A stably Bernoullian diffeomorphism that is not Anosov

Michael Shub^{∗} and Amie Wilkinson^{†}

July 7, 1998

Introduction

In this paper we construct new examples of diffeomorphisms with robust statistical properties. Our interest in these examples arose from the insight they might give into a collection of natural, optimistic and still unresolved problems in the global theory of dynamical systems.

For a compact C^{∞} manifold M with volume element μ, let Diff^{r}(M) be the space of C^{r} diffeomorphisms of M and let Diff_{μ}^{r}(M) be the elements of Diff^{r}(M) that preserve μ. Endow these spaces with the C^{r} topology, r ≥ 1.

Theorem I: Let μ be Lebesgue measure on the torus T^{3} = R^{3}/Z^{3}. There is a C^{1}-open set U ⊂ Diff_{μ}^{2}(T^{3}) such that for each g ∈ U,

1. g is Bernoulli;

2. g has Lyapunov exponents nonzero μ-a.e.;

3. g is not homotopic to an Anosov diffeomorphism.

Previously the only open sets of diffeomorphisms known to satisfy either property 1 or property 2 were the Anosov diffeomorphisms. Recall that a diffeomorphism g : M → M is Anosov if there exists a continuous, T ginvariant splitting of the tangent bundle to M,

∗Supported by an NSF grant.

†Supported in part by the IHES.

T M = E^{u} ⊕ E^{s},

and constants C > 0 and λ > 0 such that, for all n ≥ 0,

v ∈ E^{u} ⇒ ∥T g^{−n}(v)∥ ≤ Ce^{−λn}∥v∥,  and
v ∈ E^{s} ⇒ ∥T g^{n}(v)∥ ≤ Ce^{−λn}∥v∥.

Anosov diffeomorphisms are examples of uniformly hyperbolic dynamical systems; other examples include Axiom A diffeomorphisms and attractors. The ergodic theory of such systems has been studied extensively, beginning with the pioneering work of Anosov, Sinai and Bowen-Ruelle in the 1960’s and ’70’s (see, e.g., [KH]). All Anosov diffeomorphisms of T^{3} are of the special form g = ρ ∘ h ∘ ρ^{−1}, where ρ is a homeomorphism and h is a linear automorphism of T^{3}, none of whose eigenvalues has modulus 1 [Mann].

A real number λ is a Lyapunov exponent of the diffeomorphism g : M → M if there exists a nonzero vector v ∈ T M such that

lim_{n→∞} \frac{1}{n} log ∥T g^{n}(v)∥ = λ.  (1)

By Osceledets’ Theorem [Os], there is a set L ⊆ M, which has full measure with respect to any g-invariant measure, such that the limit in (1) exists for all v ∈ T_{x}M with x ∈ L. For a given x ∈ L, there are finitely many different exponents, and for some vectors, this limit may be zero. It is immediate from the definition that the Lyapunov exponents of an Anosov diffeomorphism are all nonzero. At the other extreme, an automorphism of T^{3} with 1 as an eigenvalue has an exponent 0 (everywhere). To prove Theorem I, we start with such an automorphism and perturb so that the 0 exponent becomes positive, almost everywhere. A volume-preserving diffeomorphism whose exponents are nonzero almost everywhere is called nonuniformly hyperbolic. This term was introduced by Pesin, who was able to establish many ergodic properties for nonuniformly hyperbolic systems, previously known to hold for uniformly hyperbolic ones.

The diffeomorphisms in Theorem I have some other remarkable properties which we describe below. Before turning to them and to Theorem II, we set our result in context.

One of the achievements of the theory of uniformly hyperbolic dynamical systems were the theorems of Sinai, Ruelle and Bowen on invariant measures on the attractors of a system. These attractors and measures are now called Sinai-Ruelle-Bowen measures and SRB measures (or SRB attractors), for short. They may also be called ergodic attractors.^{1}

Given f ∈ Diff^{r}(M) (not necessarily preserving μ), a closed, f-invariant set A ⊂ M and an f-invariant ergodic measure ν on A, we define B(A), the basin of A, to be the set of points x ∈ M such that f^{n}(x) → A and for every continuous function ϕ : M → R

lim_{n→∞} \frac{1}{n}(ϕ(x) + ⋯ + ϕ(f^{n}(x))) → ∫_{A} ϕ(x) dμ.

Definition: ν is an SRB measure and A is an SRB (or ergodic) attractor if the Lebesgue measure of B(A) is positive.

It follows from the definition that a diffeomorphism has at most countably many SRB measures. Sinai, Ruelle and Bowen proved that for r ≥ 2 and f an Axiom A, no-cycle diffeomorphism (see [Si], [Ru], [Bo]), almost every point in M with respect to Lebesgue measure μ is in the basin of an SRB measure, and there are only finitely many SRB measures. The first natural question is if the first part of this conclusion is a generic property in Diff^{r}(M) for r ≥ 2. (By generic we mean: containing a countable intersection of opendense sets).

Question 1: For r ≥ 2, is it true for generic f in Diff^{r}(M) that the union of the basins of the SRB attractors of f has full Lebesgue measure in M?

In the volume-preserving case Pesin [Pe] proved that for f a C^{2}, nonuniformly hyperbolic diffeomorphism, M may be written as the disjoint union of at most countably many invariant sets of positive measure on which f is ergodic. Thus for nonuniformly hyperbolic volume preserving diffeomorphisms the answer to Question 1 is “yes.” The supports of the ergodic measures are the SRB attractors. Pesin asked if nonuniform hyperbolicity was generic in Diff_{μ}^{2}(M).

1We take some of the conclusions of the theorems of Sinai, Ruelle, and Bowen as a definition and warn the reader that the use of SRB measure or attractor is not uniform in the literature. For a survey of SRB measures (using a different definition) see [You2].

Pesin’s question is answered in the negative by Cheng and Sun [CS], and Herman. In [He] (see also [Yoc]), Herman showed that on any manifold M of dimension at least 2, and for sufficiently large r, there are open sets of volume preserving C^{r} diffeomorphisms of M all of which posess positive measure sets of codimension one invariant tori, and on each torus the diffeomorphism is C^{1} conjugate to a diophantine translation. In this case all of the exponents are 0 on these tori.

Our proof gives some hope that a variant of Pesin’s original question might hold for volume-preserving diffeomorphisms: either all exponents are zero (μ-a.e.), or, as with our examples, the system may be perturbed to become stably nonuniformly hyperbolic.

Question 2a): For r ≥ 1, is it true for generic f in Diff_{μ}^{r}(M) that for almost every ergodic component of f, either all of the Lyapunov exponents of f are 0 or none are 0 (μ-a.e.)?

A special case of 2a) is 2b).

Question 2b): For r ≥ 1 does the generic ergodic diffeomorphism in Diff_{μ}^{r}(M) have either no exponent equal to 0 or all exponents equal to 0 (μ-a.e.)?

Question 2a) has an affirmative answer for 2-dimensional M in the case r = 1; Mañé has shown that the generic diffeomorphism in Diff_{μ}^{1}(M) either has all of its Lyapunov exponents zero or is an Anosov diffeomorphism. Theorem I shows that C^{1} - open sets of C^{2}, volume-preserving diffeomorphisms with nonzero exponents need not be Anosov in general.

An analogue of Question 2 for Diff^{r}(M) is the following.

Question 3: For r ≥ 1, is it true for the generic f in Diff^{r}(M) and any weak limit ν of the push forwards f_{∗}^{n}μ that almost every ergodic component of ν has some exponents not equal to 0 (ν-a.e.)? All exponents not equal to 0?

Question 2b) is closest in spirit to Theorem I, which in turn gives some credence to the possibility that 2b) is true. Question 3) might be a way to approach Question 1) along the lines of [Pe], [PS1], [PS4]. There might even be only finitely many SRB attractors, for almost every f in the generic kparameter family in Diff^{r}(M), as suggested in [PS2], or densely in Diff^{r}(M) as suggested in [Pa].

We now turn to another novel aspect of the diffeomorphisms we construct. This property concerns one of the invariant expanding Lyapunov directions; by construction, this line field is tangent to a foliation of T^{3}.

Theorem II: Let U be the set of diffeomorphisms in Theorem I. For every g ∈ U, there is a foliation W_{g}^{c} of T^{3} by C^{2} circles which is preserved by g. This foliation has the following properties:

1. There is an equivariant fibration π : T^{3} → T^{2} such that πg = hπ where h is the linear endomorphism of T^{2} given by a hyperbolic matrix A ∈ Sl(2, Z). The leaves of the foliation W_{g}^{c} are the fibers of the map π. In particular, the set of periodic leaves is dense in T^{3}.

2. There exists λ > 0 such that, for μ-almost every w ∈ T^{3}, if v ∈ T_{w}T^{3} is tangent to the leaf of W_{g}^{c} containing w, then

lim_{n→∞} \frac{1}{n} log ∥T_{w}g^{n}v∥ = λ.

3. Consequently, there exists a set S ⊆ T^{3} of full μ-measure that meets every leaf of W_{g}^{c} in a set of leaf-measure 0.

Property 3 stands in contrast to the measure-theoretic properties of other dynamically-invariant foliations, specifically the unstable foliations of a hyperbolic (or even nonuniformly hyperbolic) diffeomorphism. Unstable foliations have the property of absolute continuity: any set of full measure meets almost every leaf in a set of leafwise full measure. Absolutely continuous foliations satisfy a Fubini-type theorem; this enabled Anosov to prove ergodicity for volume-preserving hyperbolic systems. Foliations exhibiting the behavior in 3 have been referred to as “Fubini’s Nightmare,” (also “Fubini Foiled”); Katok had previously constructed an example of a dynamicallyinvariant foliation with this property, which is presented in [Mi]. Katok’s example made us aware that diffeomorphisms such as those we construct are indeed possible.

Lai-Sang Young has previously constructed open sets of C^{1} cocycles that are not uniformly hyperbolic and which have exponents nonzero [You1]. These cocycles, however, are not the natural cocycle associated to the derivative of a map.

We remark here that our construction can be slightly modified to obtain diffeomorphisms of T^{n}, for any n ≥ 3, that satisfy the conclusions of Theorems I and II. In this modification, the automorphism of T^{2} in the next section is replaced by an automorphism of T^{n−1} with one-dimensional expanding eigenspace.

We thank Michael Herman for many conversations which clarified questions 1-3 for us. Some of this material was presented and discussed in his seminar during March 1998. The question of whether perturbing a normallyhyperbolic, non-Anosov diffeomorphism could produce nonzero exponents was initially raised by Lai-Sang Young during a conversation about these questions. We thank her for reminding us that examples such as the ones we construct might exist. Numerical experiments conducted by Chai Wah Wu and later by Niels Sondergaard convinced us of their existence and we are indebted to them. We also thank Charles Pugh and Clark Robinson for useful conversations.

1 The construction

The ingredients in our construction are: a linear Anosov diffeomorphism h : T^{2} → T^{2}, circle-valued C^{3} functions φ : T^{2} → T and ψ : T → T, and vectors v_{0} ∈ R^{2}, w_{0} ∈ Z^{2}. We specify in this section how to select these ingredients; an example of a suitable choice is h(x, y) = (2x+y, x+y), φ(x, y) = sin(2πy), ψ(z) = sin(2πz), v_{0} = ((1 + \sqrt{5})/2, 1), and w_{0} = (1, 1). These define a 2parameter family f_{a,b} of diffeomorphisms of T^{3} by the formula:

f_{a,b} = g_{a} ∘ h_{b},  (2)

where

h_{b}(x, y, z) = (h(x, y), z + w_{0} · (x, y) + bφ(x, y)),

and

g_{a}(x, y, z) = ((x, y) + aψ(z)v_{0}, z).

The set U in Theorem I will be a neighborhood of f_{a,b}, for a suitable choice of parameters.

Let h : T^{2} → T^{2} be a linear Anosov diffeomorphism, i.e. a map of the form h(p) = Ap, for some hyperbolic matrix A ∈ Sl(2, Z). The expanding and contracting eigenspaces for A are both one-dimensional. Choose v_{0} ∈ R^{2} to be an expanding eigenvector for A.

The diffeomorphisms h_{b} in (2) are circle extensions of h. We describe this construction. If θ : T^{2} → T is any circle-valued function, we may construct the circle extension (or skew product) of h by θ, denoted h_{θ} : T^{3} → T^{3}, by the formula

h_{θ}(p, z) = (h(p), z + θ(p)),

for p ∈ T^{2}, and z ∈ T. It is easy to see that h_{θ} preserves μ. We will work with skew products that are stably ergodic; we shall see that they are exactly characterized by the following condition.

Definition: Let k = det(A − I). Say that θ satisfies the cocycle condition if there are no solutions to the equation

kθ(p) = Φ(h(p)) − Φ(p) + c,  (3)

with Φ ∈ C^{0}(T^{2}, T) and c ∈ T.

The cocycle condition has a simpler formulation when θ is homotopic to a constant map. Indeed, it follows from the proof of Proposition 11.1 in [BW] that for such a θ, equation (3) in the definition may equivalently be replaced by

θ(p) = Φ(h(p)) − Φ(p) + c,  (4)

where Φ ranges over maps homotopic to a constant. The following proposition is a special case of Corollary B3 in [BW].

Proposition 1.1 If θ is C^{2} and satisfies the cocycle condition, then h_{θ} is stably ergodic (in fact, stably a K-system). That is, there is a C^{1}-open neighborhood E of h_{θ} in Diff_{μ}^{2}(T^{3}) such that all g ∈ E are ergodic (K-systems).

Conversely, if h_{θ} is stably ergodic, then θ satisfies the cocycle condition.

The cocycle condition is easily seen to hold for an open and dense set of θ ∈ C^{r}(T^{2}, T), for any r ≥ 2. An example of a function satisfying this condition (for any linear Anosov h) is θ(x, y) = sin(2πx), which, in addition, is homotopic to a constant map.

Choose the function φ to be C^{3}, homotopic to a constant map and satisfying the cocycle condition. Choose w_{0} ∈ Z^{2} to be any nonzero vector, and for b ∈ R, let

φ_{b}(x, y) = w_{0} · (x, y) + bφ(x, y).

Observe that h_{b} = h_{φ_{b}}.

Lemma 1.2 If φ is homotopic to a constant map and satisfies the cocycle condition, then φ_{b} satisfies the cocycle condition, for all b ≠ 0. For these b, h_{b} is stably ergodic and stably K.

Proof: Suppose that φ_{b} does not satisfy the cocycle condition, for some b ≠ 0. Then there exist Φ ∈ C^{0}(T^{2}, T) and c ∈ T such that for all (x, y) ∈ T^{2},

kφ_{b}(x, y) = kw_{0} · (x, y) + kbφ(x, y) = Φ ∘ h(x, y) − Φ(x, y) + c, (5)

where k is the determinant of A − I.

Since kw_{0} ∈ (A − I)(Z^{2}), there exist integers r and s such that

kw_{0} · (x, y) = Ψ ∘ h(x, y) − Ψ(x, y),  (6)

with Ψ(x, y) = rx + sy. Combining (6) with (5), we obtain

Ψ ∘ h − Ψ + kbφ = Φ ∘ h − Φ + c.

Setting \overline{Φ} = Φ − Ψ, gives:

kbφ = \overline{Φ} ∘ h − \overline{Φ} + c.

Thus bφ fails to satisfy the cocycle condition. But bφ is homotopic to a constant map. By the remarks following the definition of the cocycle condition, there exist Φ̂ ∈ C^{0}(T^{2}, T), homotopic to a constant map, and c′ ∈ T such that

bφ = Φ̂ ∘ h − Φ̂ + c′.

But then

φ = b^{−1}Φ̂ ∘ h − b^{−1}Φ̂ + b^{−1}c′,

and so φ fails to satisfy the cocycle condition. This contradicts the hypothesis of the lemma.

Thus for all b ≠ 0, h_{φ_{b}} satisfies the cocycle condition, and by Proposition 1.1, the skew product h_{b} = h_{φ_{b}} is stably ergodic and stably K.□

Finally, choose ψ : T → T to be any C^{3}, nonconstant function that is homotopic to a constant map. We will also use ψ to denote the function on T^{3}:

ψ(x, y, z) = ψ(z)

and use ψ′ for the function

ψ′(x, y, z) = ψ′(z).

To summarize, we choose h to be linear Anosov, φ to be C^{3}, homotopic to a constant and satisfying the cocycle condition, ψ to be C^{3} and homotopic to a constant, v_{0} to be an expanding eigenvector for h, and w_{0} to be any nonzero vector in Z^{2}. Let f_{a,b} : T^{3} → T^{3} be defined by (2). Observe that f_{a,b} is volume-preserving, since det T f_{a,b} = 1 everywhere.

Theorems I and II follow from

Proposition 1.3 For all b > 0 sufficiently small, there exists a positive number a(b) such that, for |a| ∈ (0, a(b)), there is a neighborhood U of f_{a,b} in Diff_{μ}^{2}(T^{3}) in which the conclusions of Theorems I and II hold.

2 Proofs

In this section we prove Proposition 1.3. We first need to establish a few facts about partially hyperbolic diffeomorphisms.

2.1 Partial Hyperbolicity

A diffeomorphism g : M → M is partially hyperbolic if there exists a T ginvariant continuous splitting of the tangent bundle to M into three subbundles

T M = E_{g}^{u} ⊕ E_{g}^{c} ⊕ E_{g}^{s},

with at least one of E_{g}^{u} or E_{g}^{s} nontrivial, and constants λ_{g}^{s} < λ_{g}^{c} < μ_{g}^{c} < μ_{g}^{u}, with λ_{g}^{s} < 1 < μ_{g}^{u} such that, for all p ∈ M:

v ∈ E_{g}^{u}(p) ⇒ ∥Dg(p)v∥ ≥ μ_{g}^{u}∥v∥,

v ∈ E_{g}^{c}(p) ⇒ μ_{g}^{c}∥v∥ ≥ ∥Dg(p)v∥ ≥ λ_{g}^{c}∥v∥,  and
v ∈ E_{g}^{s}(p) ⇒ λ_{g}^{s}∥v∥ ≥ ∥Dg(p)v∥,

with respect to some Riemannian metric on M. The bundles E_{g}^{u}, E_{g}^{c}, and E_{g}^{s} are called the unstable, center, and stable bundles of g, respectively. Anosov diffeomorphisms are partially hyperbolic diffeomorphisms for which E^{c} is trivial. The property of partial hyperbolicity is open in the C^{1} topology on C^{r} diffeomorphisms of M (see [BP] or [HPS] for a discussion of partially hyperbolic diffeomorphisms).

The unstable and stable bundles of a C^{r} partially hyperbolic diffeomorphism are always integrable and tangent to foliations W_{g}^{u} and W_{g}^{s}, whose leaves are C^{r} [HPS]. A partially-hyperbolic diffeomorphism is dynamically coherent if, in addition, the bundles E_{g}^{c} ⊕ E_{g}^{u} and E_{g}^{c} ⊕ E_{g}^{s}, and E_{g}^{c} are integrable and tangent to foliations W_{g}^{uc}, W_{g}^{sc}, and W_{g}^{c}, whose leaves are at least C^{1}. If, in addition, E_{g}^{c} is C^{1}, then g is contained in an open set of dynamically coherent diffeomorphisms in Diff^{1}(M)([PS3], Theorem 2.3). Dynamical coherence that is C^{1}-stable we shall call stable dynamical coherence.

Partial hyperbolicity and dynamical coherence for compact group extensions of Anosov diffeomorphisms is well-known and is proved in [BP], Theorem 2.2. We summarize the conclusions of this theorem for circle extensions in the next lemma. Let f = f_{0,b} = h_{b} = h_{φ_{b}}, and let π_{0} : T^{3} → T^{2} be the projection π_{0}(x, y, z) = (x, y). Observe that f covers the linear automorphism h : T^{2} → T^{2}, with h ∘ π_{0} = π_{0} ∘ f. The fibers of π_{0} are circles, which foliate T^{3} trivially. The diffeomorphism f preserves this foliation, permuting the leaves isometrically.

Lemma 2.1 For all b, the skew product f = h_{φ_{b}} is partially hyperbolic and dynamically coherent. The center distribution E_{f}^{c} is tangent to the fibers of π_{0}. For i = u or s, each leaf of W_{f}^{ic} is the product of a leaf of W_{h}^{i} with a circle fiber. Each W_{f}^{i} leaf is the graph of a C^{2} function from a leaf of W_{h}^{i} into the circle T. For any z_{0} ∈ T, the translation (p, z) ↦ (p, z + z_{0}) carries W_{f}^{i} leaves to W_{f}^{i} leaves.

The Riemannian metric may be chosen so that:

λ_{f}^{s} = m^{−1},  λ_{f}^{c} = μ_{f}^{c} = 1,  and  μ_{f}^{u} = m,  (7)

where m is the expanding eigenvalue of h.

The next proposition follows from Lemma 2.1 and the theory of normallyhyperbolic diffeomorphisms in [HPS].

Proposition 2.2 For all b, for a sufficiently small, f_{a,b} is partially hyperbolic and stably dynamically coherent. The unstable, center, and stable bundles are 1-dimensional. The center-unstable bundle E_{a,b}^{u} ⊕E_{a,b}^{c} of f_{a,b} does not depend on a or b; for w ∈ T^{3},

E_{a,b}^{u} ⊕ E_{a,b}^{c}(w) = E^{uc}(w) = {(rv_{0}, s) | r, s ∈ R}.

There is a C^{1}-open neighborhood U_{0} of f_{a,b} in Diff^{2}(T^{3}), such that, for g ∈ U_{0}, the center bundle E_{g}^{c} is tangent to a foliation W_{g}^{c} that satisfies conclusion 1 of Theorem II.

If, in addition, b ≠ 0, then for a sufficiently small, f_{a,b} is stably ergodic and stably K.

Proof: Since stable ergodicity and partial hyperbolicity are C^{1}-open, it suffices to establish these properties for the skew products f_{0,b}. Fix b and let f = f_{0,b}. By Lemma 2.1, f is partially hyperbolic. By Lemma 1.2, f is stably ergodic (and stably K) if b ≠ 0.

It also follows from Lemma 2.1 that, for w ∈ T^{3},

E_{f}^{u} ⊕ E_{f}^{c}(w) = E^{uc} = {(rv_{0}, s) | r, s ∈ R},

and

E_{f}^{s} ⊕ E_{f}^{c}(w) = {(rv′_{0}, s) | r, s ∈ R},

where v′_{0} spans the contracting eigenspace for h.

Suppose now that a is small and nonzero. Then f_{a,b} is partially hyperbolic with splitting T T^{3} = E_{a,b}^{u} ⊕ E_{a,b}^{c} ⊕ E_{a,b}^{s}. For a small, E_{a,b}^{u} ⊕ E_{a,b}^{c} is very close to E^{uc}. A direct calculation shows that T f_{a,b} preserves E^{uc}. Since E_{a,b}^{u} ⊕ E_{a,b}^{c} is the fixed point of a contraction operator, it is unique. It follows that E_{a,b}^{u} ⊕ E_{a,b}^{c} = E^{uc}.

The remaining conclusions of Proposition 2.2 follow from Corollary 8.3 in [HPS]. We recall the outline of the argument here and refer the reader to [HPS], p. 116ff., for the details, including the definitions of normal hyperbolicity and plaque expansiveness.

We verify that for g ∈ Diff^{2}(T^{3}) sufficiently C^{1}-close to f = f_{0,b}, the center foliation W_{g}^{c} has the properties listed in part 1 of Theorem II. Along the way, we will show that f is stably dynamically coherent. Dynamical coherence of f and (7) imply that f is 2-normally hyperbolic at the C^{∞} foliations W_{f}^{uc}, W_{f}^{sc}, and W_{f}^{c}. Smoothness of these foliations implies that they are plaque expansive. (Alternately, the fact that f covers a hyperbolic diffeomorphism implies plaque expansiveness.) Normally hyperbolic, plaque expansive diffeomorphisms are structurally stable, in the following sense.

Theorem 2.3 ([HPS], Theorem 7.1) If f : M → M is C^{r}, r-normally hyperbolic and plaque expansive at the foliation F, then for any C^{r}-diffeomorphism g : M → M sufficiently C^{1}-close to f, there exists a foliation F′ of M and a homeomorphism H : M → M satisfying

1. H sends leaves of F′ to leaves of F,

2. fH(L) = Hg(L), for every leaf L of F, and

3. g is r-normally-hyperbolic and plaque expansive at F′

Applying Theorem 2.3 to f = f_{0,b}, we obtain that for g ∈ Diff^{2}(T^{3}) sufficiently C^{1}-close to f, there exist foliations W_{g}^{uc}, W_{g}^{sc}, and W_{g}^{c}, with C^{2} leaves, homeomorphic to the foliations W_{f}^{uc}, W_{f}^{sc}, and W_{f}^{c}, respectively. They are tangent to the distributions E_{g}^{c} ⊕ E_{g}^{u}, E_{g}^{c} ⊕ E_{g}^{s}, and E_{g}^{c}, respectively. The leaves of W_{g}^{c} are obtained by interesecting the leaves of W_{g}^{uc} and W_{g}^{sc}. Thus f is stably dynamically coherent.

We also obtain a a homeomorphism H : T^{3} → T^{3} satisfying the conclusions of Theorem 2.3, with F = W_{f}^{c} and F′ = W_{g}^{c}. Define π : T^{3} → T^{2} by π(w) = π_{0} ∘ H(w). Then the fibers of π are the leaves of W_{g}^{c}, and

π ∘ g = π_{0} ∘ H ∘ g = π_{0} ∘ f ∘ H = h ∘ π_{0} ∘ H = h ∘ π,

completing the proof.□

2.2 Smoothness of E_{a,b}^{u} inside the family f_{a,b}

In order to determine the behavior of the Lyapunov exponents of f_{a,b}, we will need to differentiate the unstable line field E_{a,b}^{u} with respect to a. We will use:

Lemma 2.4 There exists a neighborhood A of (0, 0) in R^{2}, such that, for each w ∈ T^{3}, the function

(a, b) ↦ E_{a,b}^{u}(w)

is C^{2} on A. The first two derivatives of this function depend uniformly on w.

Remark: It is not in general true that in the space of partially-hyperbolic diffeomorphisms, the map g ↦ E_{g}^{u}(w), for a fixed w, depends smoothly on g. To see that the dependence is not smooth in general, let f : T^{4} → T^{4} be the real-analytic Anosov diffeomorphism constructed by Anosov in [An] whose unstable distribution E_{f}^{u} is not C^{1}. For v ∈ T^{4}, let τ_{v} be translation in T^{4} by v. Let f_{v} = τ_{v} ∘ f ∘ τ_{−v}. The unstable distribution for f_{v} is E_{f_{v}}^{u} = τ_{v∗}E_{f}^{u}, which is the unstable distribution for f translated by τ_{v}. Then f_{v} is clearly a C^{ω} family of diffeomorphisms, but v ↦ E_{f_{v}}^{u}(w) cannot be C^{1} for all w ∈ T^{4}.

Nonetheless, there is a form of “smooth dependence” for the unstable bundle E_{g_{0}}^{u} of a C^{r} Anosov diffeomorphism g_{0}. Namely, if g is C^{r} and sufficiently C^{1}-close to g_{0}, and h_{g} = g ∘ h_{g} ∘ g_{0}^{−1} is the homeomorphism conjugating g to g_{0} (see, e.g., [KH] §18.2), then the map g ↦ E_{g}^{u}(h_{g}(w)) is C^{r−2} in a neighborhood of g_{0}, for fixed w. This is proved, for example, in [Ru2]. For partially-hyperbolic diffeomorphisms that are not structurally stable, it is less clear how to formulate an analogous statement.

In our situation, because the diffeomorphisms f_{a,b} have a common centerunstable foliation, which is C^{∞}, one can show that the unstable line field E_{a,b}^{u} varies in a C^{2} fashion along the leaves of this foliation. The difficulties we have just outlined in the general case do not arise. In fact, the same argument that gives smooth dependence of E_{a,b}^{u} along center-unstable leaves also gives smooth dependence on the parameters a, b.

Proof of Lemma 2.4: It does not affect the smoothness of e^{u} if we scale the functions φ and ψ by a positive constant. Thus we may assume that

(∥ψ∥_{0}∥v_{0}∥ + ∥φ∥_{0} + 1)^{2} < m,  (8)

where m > 1 is the unstable eigenvalue of h.

By Proposition 2.2, there is a neighborhood A_{0} of ω_{0} = (0, 0) in R^{2} such that for all ω ∈ A_{0}, f_{ω} is partially hyperbolic, stably dynamically coherent, and, for all w ∈ T^{3},

E_{ω}^{u} ⊕ E_{ω}^{c}(w) = E^{uc}(w) = {(rv_{0}, s) | r, s ∈ R}.

The splitting E_{ω_{0}}^{u} ⊕E_{ω_{0}}^{c} ⊕E_{ω_{0}}^{s} is obtained from the eigenspace decomposition of the linear map f_{ω_{0}}. The center bundle E_{ω_{0}}^{c} is spanned by ∂/∂z. Assume that A_{0} is chosen so that E_{ω}^{u} is transverse to E_{ω_{0}}^{c}.

Let W^{uc} be the C^{∞} foliation tangent to E^{uc}. The leaves of W^{uc} are smoothly permuted by f_{ω}. Let X be the disjoint union of the leaves of W^{uc}. Because the foliation structure of W^{uc} is preserved by all of the f_{ω}, there is a well-defined, C^{3} map F : A_{0} × X → A_{0} × X given by:

F(ω, w) = (ω, f_{ω}(w)).

On A_{0} × X, put the metric:

d((ω_{1}, w_{1}), (ω_{2}, w_{2})) = max{d_{A_{0}}(ω_{1}, ω_{2}), d_{X}(w_{1}, w_{2})},

where d_{X} is the induced Riemannian metric on X and d_{A_{0}}((a_{1}, b_{1}), (a_{2}, b_{2})) = max{|a_{1} − a_{2}|, |b_{1} − b_{2}|}. With respect to this metric, there exists a constant ρ such that

d(F(ω_{1}, w_{1}), F(ω_{2}, w_{2})) ≥ ρd((ω_{1}, w_{1}), (ω_{2}, w_{2})),

for all ω_{1}, ω_{2} ∈ A_{0} and w_{1}, w_{2} ∈ X. The constant ρ is the inverse of the Lipschitz norm of F^{−1}. A straightforward estimate shows that by shrinking the size of the neighborhood A_{0}, we may bring ρ arbitrarily close to:

(∥ψ∥_{0}∥v_{0}∥ + ∥φ∥_{0} + 1)^{−1}.

Let B be the trivial bundle over A_{0} × X whose fiber L_{w} = L over w is the set of all linear maps L : E_{ω_{0}}^{u}(w) → E_{ω_{0}}^{c}(w). Since E_{ω_{0}}^{u} and E_{ω_{0}}^{c} are 1-dimensional, so is L. We think of B as the product A_{0} × X × R.

With respect to the C^{∞} splitting T X = E^{uc} = E_{ω_{0}}^{u} ⊕ E_{ω_{0}}^{c}, the map T f_{ω}|_{E^{uc}} can be written:

T f_{ω} = ( \begin{matrix}A_{ω} & B_{ω} \\ C_{ω} & K_{ω} \end{matrix} ) ,

where A_{ω} : E_{ω_{0}}^{u} → E_{ω_{0}}^{u}, B_{ω} : E_{ω_{0}}^{c} → E_{ω_{0}}^{u}, C_{ω} : E_{ω_{0}}^{u} → E_{ω_{0}}^{c}, and K_{ω} : E_{ω_{0}}^{c} → E_{ω_{0}}^{c}. These maps depend in a C^{2} fashion on ω and on the basepoint w ∈ T^{3}. When ω = ω_{0}, we have B = C = 0, K = 1, and A = m.

Define a bundle map F_{♯} : B → B, covering F, by:

F_{♯}(ω, w, L) = (ω, f_{ω}(w), (C_{ω}(w) + K_{ω}(w)L)(A_{ω}(w) + B_{ω}(w)L)^{−1}).

Then F_{♯} is C^{2}, contracts fibers of B at the weakest by a factor σ ≐ m^{−1}, and has strongest base contraction by the factor ρ ≐ (∥ψ∥_{0}∥v_{0}∥+∥φ∥+1)^{−1}. These estimates depend uniformly on the size of the neighborhood A_{0}. Thus, by inequality (8), there is a neighborhood A ⊆ A_{0} of ω_{0}, in which

σρ^{−2} < 1.

By the C^{r} Section Theorem of [HPS], the unique F_{♯}-invariant section s : A × X → L is C^{2}. But the graph of s(ω, w) : E_{ω_{0}}^{u}(w) → E_{ω_{0}}^{s}(w) is precisely E_{ω}^{u}(w). We conclude that E_{ω}^{u}(w) is a C^{2} function of ω ∈ A and of w ∈ X. Since W^{uc} is a C^{∞} foliation, these estimates are uniform over w ∈ T^{3}. □

2.3 Some continuity properties of Lyapunov exponents

Let g ∈ Diff_{μ}^{1}(M) be any partially-hyperbolic diffeomorphism of a 3-manifold M whose unstable and stable bundles are 1-dimensional. Suppose that V_{g}^{u}, V_{g}^{c} and V_{g}^{s} are continuous nonvanishing vector fields everywhere tangent to the line fields E_{g}^{u}, E_{g}^{c}, and E_{g}^{s}. Then V_{g}^{u}(w), V_{g}^{c}(w) and V_{g}^{s}(w) are a basis for T_{w}M. With respect to this basis, the derivative of g at w is expressed

T g(w) = ( \begin{matrix}r_{g}^{u}(w) & 0 & 0 \\ 0 & r_{g}^{c}(w) & 0 \\ 0 & 0 & r_{g}^{s}(w) \end{matrix} ) ,

where r_{g}^{u}, r_{g}^{c}, and r_{g}^{s} are the continuous functions:

r_{g}^{u}(w) = ∥T g(w)V_{g}^{u}(w)∥ · ∥V_{g}^{u}(g(w))∥^{−1},  (9)
r_{g}^{c}(w) = ∥T g(w)V_{g}^{c}(w)∥ · ∥V_{g}^{c}(g(w))∥^{−1},  (10)
r_{g}^{s}(w) = ∥T g(w)V_{g}^{s}(w)∥ · ∥V_{g}^{s}(g(w))∥^{−1}.  (11)

Compactness implies that the functions log(r_{g}^{u}), log(r_{g}^{c}), and log(r_{g}^{s}) are integrable on M, and so we may define real-valued functions e^{u}, e^{c} and e^{s} of the diffeomorphism g by the formulas:

e^{u}(g) = ∫_{M} log r_{g}^{u}(w) dμ(w)  (12)
e^{c}(g) = ∫_{M} log r_{g}^{c}(w) dμ(w)  (13)
e^{s}(g) = ∫_{M} log r_{g}^{s}(w) dμ(w)  (14)

Lemma 2.5 The functions e^{u}, e^{c}, and e^{s} do not depend on the choice of vector fields V^{u}, V^{c} and V^{s} and are continuous in a neighborhood of g in Diff_{μ}^{1}(M).

If g is ergodic with respect to μ, then the Lyapunov exponents of g are e^{u}(g), e^{c}(g) and e^{s}(g), μ-almost everywhere.

Proof of Lemma 2.5: The exponents of g are the logarithms of the eigenvalues of the limit:

lim_{n→∞} ((T_{w}g^{n})^{t} T_{w}g^{n})^{\frac{1}{2n}}

= lim_{n→∞} ( \begin{matrix}(r_{g}^{u}(w) ⋯ r_{g}^{u}(g^{n}(w)))^{\frac{1}{n}} & 0 & 0 \\ 0 & (r_{g}^{c}(w) ⋯ r_{g}^{c}(g^{n}(w)))^{\frac{1}{n}} & 0 \\ 0 & 0 & (r_{g}^{s}(w) ⋯ r_{g}^{s}(g^{n}(w)))^{\frac{1}{n}} \end{matrix} ) .

By the Ergodic Theorem, these limits converge almost everywhere. Thus the largest Lyapunov exponent at w is

e^{u}(g, w) = lim_{n→∞} \frac{1}{n}(log(r_{g}^{u}(w)) + ⋯ + log(r_{g}^{u}(g^{n}(w))).  (15)

The function e^{u}(g, ·) is measurable, g-invariant, and ∫_{M} e^{u}(g, w)dμ(w) = ∫_{M} log(r_{g}^{u}(w))dμ(w) = e^{u}(g). Hence e^{u}(g) is the average largest Lyapunov exponent with respect to μ. Since the Lyapunov exponents of g with respect to μ are μ-a.e. unique, the function e^{u}(g) does not depend on V_{g}^{u}. Since g is partially-hyperbolic, the function g ↦ E_{g}^{u} is continuous. The vector field V_{g}^{u} can be chosen to depend continuously on g. Since e^{u}(g) depends continuously on V_{g}^{u}, it follows that e^{u} is continuous.

Suppose that g is ergodic. Then e^{u}(g, w) = e^{u}(g), μ-a.e., and e^{u} is the largest Lyapunov exponent of g, μ-a.e.

The argument is the same for e^{c} and e^{s}.□

2.4 Behavior of exponents inside the family f_{a,b}

Let A be a neighborhood of the origin in R^{2} such that f_{ω} satisfies the conclusions of Proposition 2.2 and Lemma 2.4, for all ω ∈ A. In this subsection, we choose vector fields V^{i}(ω) = V^{i}(f_{ω}) for i = u, c, s and define functions e^{i}(ω) = e^{i}(f_{ω}) according to equations (12)-(14) in the previous subsection. By Lemma 2.5, these functions depend continuously on the parameter ω and give the Lyapunov exponents of ergodic f_{ω}. Because the functions e^{i}(ω) are independent of the choice of V^{i}(ω), we may choose these vector fields so that the computation of the e^{i} is as simple as possible. Our goal is to find a parameter value ω for which f_{ω} is ergodic and e^{u}(ω), e^{c}(ω), and e^{s}(ω) are all nonzero.

The first observation, which will imply that e^{s}(ω) is the constant function on A, is that the two-dimensional center-unstable Jacobian is preserved by f_{ω}.

Lemma 2.6 Let m be the eigenvalue for v_{0}. There exists a C^{∞} 2-form α such that α is nondegenerate on E^{uc}, and for all ω ∈ A,

f_{ω}^{∗}α = mα + β_{ω},

where β_{ω} vanishes on E^{uc}:

β_{ω}(v_{1}, v_{2}) = 0, ∀v_{1}, v_{2} ∈ E^{uc}.

Proof of Lemma 2.6: Write v_{0} = (q_{1}, q_{2}) and let α = v_{0} · (dx, dy) ∧ dz = q_{1}dx ∧ dz + q_{2}dy ∧ dz. Clearly α is nondegenerate on E^{uc}. Let (a, b) = ω. One calculates that g_{a}^{∗} (dx ∧ dz) = dx ∧ dz and g_{a}^{∗} (dy ∧ dz) = dy ∧ dz, and so g_{a}^{∗}α = α. Next, note that

h_{b}^{∗}α = mα + c_{ω}(x, y)dx ∧ dy,

where c_{ω}(x, y) = m det(v_{0} ∇φ_{b}(x, y)). The form β_{ω} = c(x, y)dx∧dy vanishes on E^{uc}, since dx ∧ dy does. Finally,

f_{ω}^{∗}α = h_{b}^{∗}g_{a}^{∗}α = h_{b}^{∗}α = mα + β_{ω}.

□

We now choose V_{ω}^{u}, V_{ω}^{c}, and V_{ω}^{s}. For ω ∈ A, the line E_{ω}^{u}(w) sits inside the plane E^{uc}(w), transverse to the line spanned by (0, 0, 1). There is a unique vector in E_{ω}^{u}(w) of the form (v_{0}, r). Let V_{ω}^{u}(w) be this vector; this also defines a continuous function u_{ω} : T^{3} → R, by:

V_{ω}^{u}(w) = (v_{0}, u_{ω}(w)).

By Lemma 2.4, u_{ω}(w) is a C^{2} function of ω ∈ A. The first two derivatives depend continuously on w ∈ T^{3}.

Next, for w ∈ T^{3}, pick V_{ω}^{c}(w) tangent to E_{ω}^{c}(w) such that

α(V_{ω}^{u}(w), V_{ω}^{c}(w)) = 1.

Finally, pick V_{ω}^{s}(w) tangent to E_{ω}^{s}(w) such that

dx ∧ dy ∧ dz(V_{ω}^{u}(w), V_{ω}^{c}(w), V_{ω}^{s}(w)) = 1.

Let e^{u}(ω) = e^{u}(f_{ω}), e^{c}(ω) = e^{c}(f_{ω}), and e^{s}(ω) = e^{s}(f_{ω}) be defined by equations (12)-(14).

Lemma 2.7 The functions e^{s} and e^{u} + e^{c} are constant:

e^{s}(ω) = − log(m), e^{u}(ω) + e^{c}(ω) = log(m).

for ω ∈ A.

Proof of Lemma 2.7: For i = u, c, s, let r_{ω}^{i} = r_{f_{ω}}^{i}, be the functions defined by equations (9)-(11), using the vector fields V_{ω}^{i}.

The lemma follows if we establish that the following equations hold for all w ∈ T^{3}:

r_{ω}^{u}(w)r_{ω}^{c}(w) = m,  (16)
r_{ω}^{u}(w)r_{ω}^{c}(w)r_{ω}^{s}(w) = 1.  (17)

Using Lemma 2.6, we see that

α(T f_{ω}(w)V_{ω}^{u}(w), T f_{ω}(w)V_{ω}^{c}(w)) = f_{ω}^{∗}α(V_{ω}^{u}(w), V_{ω}^{c}(w))
= mα(V_{ω}^{u}(w), V_{ω}^{c}(w)) + β_{ω}(V_{ω}^{u}(w), V_{ω}^{c}(w))
= mα(V_{ω}^{u}(w), V_{ω}^{c}(w))
= m,

while on the other hand,

α(T f_{ω}(w)V_{ω}^{u}(w), T f_{ω}(w)V_{ω}^{c}(w)) = α(r_{ω}^{u}(w)V_{ω}^{u}(f_{ω}(w)), r_{ω}^{c}(w)V_{ω}^{c}(f_{ω}(w)))
= r_{ω}^{u}(w)r_{ω}^{c}(w)α(V_{ω}^{u}(f_{ω}(w)), V_{ω}^{c}(f_{ω}(w)))
= r_{ω}^{u}(w)r_{ω}^{c}(w),

which proves (16). Since f_{ω} preserves volume,

1 = dx ∧ dy ∧ dz(T f_{ω}(w)V_{ω}^{u}(w), T f_{ω}(w)V_{ω}^{c}(w), T f_{ω}(w)V_{ω}^{s}(w))
= r_{ω}^{u}(w)r_{ω}^{c}(w)r_{ω}^{s}(w),

proving (17).□

Lemma 2.8 For ω = (a, b) ∈ A,

e^{u}(ω) = log(m) − ∫_{T^{3}} log(1 − aψ′(w)u_{ω}(w)) dw.

Proof of Lemma 2.8: From the definition of f_{ω}, we have:

T f_{ω}(w) ( \begin{matrix}v_{0} \\ u_{ω}(w) \end{matrix} ) = ( \begin{matrix}[m + aψ′(f_{ω}(w)) [u_{ω}(w) + ∇φ_{b}(w) · v_{0}]] v_{0} \\ u_{ω}(w) + ∇φ_{b}(w) · v_{0} \end{matrix} )

= ( \begin{matrix}[m + aψ′(f_{ω}(w))r_{ω}^{u}(w)u_{ω}(f_{ω}(w))] v_{0} \\ r_{ω}^{u}(w)u_{ω}(f_{ω}(w)) \end{matrix} ) .

It follows that

r_{ω}^{u}(w) = m + aψ′(f_{ω}(w))u_{ω}(f_{ω}(w))r_{ω}^{u}(w),

and so r_{ω}^{u}(w) = m/(1 − aψ′(f_{ω}(w))u_{ω}(f_{ω}(w))). To obtain the formula, integrate log(r_{ω}^{u}(w)). □

We now study how the function e^{u}(a, b) varies when a varies, keeping b fixed.

Lemma 2.9 e^{u} is C^{2} on A, and

\frac{∂}{∂a}e^{u}(a, b) = ∫_{T^{3}} \frac{ψ′(w)u_{a,b}(w) + aψ′(w)\frac{∂u_{a,b}(w)}{∂a}}{1 − aψ′(w)u_{a,b}(w)} dw.  (18)

Proof: By Lemma 2.4, the function ω ↦ u_{ω}(w) is C^{2} on A, uniformly in w. Then by the formula in Lemma 2.8, e^{u} is C^{2} as well. Differentiating this formula with respect to a gives (18). □

Setting a = 0 in (18), we obtain:

\frac{∂}{∂a}e^{u}(0, b) = ∫_{T^{3}} ψ′(w)u_{0,b}(w) dw  (19)

By Lemma 2.1, the distribution E_{0,b}^{u} for the skew product f_{0,b} is invariant under translations of the form (x, y, z) ↦ (x, y, z + z_{0}). This implies that the function u_{0,b}(x, y, z) depends only on x and y. On the other hand, ψ′(x, y, z) = ψ′(z) depends only on z. The integral in (19) is therefore equal to

∫_{T} ψ′(z) dz ∫_{T^{3}} u_{0,b}(w) dw = 0,

since ψ is homotopic to a constant map. We have shown:

Proposition 2.10 For (a, b) ∈ A,

\frac{∂}{∂a}e^{u}(0, b) = 0

The behavior of the exponents of f_{a,b} near (0, 0) is thus determined by the second derivative of e^{u} with respect to a. An exact computation of this second derivative is difficult in general. For our purposes, it suffices to compute this derivative at (0, 0).

Proposition 2.11

\frac{∂^{2}}{∂a^{2}} e^{u}(0, 0) = −u_{0}^{2} ∫_{0}^{1} ψ′(z)^{2} dz < 0,

where u_{0} = (w_{0} · v_{0})/(m − 1).

We prove Proposition 2.11 in the final subsection.

A picture of some unstable Lyapunov exponents computed for b = 0 and small values of a is shown in Figure 2.4, where h(x, y) = (2x + y, x + y), ψ(z) = cos(2πz), and v_{0} = (λ, 1), λ = (1 + \sqrt{5})/2. A random starting point was chosen for each value of a and 100, 000 iterations were applied to the vector (λ, 1, λ). Using Propositions 2.10 and 2.11, we computed the second order Taylor expansion with respect to a at (0, 0) to be e^{u}(a, 0) ∼ log(m) − π^{2}ma^{2}, where m = (3 + \sqrt{5})/2. This approximation in fact fits the data quite well.

0.9615 0.9616 0.9617 0.9618 0.9619 0.962 0.9621 0.9622 0.9623 0.9624 0.9625 -0.006 -0.004 -0.002  0  0.002 0.004 0.006
Figure 1: e^{u}(a, b) for the family f_{a,b}, with b = 0.

2.5 Proof of Proposition 1.3

If |b| is sufficiently small, then by Lemma 2.9 and Proposition 2.11, we have

\frac{∂^{2}}{∂a^{2}} e^{u}(0, b) < 0.  (20)

By Proposition 2.10,

\frac{∂}{∂a}e^{u}(0, b) = 0.  (21)

Choose b different from 0 and satisfying (20). It follows from (20), (21), and Lemma 2.9 that for |a| sufficiently small and positive, e^{u}(a, b) is positive and strictly greater than log(m). But now by Lemma 2.7, e^{c}(a, b) = log(m)−e^{u}(a, b) > 0, and e^{s}(a, b) = − log(m). By Lemma 2.5, the Lyapunov exponents of f_{a,b} are almost everywhere equal to e^{u}(a, b), e^{c}(a, b), and e^{s}(a, b). Thus for this b, and for |a| small and positive, the Lyapunov exponents of f_{a,b} are nonzero almost everywhere.

Let f = f_{a,b}, where a and b are chosen so that f_{a,b} is stably a K-system and partially hyperbolic, the foliation W_{a,b}^{c} exists and has the properties outlined in Proposition 2.2, and the exponents of f_{a,b} are nonzero, μ − a.e.. Let U_{0} be the neighborhood of f_{a,b} in Diff_{μ}^{2}(T^{3}) given by Proposition 2.2. Every g ∈ U_{0} is a K-system, is partially hyperbolic, and has a center foliation W_{g}^{c} satisfying conclusion 1 of Theorem II. By Lemma 2.5 the exponents of g ∈ U_{0} depend continuously on g. Since f_{a,b} is nonuniformly hyperbolic, there exists a C^{1}-open neighborhood U ⊆ U_{0} of f_{a,b} in Diff_{μ}^{2}(T^{3}) so that every g ∈ U is nonuniformly hyperbolic. By a theorem of Pesin [Pe], a nonuniformly hyperbolic K-system is isomorphic to a Bernoulli shift. Therefore every g ∈ U is isomorphic to a Bernoulli shift.

It also follows from Lemma 2.5 that for g ∈ U, E_{g}^{c} is a Lyapunov direction μ-a.e. for g, with positive exponent, since g is nonuniformly hyperbolic. This implies conclusion 2 of Theorem II.

Conclusion 3 of Theorem II follows from 2. Let S be the set of w ∈ T^{3} such that E_{g}^{c}(w) is a positive Lyapunov direction. Then S has full μ-measure in T^{3}. Suppose there were a circle C ∈ W_{g}^{c} with μ_{C}(C ∩ S) > 0, where μ_{C} is Riemannian measure on C. Then the length of g^{n}(C) would be unbounded as n → ∞. But g^{n}(C) is a leaf of W_{g}^{c}, a contradiction.

Finally, g is not isotopic to an Anosov diffeomorphism, since g is isotopic to the linear map f_{0,0}, which has 1 as an eigenvalue. This completes the proof of Proposition 1.3.

2.6 Proof of Proposition 2.11

Use “f_{a}” to denote f_{a,0}, “u_{a}” to denote u_{a,0}, and “e(a)” for e^{u}(a, 0). Differentiating (18), we have

e′′(a) = ∫_{T^{3}} (\frac{ψ′(w)u_{a}(w) + aψ′(w)\frac{∂u_{a}(w)}{∂a}}{1 − aψ′(w)u_{a}(w)} )^{2}+\frac{2ψ′(w)\frac{∂u_{a}(w)}{∂a} + aψ′(w)\frac{∂^{2}u_{a}(w)}{∂a^{2}}}{(1 − aψ′(w)u_{a}(w))} dw,

and setting a = 0, we obtain

e′′(0) = ∫_{T^{3}}(ψ′(w)u_{0}(w))^{2} + 2ψ′(w)\frac{∂u_{a}(w)}{∂a}|_{a=0} dw.  (22)

The map f_{0} = f_{0,0} is linear. It is easy to see that u_{0}(w) is the constant function u_{0}(w) = u_{0} = (w_{0} · v_{0})/(m − 1) ≠ 0

For a ∈ R, w = (x, y, z) ∈ T^{3}, and u ∈ R, let

γ(a, w, u) = \frac{c + u}{m + aψ′(w)(c + u)},

where c = w_{0} · v_{0}. Note that γ(a, w, u_{a}(f_{a}^{−1}(w))) = u_{a}(w) and that for |a| sufficiently small,

u_{a}(w) = lim_{n→∞} γ(a, w, γ(a, f_{a}^{−1}(w), … γ(a, f_{a}^{−n}(w), 0) ⋯)).

We compute:

\frac{∂γ}{∂a}(a, w, u) = \frac{−ψ′(w)(c + u)^{2}}{(m + aψ′(w)(c + u))^{2}} ,

and set a = 0 to get:

\frac{∂γ}{∂a}(0, w, u) = \frac{−ψ′(w)(c + u)^{2}}{m^{2}} .

Similarly,

\frac{∂γ}{∂x}(0, w, u) = \frac{∂γ}{∂z} (0, w, u) = \frac{∂γ}{∂y} (0, w, u) = 0,

and

\frac{∂γ}{∂u}(0, w, u) = \frac{1}{m}.

We want to evaluate \frac{∂u_{a}(w)}{∂a}|_{a=0}. Since u_{a}(w) = γ(a, w, u_{a}(f_{a}^{−1}(w))), the chain rule yields:

\frac{∂u_{a}(w)}{∂a}|_{a=0} = \frac{∂}{∂a}γ(a, w, u_{a}(f_{a}^{−1}(u)))|_{a=0}
= \frac{∂γ}{∂a}(0, w, u_{0}(f_{0}^{−1}(w))) + \frac{∂γ}{∂u}(0, w, u_{0}(F_{0}^{−1}(w))) · \frac{∂u_{a}(f_{a}^{−1}(w))}{∂a}|_{a=0})
= \frac{−ψ′(w)(c + u_{0}(f_{0}^{−1}(w)))^{2}}{m^{2}} + \frac{1}{m} \frac{∂u_{a}(f_{a}^{−1}(w))}{∂a}|_{a=0}).

Recall that u_{0} is the constant function u_{0}(w) = u_{0}, and u_{0} = (c + u_{0})/m, so this expression simplifies to:

\frac{∂u_{a}(w)}{∂a}|_{a=0} = −ψ′(w)u_{0}^{2} + \frac{1}{m} \frac{∂u_{a}(f_{a}^{−1}(w))}{∂a}|_{a=0}.  (23)

Iterating (23) gives

\frac{∂u_{a}(w)}{∂a}|_{a=0} = −ψ′(w)u_{0}^{2} − \frac{ψ′(w_{−1})u_{0}^{2}}{m} − \frac{ψ′(w_{−2})u_{0}^{2}}{m^{2}} − ⋯

where w_{−j} = f_{a}^{−j}(w). Hence

∫_{T^{3}} ψ′(w)\frac{∂u_{a}(w)}{∂a}|_{a=0} dw = − ∑_{j≥0} ∫_{T^{3}} \frac{ψ′(w)ψ′(w_{−j})u_{0}^{2}}{m^{j}} dw.

The j = 0 term of this sum is − ∫ ψ′(w)^{2}u_{0}^{2} dw. Pulling out this term, we have

∫_{T^{3}}(ψ′(w)u_{0}(w))^{2} + 2ψ′(w)\frac{∂u_{a}(w)}{∂a}|_{a=0} dw =

= − ∫_{T^{3}} ψ′(w)^{2}u_{0}^{2} dw − 2 ∑_{j≥1} ∫_{T^{3}} \frac{ψ′(w)ψ′(w_{−j})u_{0}^{2}}{m^{j}} dw. (24)

Notice that

∫_{T^{3}} ψ′(w)ψ′(w_{−j})u_{0}^{2} dw = u_{0}^{2} ∫_{0}^{1} ∫_{0}^{1} ∫_{0}^{1} ψ′(z)ψ′(z − r_{j}x − s_{j}y) dx dy dz,

where r_{j} and s_{j} are integers, and r_{j} = 0 if and only if j = 0, in which case s_{j} = 0 as well. Thus, for j ≥ 1,

∫_{T^{3}} ψ′(w)ψ′(w_{−j})u_{0}^{2} dw = u_{0}^{2}r_{j}^{−1} ∫_{0}^{1} ∫_{0}^{1} ψ′(z)ψ(z − r_{j}x − s_{j}y))|_{x=0}^{x=1} dy dz
= 0.

Combining this with equations (22) and (24), we have:

e′′(0) = − ∫_{T^{3}} ψ′(w)^{2}u_{0}^{2} dw − 2 ∑_{j≥1} ∫_{T^{3}} \frac{ψ′(w)ψ′(w_{−j})u_{0}^{2}}{m^{j}} dw
= −u_{0}^{2} ∫_{0}^{1} ψ′(z)^{2} dz,

completing the proof.

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