(t) to γ_{2}(t). We may assume that γ_{2} is the constant path x_{0} and that the diameter of A([0, 1]×[0, 1]) is smaller than the injectivity radius of M. Since γ_{1} and γ_{2} are piecewise smooth, we may assume that the homotopy A(s, t) is through piecewise smooth paths. Now chop [0, 1] into subintervals at points 0 = t_{0} < t_{1} < ⋯ < t_{M} = 1 so that γ_{1} restricted to [t_{i}, t_{i+1}] lies entirely in a W_{f}^{a_{i}}-leaf , where a_{i} = u or s. For each s replace the path A(s, ·)|_{[t_{i},t_{i}+1]} by a two-legged W^{u,s}(f_{φ})-path Ã(s, ·)|_{[t_{i},t_{i}+1]}, such that:

• each W_{f}^{a}-leg of Ã(s, ·)|_{[t_{i},t_{i}+1]} is a geodesic inside W_{f}^{a};

• the W_{f}^{a_{i}}-leg of Ã(s, ·)|_{[t_{i},t_{i}+1]} is the first leg;

• Ã(s, t_{i}) = A(s, t_{i}), and Ã(s, t_{i+1}) = A(s, t_{i+1}).

Since this construction is canonical, it produces a continuous homotopy Ã(s, t) through W^{u,s}(f)-paths from γ_{1} to γ_{2}. □

We conclude this section with some notational conventions that will be used in the sequel. If α is a path, \overline{α} will be the path obtained from α by reversing direction. If β is a path whose initial point is the same as the endpoint of α, we shall denote the concatenation of α and β by α · β. If α and β are paths in M × G with the endpoint of α and the initial point of β in the same fiber, we define α ∗ β to be α · (R_{g} ∘ β), where g is chosen so that R_{g} moves the initial point of β to the endpoint of α. We can also define α ∗ β when α and β are paths in G.

8 Holonomy groups

The results in this section build on ideas developed in two papers of Brin ([B1] and [B2]). Let f_{φ} : M × G → M × G be a skew product, where f : M → M is a volume preserving C^{k−} Anosov diffeomorphism of a compact manifold, G is a Lie group with a bi-invariant metric and f : M → G is C^{k−}. We introduce several equivalence relations on M ×G. All of these equivalence relations will be invariant under f_{φ} and R_{g} for all g ∈ G. As we observed in the previous section, the set of g ∈ G such that (x, g) is equivalent to (x, e) is a group for each x ∈ M. These groups will be fundamental in the later sections of this paper.

We shall say that (x_{1}, g_{1}) ∼ (x_{2}, g_{2}) if there is a W^{u,s}(f_{φ})-path from (x_{1}, g_{1}) to (x_{2}, g_{2}). We shall say that (x, g_{1}) ∼^{∘} (x, g_{2}) if there is a W^{u,s}(f_{φ})-path from (x, g_{1}) to (x, g_{2}) whose projection to M is a null homotopic loop. Propositions 2.1 and 7.1 imply that these relations are invariant under R_{g} for all g ∈ G and each ∼ equivalence class meets {x} × G for every g ∈ G.

Definition: The holonomy group for f_{φ} at x_{0} is

H_{x_{0}}(f_{φ}) = {g ∈ G : (x_{0}, g) ∼ (x_{0}, e)}.

The restricted holonomy group for f_{φ} at x_{0} is

H_{x_{0}}^{0}(f_{φ}) = {g ∈ G : (x_{0}, g) ∼^{∘} (x_{0}, e)}.

When it is clear which diffeomorphism we are referring to, we will write H_{x_{0}} and H_{x_{0}}^{0} instead. The conjugacy class of these subgroups is independent of x_{0}. Since the basepoint x_{0} will never vary, we normally omit it from the notation.

The following proposition and its proof are adaptations of standard arguments in the theory of connections on principal fiber bundles. See, e.g., [N] for an analogous discussion.

Proposition 8.1 H and H^{0} have the following properties.

1. H^{0} is an analytic subgroup (i.e. a connected Lie subgroup) of G.

2. H^{0} is a normal subgroup of H.

3. There is a surjective homomorphism ρ : π_{1}(M, x_{0}) → H/H^{0}.

4. If f^{j}(x_{0}) = x_{0}, then ρ(f_{∗}^{j}[α]) is conjugate to ρ([α]) for all [α] ∈ π_{1}(M, x_{0}):

ρ(f_{∗}^{j}[α]) = φ_{j}(x_{0})ρ([α])φ_{j}(x_{0})^{−1}.

In particular, if H/H^{0} is abelian, then ρ(f_{∗}^{j}[α]) = ρ([α]).

5. H is a Lie subgroup of G, and H^{0} is the connected component of the identity in H.

Proof: The proof of (2) is straightforward: if (x_{0}, h) is the endpoint of a W^{u,s}(f_{φ})-path α from (x_{0}, e) whose projection π_{M} ∘ α to M is null-homotopic, and (x_{0}, g) is the endpoint of a W^{u,s}(f_{φ})-path β from (x_{0}, e), then (x_{0}, g^{−1}hg) is the endpoint of the W^{u,s}(f_{φ})-path:

γ = β ∗ α ∗ \overline{β} = β · (R_{g} ∘ α) · \overline{(R_{g^{−1}hg} ∘ β)}

from (x_{0}, e). Then g^{−1}hg ∈ H^{0}, since

π_{M} ∘ γ = (π_{M} ∘ β) · (π_{M} ∘ α) · \overline{π_{M} ∘ β},

which is null-homotopic, since π_{M} ∘ α is null-homotopic.

We now show (1). For any element g ∈ H^{0}, there is a W^{u,s}(f_{φ})-path γ : [0, 1] → M × G such that γ(0) = (x_{0}, e), γ(1) = (x_{0}, g), and such that π_{M} ∘ γ is homotopic to the trivial loop through a homotopy fixing the basepoint x_{0}.

By Proposition 7.2, we may choose a homotopy through W^{u,s}(f)-loops in M based at x_{0} from the trivial loop to π_{M} ∘γ. By Proposition 7.1, each path in this homotopy has a lift to a W^{u,s}(f_{φ})-path in in M × G that starts at (x_{0}, e) and ends in π_{M}^{−1}(x_{0}). This gives us a continuous map A : [0, 1] × [0, 1] → M × G such that, for each s ∈ [0, 1], the path A(s, ·) is a W^{u,s}(f_{φ})-path in M × G starting at (x_{0}, e), covering a null-homotopic loop in M and ending in the fiber over x_{0}. Moreover A(0, ·) is the trivial loop at (x_{0}, e) and A(1, 1) = γ(1) = (x_{0}, g). The path π_{G} ∘ A(·, 1) is a path in H^{0} from e to g. Thus H^{0} is path-connected. By a theorem of Kuranashi-Yamabe [Y], a path-connected subgroup of a Lie group is an analytic Lie subgroup. This proves (1).

By Proposition 7.2, each element of π_{1}(M, x_{0}) is represented by a W^{u,s}(f)- loop based at x_{0}, and homotopic W^{u,s}(f)-paths are W^{u,s}(f)-homotopic. Let α be a W^{u,s}(f)-loop in M that represents the class [α] ∈ π_{1}(M, x_{0}). Let α_{e} be the lift of α given by Proposition 7.1, satisfying α_{e}(0) = (x_{0}, e). Define ρ([α]) by:

ρ([α]) = p ∘ π_{G}(α_{e}(1)),

where p : G → H/H^{0} is the canonical projection. Then ρ is well-defined, for if α′ ∈ [α] is another representative of the same based homotopy class, then α · \overline{α′} is null-homotopic, implying that π_{G}(α_{e}(1))π_{G}(α′_{e}(1))^{−1} ∈ H^{0}. This defines a surjective homomorphism ρ : π_{1}(M, x_{0}) → H/H^{0}, proving (3).

To show (4), suppose that f^{j}(x_{0}) = x_{0}, and let α be a W^{u,s}(f)-loop based at x_{0}. Then f^{j} ∘ α is a loop based at x_{0} and, by Proposition 7.1, the W^{u,s}(f_{φ})-loop f_{φ}^{j} ∘ α_{e} is a lift of f^{j} ∘ α, with

π_{G} (f_{φ}^{j} ∘ α_{e}(0)) = φ_{j}(x_{0})

and

π_{G} (f_{φ}^{j} ∘ α_{e}(1)) ∈ φ_{j}(x_{0})ρ([α]).

Thus

ρ([f^{j} ∘ α]) = φ_{j}(x_{0})ρ([α])φ_{j}(x_{0})^{−1},

which proves (4).

Let H′ be the connected component of the identity e in H. Clearly H^{0} ⊆ H′. Note that H′ is also a connected Lie subgroup of G, being path-connected. Since ρ is surjective and π_{1}(M) is countable, so is H/H^{0}, and therefore, so is H′/H^{0}. This implies that H is open in H′, and so H = H′, proving (5). □

Observe that if γ_{1} and γ_{2} are two W^{u,s}(f_{φ})-paths that begin at (x_{0}, e) and end at two points (x, g_{1}) and (x, g_{2}) in the same fiber of M ×G, then γ_{1} ·R_{g_{2}^{−1}g_{1}} \overline{γ_{2}} is a W^{u,s}(f_{φ})-path from (x_{0}, e) to (x_{0}, g_{2}^{−1}g_{1}), and hence g_{2}^{−1}g_{1} ∈ H. It follows that the projections to G of the endpoints of all the W^{u,s}(f_{φ})-paths from (x_{0}, e) to {x} × G lie in a single coset belonging to G/H. Moreover the endpoints of these paths fill up the entire coset, since if γ is a W^{u,s}(f_{φ})-path from (x_{0}, e) to (x, g) and σ is a W^{u,s}(f_{φ})-path from (x_{0}, e) to (x_{0}, h) for some h ∈ H, then σ · R_{h}γ is a W^{u,s}(f_{φ})-path from (x_{0}, e) to (x_{0}, gh). Thus there is a well defined map

Φ : M → G/H

such that Φ(x) is the set of endpoints of the W^{u,s}(f_{φ})-paths from (x_{0}, e) to {x} × G. More generally, for any g_{0} ∈ G, the set of endpoints of the W^{u,s}(f_{φ})- paths from (x_{0}, g_{0}) to {x} × G is {x} × Φ(x)g_{0}.

We cannot always talk about smoothness of the map Φ, because G/H is not a manifold unless H is closed. However we have:

Proposition 8.2 Let \overline{Φ} : M → G/\overline{H} be the composition of Φ with the natural projection G/H → G/\overline{H}. Then \overline{Φ} is C^{k−}. In particular Φ itself is C^{k−} if H is closed.

Proof: By a theorem of Journé [J], it suffices to show that \overline{Φ} is uniformly C^{k−} along the leaves of the foliations W_{f}^{u} and W_{f}^{s}.

Let B(x, δ) be the geodesic ball around x in the metric that we chose on M in Section 2. There is a small enough δ > 0 such that for each x ∈ M every y ∈ B(x, δ) can be reached from x along a short W^{u,s}(f)-path with two legs, the first in W_{f}^{s}(x) and the second in W_{f}^{u}(y). Let Γ_{x}(y) be the endpoint of the lift of this path to a W^{u,s}(f_{φ})-path starting at (x, e).

Let W_{f}^{s}(x; δ) and W_{f}^{u}(x; δ) be the components of x in W_{f}^{s}(x) ∩ B(x, δ) and W_{f}^{u}(x) ∩ B(x, δ) respectively and define Γ_{x}^{s} and Γ_{x}^{u} to be the restrictions of Γ_{x} to W_{f}^{s}(x; δ) and W_{f}^{u}(x; δ) respectively. It follows from Propositions 2.1 and 7.1 that the functions Γ_{x}^{s} and Γ_{x}^{u} are C^{k−} and vary continuously with x in the C^{k−} topology.

Observe that if y ∈ W_{f}^{u}(x_{0}; δ) and z ∈ W_{f}^{s}(x_{0}; δ), then

\overline{Φ}(y) = Γ_{x_{0}}^{u}(y)\overline{Φ}(x) and \overline{Φ}(z) = Γ_{x_{0}}^{s}(z)\overline{Φ}(x).

The desired smoothness of \overline{Φ} follows immediately. □

Corollary 8.3 The equivalence relation ∼ is continuous.

Proof: The desired functions Γ_{x} were constructed in the previous proof. They are obviously continuous. □

8.1 The ergodic isotropy subgroup

We now introduce two further equivalence relations on M × G. We shall say that (x_{1}, g_{1}) ˆ∼ (x_{2}, g_{2}) if (x_{1}, g_{1}) ∼ f_{φ}^{j}(x_{2}, g_{2}) for some integer j. Observe that ˆ∼ is continuous, because ∼ is continuous by Corollary 8.3. We define ≈ to be the closure of the relation ˆ∼. Let Ĥ = {g ∈ G : (x_{0}, e) ˆ∼ (x_{0}, g)} and K = {g ∈ G : (x_{0}, e) ≈ (x_{0}, g)}. Then K is the closure of Ĥ by Lemmas 6.1 and 6.2. Since it is closed, K is a Lie subgroup of G. Brin [B2] proved that the ≈ equivalence classes are the ergodic components for the action of f_{φ} on M × G. For this reason, we call K the ergodic isotropy subgroup at x_{0}.

Proposition 8.4 \overline{H^{0}} and \overline{H} are normal subgroups of K.

Proof: \overline{H^{0}} is normal because it is the connected component of the identity in K. To see this, recall from Proposition 8.1 that H^{0} is the connected component of the identity in the Lie subgroup H, and observe that H has countable index in Ĥ. Hence H^{0} is a connected Lie subgroup with countable index in Ĥ and so \overline{H^{0}} is a connected Lie subgroup with countable index in K.

In order to show that \overline{H} is normal in K, it suffices to show that H is a normal subgroup of Ĥ. Let h ∈ H and ĥ ∈ Ĥ. Then there is a W^{u,s}(f_{φ})-path σ from (x_{0}, e) to f_{φ}^{j}(x_{0}, ĥ) = (f^{j}x_{0}, φ_{j}(x_{0})ĥ), where

φ_{j}(x_{0}) = φ(f^{j−1}x_{0})φ(f^{j−2}x_{0}) ⋯ φ(fx_{0})φ(x_{0}).

We claim that the W^{u,s}(f_{φ})-path

γ ∗ (f_{φ}^{j} ∘ σ) ∗ \overline{γ} = γ · Rĥ(f_{φ}^{j} ∘ σ) · R_{ĥ^{−1}hĥ}\overline{γ}

joins (x_{0}, e) to (x_{0}, ĥ^{−1}hĥ), which implies ĥ^{−1}hĥ ∈ H. Indeed f_{φ}^{j} ∘ σ joins (f^{j}x_{0}, ϕ_{j}(x_{0})) to (f^{j}x_{0}, ϕ_{j}(x_{0})h) and hence Rĥ(f_{φ}^{j} ∘ σ) joins (f^{j}x_{0}, ϕ_{j}(x_{0})ĥ) to (f^{j}x_{0}, ϕ_{j}(x_{0})hĥ). Similarly, \overline{γ} joins (f^{j}x_{0}, ϕ_{j}(x_{0})ĥ) to (x_{0}, e) and hence R_{ĥ^{−1}hĥ}\overline{γ} joins (f^{j}x_{0}, ϕ_{j}(x_{0})hĥ) to (x_{0}, ĥ^{−1}hĥ). □

Lemma 8.5 K/\overline{H} is abelian.

Proof: The map j ↦ {ĥ ∈ Ĥ : (x_{0}, e) ∼ f_{φ}^{j}(x_{0}, ĥ)} is a surjective homomorphism from Z to Ĥ/H and ĥH ↦ ĥ\overline{H} is a homomorphism from Ĥ/H to K/\overline{H} whose image is dense. □

The holonomy and isotropy subgroups behave naturally under quotients:

Lemma 8.6 Let N be a normal subgroup of a Lie group G and let f_{ψ} : M × N\G → M × N\G be an algebraic factor of f_{φ}. Let p : G → N\G be the canonical projection. Then H_{x}^{0}(f_{ψ}) = p(H_{x}^{0}(f_{φ})), H_{x}(f_{ψ}) = p(H_{x}(f_{φ})), and K_{x}(f_{ψ}) = p(K_{x}(f_{φ})).

Proof: This will follow easily if we can show that

W_{f_{ψ}}^{a}(x, p(g)) = (id × p)(W_{f_{φ}}^{a}(x, g)),

for a = u or s. We consider a = s; the other case is similar. If (x′, g′) ∈ W_{f_{φ}}^{s}(x, g), we have dist(f_{φ}^{n}(x′, g′), f_{φ}^{n}(x, g)) → 0 as n → ∞ and hence

dist(f_{ψ}^{n}(x′, p(g′)), f_{ψ}^{n}(x, p(g))) → 0 as n → ∞.

It follows that (id × p)(W_{f_{φ}}^{s}(x, g)) ⊂ W_{f_{ψ}}^{s}(x, p(g)). On the other hand, it easy to see that (id × p)(W_{f_{φ}}^{s}(x, g)) and W_{f_{φ}}^{s}(x, g) both lie in W_{f}^{s}(x) × N\G and both contain exactly one point in {x′} × N\G for each x′ ∈ W_{f}^{s}(x′). Hence W_{f_{ψ}}^{s}(x, p(g)) = (id × p)(W_{f_{φ}}^{s}(x, g)).□

9 Engulfing and the proof of Theorem A

Let f_{φ} : M × G → M × G be as in the previous section and assume in addition that the group G is compact. Recall that we are using a bi-invariant metric on G. With this choice of metric the Lie group and Riemannian exponential maps coincide.

Choose a point x_{0} ∈ M and let H^{0} = H_{x_{0}}^{0} be the restricted holonomy group at x_{0} as defined in the previous section. The main goal of this section is to prove

Theorem 9.1 If H^{0} = G, then f_{φ} is stably ergodic.

Theorem A follows from this theorem and a result of Brin that we present at the end of this section. Note that if H^{0} = G, then H_{x}^{0} = G for every x ∈ M; this is evident from Proposition 8.1. The results in Section 12 imply that the converse of Theorem 9.1 holds when M is an infranilmanifold, but we do not know whether the converse of Theorem 9.1 holds without this additional hypothesis.

Proof of Theorem 9.1: Since H^{0} = G, every x ∈ M has the property that every point in {x} × G can reached from (x, e) along a W^{u,s}(f_{φ})-path. It follows from this and the fact that every x ∈ M can be reached from x_{0} along a W^{u,s}(f)- path that every (x, g) ∈ M × G can be reached from (x_{0}, e) along a W^{u,s}(f_{φ})- path. By Corollaries 2.2 and 5.3, it suffices to prove that (x_{0}, e) can be centrally engulfed from (x_{0}, e).

For this, it will suffice to find a continuous map Ψ : D^{c} × [0, 1] → M × G, where D^{c} is homeomorphic to the closed disc of dimension c = dim(G), with the following properties:

1. for each z ∈ D^{c}, the path ψ_{z}(·) = Ψ(z, ·) is a W^{u,s}(f_{φ})-path with ψ_{z}(0) = (x_{0}, e) and ψ_{z}(1) ∈ {x_{0}} × G;

2. there is a constant C such that each path ψ_{z} has at most C legs.

3. ψ_{z}(1) ≠ (x_{0}, e) for any z ∈ ∂D^{c};

4. the map H_{c}(D^{c}, ∂D^{c}) → H_{c}(G, G \ {e}) induced by z → π_{G}(ψ_{z}(1)) is nontrivial.

Let us call a map h : Z → G achievable if it is the “endpoint map” of a continuous family of W^{u,s}(f_{φ})-paths that begin at (x_{0}, e), end in {x_{0}} × G and project to null-homotopic loops in M. More precisely, h is achievable if there are a continuous function H : Z × [0, 1] → M × G and a positive integer C such that, for each z ∈ K, the path h_{z} : [0, 1] → M × G defined by h_{z}(t) = H(z, t) is a W^{u,s}(f_{φ})-path from (x_{0}, e) to (x_{0}, h(z)) with at most C legs that is the lift of a null homotopic loop in M.

The notion of achievable map allows us to reformulate the desired properties of Ψ. What we want is an achievable map ψ : (D^{c}, ∂D^{c}) → (G, G \ {e}) that induces a nontrivial map on c-dimensional homology. Moreover it is not necessary that ψ itself be achievable; it will suffice if ψ can be approximated arbitrarily closely in the C^{0} topology by achievable maps.

Definition: A map h : Z → G is approximable if for each ε > 0 there is an achievable map h^{ε} : Z → G such that dist_{C^{0}}(h, h^{ε}) < ε.

Lemma 9.2 Let h_{i} : Z_{i} → G, 1 ≤ i ≤ k, be approximable. Then the product map

(z_{1}, …, z_{k}) ↦ h_{k}(z_{k})h_{k−1}(z_{k−1}) ⋯ h_{2}(z_{2})h_{1}(z_{1})

is approximable.

Proof: Since the map

C^{0}(Z_{1}, G) × ⋯ × C^{0}(Z_{k}, G) → C^{0}(Z_{1} × ⋯ × Z_{k}, G)

that takes (h_{1}, …, h_{k}) to (z_{1}, …, z_{k}) ↦ h_{k}(z_{k})h_{k−1}(z_{k−1}) ⋯ h_{2}(z_{2})h_{1}(z_{1}) is continuous, it suffices to show that a product of achievable maps is achievable. This in turn reduces to showing that if h′ : Z′ → G and h′′ : Z′′ → G are achievable, then so is the map h : Z′ × Z′′ → G defined by h(z′, z′′) = h′′(z′′)h′(z′). But this is true, since we can choose the W^{u,s}(f_{φ})-path h_{(z′,z′′)} from (x_{0}, e) to (x_{0}, h_{(z′,z′′)}) whose existence is required by the definition of achievability to be h′ ∗ h′′ = h′_{z′} · R_{h′(z′)}h′′_{z′′}. □

Suppose now that we knew that the geodesic arc, σ_{v} : [0, 1] → G, σ_{v}(s) = exp(sv), was approximable for all v ∈ 𝔤. Then the map

ψ(z_{1}, …, z_{c}) = exp(z_{c}v_{c}) exp(z_{c−1}v_{c−1}) ⋯ exp(z_{2}v_{2}) exp(z_{1}v_{1})

is approximable for any choice of v_{1}, …, v_{c} ∈ 𝔤. Moreover ψ will be a diffeomorphism of [−1, 1]^{c} onto a neighborhood of e if we choose the v_{i} to be short enough and to form a basis for 𝔤 that is orthogonal with respect to the bi-invariant metric.

Thus we could obtain the desired central engulfing of (x_{0}, e) from (x_{0}, e) by taking D^{c} = [−1, 1]^{c} and ψ as above, if we knew that the geodesic arc σ_{v} is approximable for every v ∈ 𝔤. Theorem 9.1 now follows from the next proposition.□

Proposition 9.3 The geodesic arc σ_{v} is approximable if and only if v is tangent to H^{0}.

Proof: Let us call v ∈ 𝔤 approximable if σ_{v} is approximable. The first step will be to show that if v is approximable, then so is any multiple of v. Before doing this, we note some elementary properties of approximable paths (i.e. approximable maps from [0, 1] to G) that will needed. Recall that if α and β are paths in G, then α ∗ β is the path formed by concatenating α with R_{g} ∘ β, where g is chosen so that R_{g} moves the initial point of α to the endpoint of β. If the paths α and β are approximable paths, so is α ∗ β. Also any subpath, any right translate and any reparametrization (even direction reversing) of an approximable path is again approximable.

Lemma 9.4 If v ∈ 𝔤 is approximable, then λv is approximable for all λ ∈ R.

Proof: The vector 0 is approximable, because σ_{0} is achievable, since σ_{0}(t) = e for 0 ≤ t ≤ 1.

Observe that (with suitable reparametrization) σ_{λv} is a subpath of σ_{v} for any λ ∈ (0, 1) and σ_{2v} = σ_{v} ∗ σ_{v}. Moreover σ_{−v} is (after an orientation reversing reparametrization) a right translate of σ_{v}, since σ_{−v}(t) = σ_{v}(1 − t) exp(−v).

If v is approximable, then so are λv for any λ ∈ [0, 1), 2v and −v. The lemma follows easily.

The lemma tells us that if a vector v ∈ 𝔤 is approximable, then every geodesic segment contained in the one parameter subgroup tangent to v is approximable. It follows that if v_{1}, …, v_{k} are approximable vectors, then the geodesic polygon σ_{v_{1}} ∗ ⋯ ∗ σ_{v_{k}} is approximable.

We now formulate a criterion for a vector v ∈ 𝔤 to be approximable. Let r_{inj} be the injectivity radius of G with respect to our bi-invariant metric. Then the exponential map is a diffeomorphism from the open ball B in 𝔤 of radius r_{inj} to the geodesic ball U of radius r_{inj} about e in G. Let exp^{−1} : U → B be the inverse of exp : B → U.

Lemma 9.5 A nonzero vector v ∈ 𝔤 is approximable if there is a sequence of approximable paths γ_{n} : [0, 1] → G with γ_{n}(0) = e such that:

1. there is a sequence c_{n} ↘ 0 such that dist(γ_{n}(t), e) < c_{n} for 0 ≤ t ≤ 1;

2. if w_{n} = exp^{−1}γ_{n}(1) for those n such that c_{n} < r_{inj}, then \frac{∥v∥}{∥w_{n}∥}w_{n} → v.

Proof: The path γ_{n}^{k} = \overbrace{γ_{n} ∗ ⋯ ∗ γ_{n}}^{k} is approximable for any positive integer k. Note that σ_{n} is 2c_{n}-approximated by γ_{n} because both paths lie in the geodesic ball of radius c_{n} around e. It follows that σ_{kw_{n}} is 2c_{n}-approximated by γ_{n}^{k} for any k. Let k_{n} be the first integer larger than ∥v∥/∥w_{n}∥. Then the paths γ_{n}^{k_{n}} converge in the C^{0} topology to σ_{v}. Since a C^{0} limit of approximable paths is approximable, σ_{v} is approximable.

Lemma 9.6 The set of approximable vectors is a Lie subalgebra of 𝔤.

Proof: In view of Lemma 9.4, we need to show that if v and w are approximable, then v + w and [v, w] are approximable. Recall (see e.g. Lemma 11.6 in Ch. II §1 of [H]) that for any vectors v, w ∈ 𝔤 we have

exp((v + w)/n) = exp(v/n) exp(w/n) + O(1/n^{2})

and

exp([v, w]/n^{2}) = exp(−v/n) exp(−w/n) exp(v/n) exp(w/n) + O(1/n^{3}).

Let α_{n} = σ_{w/n} ∗σ_{v/n} and β_{n} = σ_{w/n} ∗σ_{v/n} ∗σ_{−w/n} ∗σ_{−v/n}. These paths are approximable if v and w are approximable, and we have α_{n}(1) = exp(v/n) exp(w/n) and

β_{n}(1) = exp(−v/n) exp(−w/n) exp(v/n) exp(w/n).

It follows from Lemma 9.5 that if v and w are approximable, then v + w and [v, w] are both approximable.

Let 𝔤_{app} be the Lie algebra defined by the previous lemma and 𝔤_{app}^{⊥} its orthogonal complement. Let G_{app} be the subgroup tangent to 𝔤_{app}. Proposition 9.3 follows immediately from the next proposition. □

Proposition 9.7 G_{app} = H^{0}.

Proof: We begin by showing that G_{app} ⊂ H^{0}. As above, let B be the open ball of radius r_{inj} about 0 in 𝔤 and U the image of B under the exponential map. We shall show that σ_{v} lies in H^{0} for any v ∈ 𝔤_{app} with ∥v∥ < r_{inj}/2. Since H^{0} is a Lie subgroup, this immediately implies that G_{app} ⊂ H^{0}. If v ∈ 𝔤_{app}, the geodesic segment σ_{v} can be approximated arbitrarily closely in the C^{0} topology by admissible paths. Since right translations carry admissible paths to admissible paths, we see that σ_{v} can be approximated arbitrarily closely by admissible paths that start at e. The definition of an admissible path tells us that an admissible path which starts at e must lie in H^{0}. If we now assume that ∥v∥ < r_{inj}/2, we can conclude that σ_{v} can be approximated arbitrarily closely by paths in H^{0} ∩U. Such paths must lie in the component C of H^{0} ∩ U that contains e. Since C is a closed subset of U (even though C is not closed in G), we see that σ_{v} lies in C and therefore in H^{0}.

Now we want to show that G_{app} = H^{0}. Suppose not. Then there is g ∈ H^{0} \ G_{app}. Since g ∈ H^{0}, there is an admissible path α : [0, 1] → G with α(0) = e and α(1) = g. There must be a t_{0} ∈ [0, 1) and a sequence τ_{n} ↘ 0 such that α(t_{0}) ∈ G_{app} and α(t_{0} +τ_{n}) ∉ G_{app} for each n. Then β(t) = α(t−t_{0})α(t_{0})^{−1} is an admissible path with β(0) = e and β(τ_{n}) ∉ G_{app} for each n.

For r > 0, let B_{r} = {v ∈ 𝔤 : ∥v∥ < r}. Let K = exp(𝔤_{app} ∩ B_{r_{0}}) and N = exp(𝔤_{app}^{⊥} ∩ B_{r_{0}}) with r_{0} < r_{inj} chosen small enough so that there is at most one point in Kg ∩ N for each g ∈ G. For all large enough n, β(τ_{n}) will be close enough to e so that Kβ(τ_{n}) ∩ N ≠ ∅ and therefore contains a unique point g_{n}.

Observe now that g_{n} is joined to e by an approximable path γ_{n} consisting of the arc of β from e to β(τ_{n}) followed by the geodesic segment in Kβ(τ_{n}) from β(τ_{n}) to g_{n}. Both β(τ_{n}) and g_{n} approach e as n → ∞. By passing to a subsequence, we may assume that the sequence of unit vectors exp^{−1}(g_{n})/∥ exp^{−1}(g_{n})∥ converges to a unit vector u. Since u_{n} ∈ 𝔤_{app}^{⊥} for each n, we have u ∈ 𝔤_{app}^{⊥}. On the other hand, it is obvious that the sequence of paths γ_{n} satisfies the hypotheses of Lemma 9.5. Hence u is approximable. This contradiction shows that G_{app} = H^{0}.

Theorem A is an immediate consequence of Theorem 9.1 and the following result of Brin, which shows that the condition H^{0} = G is dense in Ext^{k−}(f, G).

Theorem 9.8 [[B1], Proposition 2.3] Let f_{φ} be a C^{k−}, compact G-extension of an Anosov diffeomorphism f : M → M. Then for any δ > 0, there exists a C^{k−}, G-extension f_{ψ} such that:

• d_{C^{k−}}(f_{φ}, f_{ψ}) < δ, where j is the greatest integer less than k;

• H^{0}(f_{ψ}) = G.

We sketch the proof. First we prove the slightly easier result that φ can be perturbed so as to make H = G. We may choose the basepoint x_{0} of our holonomy groups to be a periodic point, since periodic points for f are dense in M by the Anosov closing lemma. There are infinitely many distinct homoclinic orbits for x_{0}; indeed both W^{u}(x_{0}) and W^{s}(x_{0}) are dense in M, which is a basic set for f. Let z_{1}, …, z_{c}, where c = dim G be homoclinic points for x_{0} that belong to distinct orbits of f. For each z_{i} choose a neighborhood U_{i} with the properties that x_{0} ∉ U_{i} and f^{k}z_{j} ∈ U_{i} if and only if j = i and k = 0. Now choose for each i a two-legged W^{u,s}(f)-loop formed by a path from x_{0} to z_{i} in W^{u}(x_{0}) and a path from z_{i} to x_{0} in W^{s}(x_{0}). Let γ_{i}, 1 ≤ i ≤ c, be the lifts of these loops to W^{u,s}(f_{φ})-paths that begin at (x_{0}, e) and end in {x_{0}} × G. If φ̃ is a perturbation of φ, let γ̃_{i} be the W^{u,s}(fφ̃)-path that starts at (x_{0}, e) and has the same projection to M as γ_{i}. Observe that if φ and φ̃ differ only in U_{i}, then γ_{j} and γ̃_{j} both have the same endpoint for each j ≠ i. It is not difficult to perturb φ within U_{i} so as to move the endpoint of γ̃_{i} in any desired direction within {x_{0}} × G. Consequently we can arrange that the projections to G of the endpoints of the paths γ̃_{1}, …, γ̃_{c} do not lie in any subgroup of G with dimension less than c. Since these projections belong to H(fφ̃), we obtain H(fφ̃) = G.

In order to adapt the preceding argument so that it applies to H^{0}, we need to change the paths γ_{i} so that their projections to M are null homotopic. To this end, we choose periodic points x_{i}, 1 ≤ i ≤ c, close to x_{0}, and for each i a pair of heteroclinic points z_{i}^{0} and z_{i}^{1} such that there are short paths from x_{0} to z_{i}^{0} in W^{u}(x_{0}), from z_{i}^{0} to x_{i} in W^{s}(x_{i}), from x_{i} to z_{i}^{1} in W^{u}(x_{i}), and from z_{i}^{1} to x_{0} in W^{s}(x_{0}). This loop is null homotopic and its lift to a W^{u,s}(f_{φ})-path starting at (x_{0}, e) is the new γ_{i}. Of course we ensure that all of the points were chosen from different orbits of f. The sets U_{i} are now neighborhoods of the points z_{i}^{0} with the properties that none of the periodic points x_{0}, x_{1}, …, x_{c} is in any U_{i} and f^{k}z_{j}^{a} ∈ U_{i} if and only if j = i, a = 0 and k = 0. After these changes, the argument proceeds as before.

10 Reduction and algebraic conjugacy

In the following sections we shall frequently form algebraic factors in two special ways.

The first is a process that we call reduction. Suppose N is a closed normal subgroup of G and p : G → N\G = G/N is the projection. Then the skew product f_{p∘φ} : M × N\G → N\G is the algebraic factor of f_{φ} induced by the constant map Φ(x) = N from M to G/N. By Lemma 8.6, we have K(f_{p∘φ}) = p(K(f_{φ})), H(f_{p∘φ}) = p(H(f_{φ})) and H^{0}(f_{p∘φ}) = p(H^{0}(f_{φ})).

The second process is algebraic conjugacy. In this case we quotient by the trivial group. Suppose φ : M → G and ψ : M → G are cohomologous with respect to f, i.e. there is a function Φ : M → G such that

ψ(x) = Φ(fx)^{−1}ϕ(x)Φ(x).

Let π_{Φ}(x, g) = (x, Φ(x)^{−1}g). Then the following diagram commutes.

[figure: M × G; M × G; M × G; M × G; f_{φ}; f_{ψ}; π_{Φ}; π_{Φ}]

This says that f_{ψ} is the algebraic factor of f_{φ} induced by Φ : M → G/{e} = G. Similarly f_{φ} is the algebraic factor of f_{ψ} induced by the map x ↦ Φ(x)^{−1}.

When the group G is abelian, we shall use additive notation. The condition that φ and ψ are cohomologous becomes

ψ(x) = φ(x) − Φ(f(x)) + Φ(x)

and we have π_{Φ}(x, g) = g − Φ(x).

We shall make several successive reductions and algebraic conjugacies. This is permissible, since it is an easy exercise to show that an algebraic factor of an algebraic factor is again an algebraic factor.

11 The Parry-Pollicott reduction

This section is preparatory to the proof of Theorem B.

Not every function φ : M → G can be lifted to a function from M to the universal cover of G; the group φ_{∗}(π_{1}(M)) is an obstruction. Functions that can be lifted are much easier to work with, especially when one is interested in perturbing them in a specified manner. Parry and Pollicott [PP] have shown that, when G is abelian, and under certain assumptions on f, every skew product f_{ϕ} has an algebraic factor f_{ψ} such that ψ can be lifted. We recall their argument here.

Let M be an infranilmanifold. Then, as was shown in [Ma1], f^{∗} is a hyperbolic linear automorphism of the torsion free part of H^{1}(M, Z). It follows that

(f^{∗} − Id)(H^{1}(M, Z)) is a finite index subgroup of H^{1}(M, Z) (∗).

The next proposition is from [PP].

Proposition 11.1 [PP] Let f_{φ} : M × G → M × G be a C^{k−} skew product such that f satisfies property (∗) and the group G is a torus. Then f_{φ} has an algebraic factor f_{ψ} : M × G_{0} → M × G_{0} such that G_{0} is the quotient of G by a finite subgroup and ψ : M → G_{0} is homotopic to a constant map.

Since ψ is homotopic to the identity, it can be lifted. After choosing an identification of G_{0} with the standard torus T^{d}, where d = dim G_{0}, we can express ψ in the form ψ(x) = (exp(ir_{1}(x)), …, exp(ir_{d}(x))), where r_{i} : M → R, i = 1, …, d.

Proof of Proposition 11.1: We shall consider the special case G = T. The general result then follows by expressing G as a product of circles and applying the special case to each component of φ.

We use additive notation for T.

The set C^{0}(M, T) of continuous functions from M to T is an abelian group under pointwise addition. The Bruschlinsky group π^{1}(M) is C^{0}(M, T) modulo homotopy equivalence. We could just as well have defined π^{1}(M) to be C^{k−} functions modulo C^{k−} homotopy; the proof is a standard exercise.

There is an isomorphism from π^{1}(M) to H^{1}(M, Z), defined as follows. Fix a choice of generator ζ for H^{1}(T, Z) (which amounts to choosing an orientation of T). Observe that if φ_{1} and φ_{2} represent the same class in π^{1}(M), then φ_{1}^{∗}ζ = φ_{2}^{∗}ζ. Thus there is a well defined homomorphism

i_{ζ} : π^{1}(M) → H^{1}(M, Z)

that takes φ to φ^{∗}ζ. In fact i_{ζ} is an isomorphism. For a proof, see e.g. Chapter II.7 of [Hu].

Let ω = i_{ζ}[φ] and let m be the index of (f^{∗}−Id)H^{1}(M, Z) in H^{1}(M, Z). Since mω is in (f^{∗} − Id)H^{1}(M, Z), there is an ω′ ∈ H^{1}(M, Z) such that (f^{∗} − Id)ω′ = mω. Pick any C^{k−} representative Φ ∈ i_{ζ}^{−1}ω′ and note that

i_{ζ}[Φ ∘ f − Φ] = (f^{∗} − Id)ω′ = mω = i_{ζ}[mφ].

The functions Φ ∘ f − Φ and mφ are therefore homotopic, and so their difference \overline{ψ} = mφ − Φ ∘ f + Φ is homotopic to a constant.

Observe that mφ is algebraically conjugate to \overline{ψ}.

Now let Γ_{m} be the finite subgroup of T with m elements. The group G_{0} in the statement of the proposition will be T/Γ_{m}. For t ∈ T, let p_{m}(t) = t + Γ_{m} and q_{m}(t) = t/m + Γ_{m}. The map p_{m} : T → T/Γ_{m} is the natural projection and q_{m} : T → T/Γ_{m} is a group isomorphism. Set ψ = q_{m}∘\overline{ψ}. Then ψ is homotopic to a constant and is algebraically conjugate to q_{m} ∘ mφ; the conjugacy is induced by q_{m} ∘ Φ. But q_{m} ∘ mφ = p_{m} ∘ φ, and f_{p_{m}∘φ} is the factor of φ obtained by reducing by Γ_{m}.

Thus, if we reduce f_{φ} by the finite subgroup Γ_{m}, we obtain a skew product that is algebraically conjugate to the skew product f_{ψ}, and ψ : M → T/Γ_{m} = G_{0} is homotopic to a constant. It follows immediately that f_{φ} has an algebraic factor with the desired property. □

Remark: Instead of mφ, it would have sufficed to consider m′φ, where m′ is the smallest positive integer such that m′ω ∈ (f^{∗} − Id)H^{1}(M, Z). By taking φ = mχ, we see that, for every χ : M → T^{d}, mχ is cohomologous to a function which is homotopic to a constant map.

12 Proof of Theorem B

In this section, f_{φ} : M × G → M × G is a skew product, where f : M → M is a volume preserving C^{k−} Anosov diffeomorphism of a compact infranilmanifold, G is a compact Lie group and f : M → G is C^{k−}. We choose a point x_{0} ∈ M and consider the groups H^{0} = H_{x_{0}}^{0}(f_{φ}), H = H_{x_{0}}(f_{φ}), and K = K_{x_{0}}(f_{φ}) that were defined in Section 8.

From Theorem 9.1, we know that H^{0} = G implies that f_{φ} is stably ergodic. Theorem B follows from

Theorem 12.1 If H^{0} ≠ G then f_{φ} contains an algebraic factor of type (1), (2) or (3), and so is not stably ergodic.

As we remarked after the statement of Theorem B, it is easy to see that factors of types (1), (2) or (3) are not stably ergodic. Thus it follows from Theorem 12.1 that f_{φ} is stably ergodic only if H^{0} ≠ G.

The proof of Theorem 12.1 breaks naturally into four parts.

1. If K ≠ G then f_{φ} has an algebraic factor of type (1).

2. If K = G and \overline{H} ≠ G then f_{φ} has an algebraic factor of type (2).

3. If \overline{H} = G, then \overline{H^{0}} = G.

4. If \overline{H^{0}} = G and H^{0} ≠ G then f_{φ} has an algebraic factor of type (3).

We remark here that the proofs of first two parts do not use the hypothesis that M is an infranilmanifold. Only the last two parts (Propositions 12.4 and 12.7) require the extra hypothesis.

Proposition 12.2 If K ≠ G then f_{φ} has an algebraic factor of type (1).

Proof: Let Φ : M → G be the function given by Proposition 8.2 and let Φ_{0} = p ∘ Φ : M → G/K, where p : G → G/K is the canonical projection. Then Φ_{0} is C^{k−} and maps ≈ equivalence classes in M × G to constant sections of M × K\G, where ≈ is the equivalence relation defined in Section 8.1. The diagram

[figure: M × G; M × G; M × K\G; M × K\G; f_{φ}; (x, g) ↦ (x, Φ_{0}(x)^{−1}g); (x, g) ↦ (x, Φ_{0}(x)^{−1}g)]

induces a map from M × K\G → M × K\G that is an algebraic factor of f_{φ}. Since f_{φ} maps each ≈ equivalence class into itself, this induced map must be f × Id. Thus f_{φ} has an algebraic factor of type 1. □

Proposition 12.3 If K = G and \overline{H} ≠ G then f_{φ} has an algebraic factor of type (2).

Proof: By Proposition 8.4, we know that \overline{H} is a normal subgroup of K. Since we are assuming that K = G, this implies that \overline{H} is normal in G. Thus we can reduce by \overline{H} to obtain a skew product f_{φ_{1}} : M × K/\overline{H} → M × K/\overline{H} that has H(f_{φ_{1}}) = \overline{H}/\overline{H} = \{e\}. The group K/\overline{H} is abelian by Lemma 8.5. Hence we can reduce again, this time by a closed codimension 1 subgroup of K/\overline{H}, to obtain a skew product f_{φ_{2}} : M × T → M × T that has H(f_{φ_{2}}) = {e}.

We now construct an algebraic conjugacy from f_{φ_{2}} to f × R_{α} for a suitable α ∈ T. Note that f × R_{α} is the skew product induced by the constant map that takes all points in M to α.

Recall the equivalence relation ∼ introduced in Section 8 and its properties. Since H(f_{φ_{2}}) = {e}, each of the ∼ equivalence classes for f_{φ_{2}} intersects each fiber of M × T in a single point, and is therefore a graph. In particular, there is a function Φ : M → T such that the equivalence class of (x_{0}, e_{0}) is the graph of Φ. It follows from Proposition 8.2 that Φ is C^{k−}.

Both f_{φ} and translations of M × T carry ∼ equivalence classes into ∼ equivalence classes. Thus if E is the ∼ equivalence class of (x_{0}, e), there is α ∈ T such that f_{φ_{2}}E = R_{α}E. Now for any x ∈ M, we can work out in two different ways where f_{φ_{2}}E intersects {f(x)} × T. We obtain

(f(x), Φ(f(x)) + α) = (f(x), Φ(x) + φ_{2}(x)).

Hence φ_{2}(x) = Φ(f(x)) − Φ(x) + α, and the following diagram commutes.

[figure: M × T; M × T; M × T; M × T; f_{φ_{2}}; f × R_{α}; (x, g) ↦ (x, g − Φ(x)); (x, g) ↦ (x, g − Φ(x))]

Thus f × R_{α} is an algebraic factor of f_{φ_{2}} and hence of f_{φ}. It is a factor of type 2. □

In the remaining propositions we use the assumption that M is an infranilmanifold. Since all Anosov diffeomorphisms of infranilmanifolds have fixed points, we may now assume that the point x_{0}, where our holonomy groups are based, is a fixed point of f. This is convenient in our proof of Lemma 12.6.

Proposition 12.4 If \overline{H} = G, then \overline{H^{0}} = G.

Proof: We suppose that \overline{H^{0}} ≠ G and derive a contradiction. It follows from Proposition 8.1 (part 2) that \overline{H^{0}} is a normal subgroup of \overline{H} = G. Thus we can reduce by \overline{H^{0}} to obtain a skew product f_{φ_{1}} : M × G_{1} → M × G_{1}, where G_{1} = \overline{H^{0}}\G. Let H_{1}^{0} = H^{0}(f_{φ_{1}}) and H_{1} = H(f_{φ_{1}}). By Lemma 8.6,

H_{1}^{0} = H^{0}(f_{φ_{1}}) = p(H^{0}) = {e},

while

\overline{H_{1}} = \overline{H(f_{φ_{1}})} = p(\overline{H}) = p(G) = G_{1}.

By Proposition 8.1, there is a surjective homomorphism

ρ : π_{1}(M) → H_{1}/H_{1}^{0} = H_{1}.

Since M is an infranilmanifold, π_{1}(M) has a normal nilpotent subgroup of finite index, and so H_{1} has a normal nilpotent subgroup N of finite index. Since \overline{H_{1}} = G_{1}, the closure of N is a normal nilpotent subgroup of finite index in G_{1}. Since \overline{N} is compact, it contains a torus of finite index, and so G_{1} contains a torus of finite index. But G_{1} is connected, so G_{1} is a torus, which has positive dimension by our assumption that \overline{H^{0}} ≠ G. After reducing by a closed codimension 1 subgroup of G_{1}, we may assume that G_{1} is a circle.

We will need:

Lemma 12.5 Let f : M → M be an Anosov diffeomorphism of a compact infranilmanifold and let f_{φ_{1}} : M × T → M × T be a skew product such that H^{0}(f_{φ_{1}}) = {0}. Then H(f_{φ_{1}}) is finite and is trivial if φ is homotopic to a constant.

Proof: By Proposition 11.1, f_{φ_{1}} has an algebraic factor f_{ψ} : M × T → M × T, where ψ(x) = exp(ir(x)), for some function r : M → R. Furthermore, H(f_{ψ}) = H(f_{φ_{1}})/F, where F is finite. If φ is homotopic to a constant, we can take ψ = φ. Thus it suffices to show that H(f_{ψ}) is trivial.

Let p_{1} : R → T be the projection p_{1}(x) = exp(ix). Then ψ = p_{1} ∘ r, and the skew product f_{r} : M ×R → M ×R has f_{ψ} as a factor. According to Lemma 8.6, H(f_{ψ}) = p_{1}(H(f_{r})) and p_{1}(H^{0}(f_{r})) = H^{0}(f_{ψ}). Since H^{0}(f_{ψ}) is trivial, H^{0}(f_{r}) is a proper analytic subgroup of R, and therefore H^{0}(f_{r}) = {0}. Lemma 12.5 now follows from the next lemma.□

Lemma 12.6 Let f : M → M be an Anosov diffeomorphism of a compact infranilmanifold and let f_{r} : M × R → M × R be a skew product such that H^{0}(f_{r}) = {0}. Then H(f_{r}) = {0}.

Proof: Since H^{0}(f_{r}) is trivial, part 3 of Proposition 8.1 gives us a surjective homomorphism ρ : π_{1}(M) → H(f_{r}). Since H(f_{r}) ⊂ R, H(f_{r}) is an abelian group and ρ must factor through H_{1}(M, Z). Since R is also torsion-free, ρ must actually factor through the torsion-free part of H_{1}(M, Z). Thus ρ induces a linear map L_{ρ} on H_{1}(M, R), which in turn corresponds to a class ω_{ρ} ∈ H^{1}(M, R).

Recall that the base x_{0}, at which we are computing the holonomy groups, is a fixed point of f. Since H(f_{r}) is abelian, part 4 of Proposition 8.1 tells us that ρ ∘ f_{∗} = ρ.

It follows immediately that f^{∗}ω_{ρ} = ω_{ρ}. Since M is an infranilmanifold, the map (f^{∗} − I) : H^{1}(M, R) → H^{1}(M, R) is invertible. This implies that ω_{ρ} = 0. Hence ρ is the trivial homomorphism and H(f_{r}) = {0}. □

We now complete the proof of Proposition 12.4. Applying Lemma 12.5 to f_{φ_{1}}, we obtain that H_{1} is finite. This contradicts the fact that \overline{H_{1}} = G_{1}. Going back to f_{φ}, we must have \overline{H^{0}} = G. □

Proposition 12.7 If \overline{H^{0}} = G and H^{0} ≠ G then f_{φ} has an algebraic factor of type (3).

Proof: Our first step is to reduce to the case where G is a torus. A similar reduction appears in Field-Parry [FP]. Let G′ be the derived subgroup of G, i.e. the analytic subgroup whose Lie algebra is [𝔤, 𝔤].

Lemma 12.8 G′ is a closed normal subgroup. If D is a dense analytic subgroup of G, then G′ ⊂ D.

Note that H^{0} is a dense analytic analytic subgroup by Proposition 8.1 and our hypothesis that \overline{H^{0}} = G.

Proof of Lemma 12.8: Let 𝔷 be center of 𝔤 and let 𝔤′ be the orthogonal complement of 𝔷 with respect to our bi-invariant inner product. Bi-invariance means that the inner product is Ad-invariant. Differentiating shows that for any X, Y, Z ∈ 𝔤 we have

⟨[Y, X], Z⟩ + ⟨X, [Y, Z]⟩ = 0.

If Z ∈ 𝔷, we obtain

⟨[X, Y], Z⟩ = ⟨X, [Y, Z]⟩ = 0

for any X, Y ∈ 𝔤. Hence [𝔤, 𝔤] ⊂ 𝔤′ and 𝔤′ is an ideal of 𝔤 (as is 𝔷). The center of 𝔤′ is trivial because an element of the center of 𝔤′ would also commute with all the elements of 𝔷 and would have to belong to 𝔷. On the other hand, the Killing form on 𝔤′ is negative semidefinite, because 𝔤′ is an ideal of 𝔤, which is the Lie algebra of a compact Lie group. It follows that 𝔤′ is semisimple. Hence 𝔤′ = [𝔤′, 𝔤′]; see e.g. section 5.2 of [Hum].

Thus 𝔤′ is a semisimple ideal and 𝔤′ = [𝔤, 𝔤]. It follows immediately that G′ is a compact semisimple Lie group. In particular G′ is closed and normal as a subgroup of G.

Now suppose that D is a dense analytic subgroup of G. Let A be the analytic subgroup of G tangent to 𝔷. Then A is normal and G/A is compact and semisimple, because its Lie algebra is isomorphic to 𝔤′. The projection D_{A} of D to G/A is a dense analytic subgroup of G/A. Since D_{A} is dense, any element of the center of D_{A} must commute with all elements of G/A. The center of G/A is finite, so D_{A} must also have finite center and the Lie algebra of D_{A} must have trivial center. On the other hand, D_{A} supports a bi-invariant metric because we can take a bi-invariant metric on G/A and restrict it to D_{A}. These two properties imply that D_{A} is compact (see e.g. Corollary 21.4 in [Mi]). Thus D_{A} must be the whole of G/A since it is both closed and dense. Consequently the Lie algebra 𝔡 of D contains a complement to 𝔷. But this implies that [𝔡, 𝔡] = [𝔤, 𝔤]. Hence [𝔤, 𝔤] ⊂ 𝔡 and G′ ⊂ D. □

It follows from the previous lemma that we can reduce by G′. This yields a skew product f_{φ_{1}} : M × G_{1} → M × G_{1}, where G_{1} = G/G′, which is a torus since it is a connected abelian Lie group. Lemma 8.6 tells that \overline{H^{0}}(f_{φ_{1}}) and H^{0}(f_{φ_{1}}) are the projections to G_{1} of \overline{H^{0}} and H^{0} respectively. Hence \overline{H^{0}}(f_{φ_{1}}) = G_{1}. We also have H^{0}(f_{φ_{1}}) ≠ G_{1}. For we have G′ ⊂ H^{0} by Lemma 12.8, and if H^{0} projected to the whole of G_{1}, that would force H^{0} = G.

Proposition 12.7 now follows by applying the next lemma to f_{φ_{1}}. □

Lemma 12.9 Let f : M → M be an Anosov diffeomorphism of a compact infranilmanifold M, and let f_{φ} : M × T^{d} → M × T^{d} be a skew product such that H^{0} ≠ G, but \overline{H^{0}} = G. Then f_{φ} has an algebraic factor of type (3).

Proof: By Proposition 11.1, f_{φ} has an algebraic factor f_{ψ} : M × T^{d} → M × T^{d}, where ψ(x) = (exp(ir_{1}(x)), …, exp(ir_{d}(x))). We shall show below that there are real numbers λ_{1}, …, λ_{d}, β, and a C^{k−} function Φ : M → R such that for all x ∈ M,

λ_{1}r_{1}(x) + ⋯ + λ_{d}r_{d}(x) = Φ(f(x)) − Φ(x) + β.

From this it follows that ψ is cohomologous to a map χ : M → T^{d} such that χ(x) = (exp(is_{1}(x)), …, exp(is_{d}(x))) and

λ_{1}s_{1}(x) + ⋯ + λ_{d}s_{d}(x) = β.

Indeed, if λ_{j} ≠ 0, we can take s_{j}(x) = r_{j}(x) − λ_{j}^{−1}[Φ(f(x)) − Φ(x)] and s_{i}(x) = r_{i}(x) for i ≠ j. Then f_{ψ} is algebraically conjugate to f_{χ}, which is the desired algebraic factor of type (3).

Let p_{1} : R^{d} → T^{d} be the projection

p_{1}(θ_{1}, …, θ_{m}) = (exp(iθ_{1}), …, exp(iθ_{d})),

and let r : M → R^{d} be the function

r(x) = (r_{1}(x), …, r_{d}(x)).

Then ψ = p_{1} ∘ r, and the skew product f_{r} : M × R^{d} → M × R^{d} has f_{ψ} as a factor. Let H_{1} = H(f_{r}), and let H_{1}^{0} = H^{0}(f_{r}). According to Lemma 8.6, p(H_{1}^{0}) = H^{0} ≠ G, and so H_{1}^{0} is a proper subgroup of R^{d}, i.e., a proper subspace of R^{d}. Let Λ : R^{d} → R be a nontrivial linear map such that H_{1}^{0} ⊂ ker Λ.

Let s = Λ ∘ r. For the skew-product f_{s} : M × R → M × R, we have, by Lemma 8.6, that

H^{0}(f_{s}) = Λ(H_{1}^{0}) = {0}.

By Lemma 12.6, it follows that H(f_{s}) = {0}. Let Φ : M → R be the C^{k−} function defined by applying Proposition 8.2 to f_{s}. As in the proof of Proposition 12.3, we see that there is a real number β such that

s(x) = Φ(f(x)) − Φ(x) + β.

In other words,

Λ ∘ r(x) = Φ(f(x)) − Φ(x) + β,

and there are real numbers λ_{1}, …, λ_{d} such that

λ_{1}r_{1}(x) + ⋯ + λ_{d}r_{d}(x) = Φ(f(x)) − Φ(x) + β.

□

Proof of Corollary B3: Suppose that mφ = Φ ∘ f − Φ + c. Then the proof of Proposition 12.3 shows that f_{mφ} is algebraically conjugate to f × R_{c}, Since f_{mφ} is an algebraic factor of f_{φ} by Proposition 11.1, it follows from Lemma 3.1 that f_{φ} is not stably ergodic.

Conversely, if f_{φ} is not stably ergodic, then H^{0}(f_{φ}) ≠ T and hence H^{0}(f_{φ}) = {e}. Since f_{mφ} is an algebraic factor of f_{φ}, we obtain H^{0}(f_{mφ}) = {e}, by Lemma 8.6. By the remark after Proposition 11.1, mφ is cohomologous to a function ψ : M → T that is homotopic to a constant. Since f_{mφ} and f_{ψ} are algebraically conjugate, we have H^{0}(f_{ψ}) = {e}. Hence H(f_{ψ}) = {e} by Lemma 12.5. It now follows from the proof of Proposition 12.3 that ψ is cohomologous to a constant. Hence mφ is cohomologous to the same constant.□

References

[AKS] R. Adler, B. Kitchens, M. Shub, Stably ergodic skew products, Discrete and Continuous Dynamical Systems 2 (1996), 349–350.

[B1] M. Brin, Topological transitivity of one class of dynamical systems and flows of frames on manifolds of negative curvature, Func. Anal. Appl. 9 (1975), no. 1, 9–19.

[B2] M. Brin, The topology of group extensions of C systems, Mat. Zametki 18 (1975), no. 3, 453–465.

[BP] M. Brin and Ja. Pesin, Partially hyperbolic dynamical systems, Math. USSR Izvestija 8 (1974) no. 1, 177–218.

[BPW] K. Burns, C. Pugh and A. Wilkinson, Stable ergodicity and Anosov flows, to appear in Topology.

[CE] J. Cheeger and D. Ebin, Comparison Theorems in Riemannian Geometry, North Holland/American Elsevier, 1975.

[FP] M. Field and W. Parry, Stable ergodicity of skew extensions by compact Lie groups, Topology 38 (1999) No 1, 167–187.

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

[H] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, 1978.

[HPS] M. Hirsch, C. Pugh, M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, 1977.

[Hu] S.-T. Hu, Homotopy Theory, Academic Press, 1959.

[Hum] J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer Verlag, 1972.

[J] J.-L. Journé, A regularity lemma for functions of several variables, Rev. Mat. Iberoamericana 4 (1988), no. 2, 187–193.

[KK] A. Katok and A. Kononenko, Cocycles’ stability for partially hyperbolic systems, Math. Res. Lett. 3 (1996), no. 2, 191–210.

[Ma1] A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math. 96 (1974), 422–429.

[Ma2] A. Manning, Anosov diffeomorphisms on nilmanifolds, Proc. Amer. Math. Soc. 38 (1973), 423–426.

[Mi] J. Milnor, Morse Theory, Princeton University Press, 1969.

[N] K. Nomizu, Lie Groups and Differential Geometry, Mathematical Society of Japan, 1956.

[P] W. Parry, Entropy and Generators in Ergodic Theory, W. A. Benjamin, Inc. 1969.

[PP] W. Parry and M. Pollicott, Stability of mixing for toral extensions of hyperbolic systems, Tr. Mat. Inst. Steklova 216 (1997), Din. Sist. i Smezhnye Vopr., 354–363.

[PS1] C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity, J. of Complexity 13 (1977), 125–179.

[PS2] C. Pugh and M. Shub, Stable ergodicity and julienne quasiconformality, September, 1997 preprint.

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

[Wa] C. Walkden, Stable ergodic properties of cocycles over hyperbolic attractors, 1997 preprint.

[Wi] A. Wilkinson, Stable ergodicity of the time one map of a geodesic flow, Ergod. Th. and Dynam. Syst. 18 (1998), no. 6, 1545–1588.

[Y] H. Yamabe, On an arcwise connected subgroup of a Lie group, Osaka Math. J. 2 (1950), 13–14.

Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, USA. E-mail addresses: burns@math.nwu.edu, wilkinso@math.nwu.edu