EVERY COMPACT MANIFOLD CARRIES A HYPERBOLIC ERGODIC FLOW

HUYI HU, YAKOV PESIN, ANNA TALITSKAYA

Dedicated to Anatole Katok on the occasion of his 60th birthday.

Abstract. We show that every compact smooth Riemannian manifold M of dimM ≥ 3, admits a volume-preserving Bernoulli flow with non-zero Lyapunov exponents except for the Lyapunov exponent along the direction of the flow.

1. Introduction

In [DP], Dolgopyat and Pesin obtained an affirmative solution to the long-standing problem of whether a compact smooth Riemannian manifold admits a volume-preserving ergodic (Bernoulli) diffeomorphism with non-zero Lyapunov exponents. In this paper we discuss a continuous time version of this problem and we prove the following result.

Theorem 1. Given a compact smooth Riemannian manifold M of dimM ≥ 3, there exists a C^{∞} flow f^{t} such that for each t ≠ 0,

(1) f^{t} preserves the Riemannian volume μ on M;

(2) f^{t} has non-zero Lyapunov exponents (except for the exponent along the flow direction) at m-almost every point x ∈ M;

(3) f^{t} is a Bernoulli diffeomorphism.

Acknowledgment. The final version of this paper was written when Ya. P. was visiting Research Institute for Mathematical Science (RIMS) at Kyoto University. Ya. P. wish to thank RIMS for hospitality.

2. Construction of diffeomorphisms by Katok and Brin

In our construction of the flow we use a special diffeomorphism of the two dimensional unit disc 𝒟^{2} constructed by Katok in [K]. We summarise the description and properties of this diffeomorphism in the following proposition.

Proposition 2. There exists a C^{∞} diffeomorphism g : 𝒟^{2} → 𝒟^{2} with the following properties:

(1) g preserves area on 𝒟^{2};

(2) g has non-zero Lyapunov exponents almost everywhere;

(3) g is a Bernoulli map; (4) d^{k}(g − id)|∂𝒟^{2} = 0 for any k ≥ 0, i.e., on the boundary of the disk g is the identity

Key words and phrases. Hyperbolic measure, Lyapunov exponents, stable ergodicity, accessibility.

H. H. was partially supported by NSF grants DMS-0196234 and DMS-0240097; Ya. P. and A. T. were partially supported by the National Science Foundation grant DMS-0088971 and U.S.-Mexico Collaborative Research grant 0104675.

map and has all its derivatives zero.

Let us sketch the construction of the diffeomorphism g. We begin with the automorphism

g_{0} of the torus 𝒯^{2} given by the matrix ( \begin{matrix}13 & 8 \\ 8 & 5 \end{matrix} ). The map g_{0} has four fixed points

q_{1} = (0, 0) , q_{2} = (\frac{1}{2}, 0) , q_{3} = (0, \frac{1}{2}) , q_{4} = (\frac{1}{2}, \frac{1}{2}) .

In a small neighborhood D_{r}^{i} = {(s_{1}, s_{2}) : s_{1}^{2} + s_{2}^{2} ≤ r} of each q_{i}, 0 < r < 1, the map g_{0} is the time-1 map of the flow given by

ṡ_{1} = −(log α)s_{1}, ṡ_{2} = (log α)s_{2},

where α > 1 is the larger eigenvalue of g_{0} and {s_{1}, s_{2}} is the coordinate system in a neighborhoods of each q_{i} generated by the eigenvectors of g_{0}.

Then we consider the map g_{1} that is conjugate to g_{0} via a conjugacy ϕ_{0} that slows down the motion near q_{i}. More precisely, g_{1} is the time-1 map of the flow given by

(2.1)  ṡ_{1} = −(log α)s_{1}ψ(s_{1}^{2} + s_{2}^{2}), ṡ_{2} = (log α)s_{2}ψ(s_{1}^{2} + s_{2}^{2})

in D_{r}^{i}, and g_{1} = g_{0} otherwise, where ψ is a C^{∞} function except at zero and such that ψ(0) = 0, ψ(ξ) ≥ 0 for ξ ≥ 0, ψ(ξ) = 1 for ξ ≥ r and

∫_{0}^{r} \sqrt{\frac{1}{ψ(ξ)}} dξ < ∞.

The map g_{1} preserves a probability measure dν = κ_{0}^{−1}κ dm, where m is area and the density κ is a positive C^{∞} function defined by the formula

κ(s_{1}, s_{2}) = \begin{cases}(ψ(s_{1}^{2} + s_{2}^{2}))^{−1} & if (s_{1}, s_{2}) ∈ D_{r}^{i}. \\ 1 & otherwise. \end{cases}

Here

κ_{0} = ∫_{𝒯^{2}} κ dm.

Note that κ is infinite at q_{i}.

Define the map ϕ_{1} by the formula

ϕ_{1}(s_{1}, s_{2}) = \frac{1}{\sqrt{κ_{0}(s_{1}^{2} + s_{2}^{2})}} (∫_{0}^{s_{1}^{2}+s_{2}^{2}} \frac{du}{ψ(u)})^{1/2} (s_{1}, s_{2})

near each q_{i} and then extend the map to 𝒯^{2} in such a way that ϕ_{1} is C^{∞}, commutes with the involution J(t_{1}, t_{2}) = (1−t_{1}, 1−t_{2}) on 𝒯^{2}, and satisfies (ϕ_{1})_{∗}ν = m. Hence, g_{2} = ϕ_{1}∘g_{1}∘ϕ_{1}^{−1} is a C^{∞} area preserving map.

Let ϕ_{2} : 𝒯^{2} → S^{2} be a double branched covering that satisfies ϕ_{2} ∘ J = ϕ_{2}, (ϕ_{1})_{∗}m = m, and C^{∞} everywhere except for the points q_{i}, where it branches and near q_{i},

ϕ_{2}(s_{1}, s_{2}) = \frac{1}{\sqrt{s_{1}^{2} + s_{2}^{2}}} (s_{1}^{2} − s_{2}^{2}, 2s_{1}s_{2}).

The map g_{3} = ϕ_{2} ∘ g_{2} ∘ ϕ_{2}^{−1} is a C^{∞} diffeomorphism of the sphere 𝒮^{2}.

Finally, let ϕ_{3} be a C^{∞} map that blows up the point q_{4} into a circle and makes g = ϕ_{3} ∘ g_{3} ∘ ϕ_{3}^{−1} to be the desired map of the disk. We refer the reader to [K] (see also [DP]) for more details.

Note that the map g_{1} preserves measure ν and the maps g_{2}, g_{3} and g preserve area. The desired result follows.

We call the map g Katok’s map. We need some additional properties of this map.

Since g has non-zero Lyapunov exponents, for almost every x there are global stable and unstable manifolds, 𝒲_{g}^{s}(x) and 𝒲_{g}^{u}(x), at x.

Let Q = {q_{1}, q_{2}, q_{3}} ∪ ∂𝒟^{2} be the discontinuity set of g.

Proposition 3. The following properties hold:

(1) periodic points of g are everywhere dense;

(2) g possesses two one-dimensional continuous foliations which are extensions of the stable and unstable global foliations W_{g}^{s}(x) and W_{g}^{u}(x); we will use the same notations for these foliations;

(3) there exist neighborhoods U ⊂ U_{1} of ∂𝒟^{2} and a vector field V in U_{1} which generates an area-preserving flow g^{t}: U → 𝒟^{2}, −2 < t < 2 for which g|U = g^{1}.

Proof. Note that the map g_{1} is topologically conjugate to g_{0} and Statement 1 follows. Statements 2 is proved in [K] (see Lemma 4.1). We prove the last statement. By construction, near each point q_{i}, the map g_{1} is the time-1 map of a vector field V_{1}. Moreover, near each q_{i} we have V_{1}(−x) = −V_{1}(x) for any x, see (2.1). It follows that the maps g_{2} and g_{3} near q_{i} are the time-1 maps of the vector fields given by V_{2} = dϕ_{1} ∘ V_{1} ∘ ϕ_{1}^{−1} and V_{3} = dϕ_{2} ∘ V_{2} ∘ ϕ_{2}^{−1} respectively. Here we should stress that V_{3} is well defined even though ϕ_{2} is a two to one map. In fact, near q_{i} we have ϕ_{1}(−x) = −ϕ_{1}(x) and ϕ_{2}(−x) = ϕ_{2}(x). This gives V_{2}(−x) = −V_{2}(x) and therefore, for any y near q_{i}, ϕ_{2}^{−1}(y) has two preimages x and −x at which

(dϕ_{2})_{−x}(V_{2}(−x)) = (dϕ_{2})_{−x}(−V_{2}(x)) = (dϕ_{2})_{x}(V_{2}(x)).

Now we see that g is the time-1 map of the vector field V = dϕ_{3} ∘ V_{2} ∘ ϕ_{3}^{−1}. □

The next result shows that the map g is diffeotopic to the identity map.

Proposition 4. There exists a map G : 𝒟^{2} × [0, 1] → 𝒟^{2} with the following properties: (1) G(x, t) is C^{∞} in (x, t);

(2) G(·, 0) = id and G(·, 1) = g;

(3) G(x, t) = g^{t}(x) for any x ∈ U and t ∈ [0, 1], where g^{t}(x) is the flow in Proposition 3;

(4) for any t ∈ [0, 1] the map G(·, t) : 𝒟^{2} → 𝒟^{2} is an area-preserving diffeomorphism;

(5) d^{k}G(x, 1) = d^{k}G(g(x), 0) for any k ≥ 0.

Proof. Recall that in the neighborhood U of the boundary of 𝒟^{2} the map g is the time-1 map of the flow generated by the vector field V. We extend V to a smooth vector field V̂ on the whole 𝒟^{2}, and let g^{t} be the flow generated by V̂. Note that g|U = g^{1}|U. We need the following result of Smale (see [S], Theorem B):

Lemma 5. Let 𝒜 be the space of C^{∞} diffeomorphisms of the unit square which are equal to the identity in some neighborhood of the boundary. Endow 𝒜 with the C^{r} topology, 1 < r ≤ ∞. Then 𝒜 is contractible to a point.

The statement also holds if the unit square is replaced by the unit disk. Applying the result to the diffeomorphism g ∘ g^{−1}, which is equal to the identity on U, we obtain a homotopy G̃ : 𝒟^{2} ×[0, 1] → 𝒟^{2} such that G̃(·, 0) = id |𝒟^{2} and G̃(·, 1) = g^{−1} ∘g. Moreover, G̃ is C^{∞} in (x, t), i.e., G̃ is a diffeotopy in 𝒜 (see [S], Theorem 4). Therefore, for each t ∈ [0, 1], there is a neighborhood U_{t} of ∂𝒟^{2} such that G̃(·, t)|U_{t} = id |U_{t}. One can show that the set

U = int ⋂_{t∈[0,1]} U_{t}

is not empty and is a neighborhood of ∂𝒟^{2}. Denote g^{t} = G̃(·, t). It follows that the map G(·, t) = g^{t} ∘ g^{t} satisfies Statements 1- 3 of the proposition. To prove Statements 4 and 5, we need the following lemma.

Lemma 6. Let {Ω_{0}^{t}} and {Ω_{1}^{t}} be two families of volume forms on 𝒟^{2} that are C^{∞} in (x, t). Assume that Ω_{0}^{t}|U = Ω_{1}^{t}|U for any t and Ω_{0}^{t} = Ω_{1}^{t} for t ∈ [0, ε) ∪ (1 − ε, 1]. Then there exists a map Ḡ : 𝒟^{2} × [0, 1] → 𝒟^{2} with the following properties:

(1) Ḡ(x, t) is C^{∞} in (x, t);

(2) Ḡ(·, 0) = Ḡ(·, 1) = id;

(3) for any t ∈ [0, 1] the map Ḡ(·, t) : 𝒟^{2} → 𝒟^{2} is a diffeomorphism with Ḡ(·, t)^{∗}Ω_{1}^{t} =

Ω_{0}^{t};
(4) Ḡ(x, t) = x for any t ∈ [0, 1] and x in some neighborhood U′ ⊂ U of ∂𝒟^{2}.

Proof. The argument is a modification of the proof of Moser’s theorem ([M]). We follow here the approach in [KH] (see Theorem 5.1.27). Let Ω^{t} = Ω_{1}^{t} − Ω_{0}^{t} and Ω_{s}^{t} = Ω_{0}^{t} + sΩ^{t} for s ∈ [0, 1]. We know that Ω^{t}|U = 0. We construct a family of one forms ℋ^{t} such that ℋ^{t} is C^{∞} in (x, t), dℋ^{t} = Ω^{t} and ℋ^{t}|U′ = 0 for some neighborhood U′ ⊂ U of ∂𝒟^{2}. Consider a Euclidean coordinate system (x_{1}, x_{2}) such that 𝒟^{2} = {(x_{1}, x_{2}) : x_{1}^{2} + x_{2}^{2} ≤ 1}. We have

Ω^{t} = ρ^{t}(x)dx_{1} ∧ dx_{2}

and

Ω_{s}^{t} = ρ_{s}^{t}(x)dx_{1} ∧ dx_{2}

with ρ_{s}^{t} > 0. Note that ρ^{t}|_{U} = 0 and ∫ ρ^{t} dx = 0 for any t ∈ [0, 1]. Given a C^{∞} function θ^{t} = θ^{t}(x), x ∈ 𝒟^{2}, t ∈ [0, 1], let

E_{1}(θ^{t})(a) = ∫_{𝒟^{2} ∩{x_{2}=a}} θ^{t}(x_{1}, a)dx_{1}

and

E_{2}(θ^{t})(a) = ∫_{𝒟^{2} ∩{x_{1}=a}} θ^{t}(a, x_{2})dx_{2}

be the expectation of θ^{t} along the lines x_{2} = constant and x_{1} = constant. We choose the function θ^{t} such that

E_{1}(θ^{t}) = E_{1}(ρ^{t}), E_{2}(θ^{t}) = 0,

and θ^{t}|U′ = 0 where U′ ⊂ U is a neighborhood of ∂𝒟^{2}. Such θ^{t} exists. Indeed, choose any positive C^{∞} function δ such that ∫ δdx_{1} = 1, the support supp δ ⊂ (−ε, ε) for some small ε > 0, and then set θ^{t}(x_{1}, x_{2}) = δ(x_{1})E_{1}(ρ^{t})(x_{2}). Now let

a^{t}(x_{1}, x_{2}) = − ∫_{𝒟^{2} ∩{(x_{1},y):y<x_{2}}} θ^{t}(x_{1}, y) dy,
b^{t}(x_{1}, x_{2}) = ∫_{𝒟^{2} ∩{(y,x_{2}):y<x_{1}}}[ρ^{t}(y, x_{2}) − θ^{t}(y, x_{2})] dy.

The 1-forms

ℋ^{t}(x_{1}, x_{2}) = a^{t}(x_{1}, x_{2})dx_{1} + b^{t}(x_{1}, x_{2})dx_{2}

satisfy the desired requirements.

For each s ∈ [0, 1], consider the vector field

V_{s}^{t} = −\frac{b^{t}}{ρ_{s}^{t}} \frac{∂}{∂x_{1}} + \frac{a^{t}}{ρ_{s}^{t}} \frac{∂}{∂x_{2}} .

It is well defined since ρ_{s}^{t} > 0 on 𝒟^{2} and it is C^{∞} in (x, t, s). We also have V_{s}^{t}|U′ = 0 and Ω_{s}^{t}(V_{s}^{t}, ·) = −ℋ^{t}. Let (Ḡ^{t})^{s}: 𝒟^{2} → 𝒟^{2} be the solution of the differential equation \frac{dx}{ds} = V_{s}^{t}(x) satisfying the initial condition (Ḡ^{t})^{0} = id. We have that

(dḠ^{t})^{1}^{∗}Ω_{1}^{t} = (dḠ^{t})^{0}^{∗}Ω_{0}^{t} = Ω_{0}^{t}

(see [KH], Theorem 5.1.27 for more details). So Ḡ^{t} = (Ḡ^{t})^{1} is the desired map. □

We proceed with the proof of the proposition. Consider the map Ḡ as in the lemma with Ω_{0}^{t} = dx_{1} ∧ dx_{2} on 𝒟^{2} for any t ∈ [0, 1] and Ω_{1}^{t} = dg̃^{t∗}dĝ^{t∗}Ω_{0}^{t}. Let ḡ^{t} = Ḡ(·, t). Then the map G : 𝒟^{2} × [0, 1] → 𝒟^{2},

G(x, t) = ḡ^{t} ∘ ĝ^{t} ∘ g̃^{t}

satisfies Statements 1 – 4 of the proposition. If necessary one can change the map G̃(·, t) in a small neighborhood of the set 𝒟^{2} × 0 and 𝒟^{2} × 1 so that it will also satisfy Statement 5. For example, one can choose G̃ in such a way that

G̃(Ĝ(x, t), t) = G̃(Ĝ(x, 0), 0), t ∈ [0, ε), G̃(Ĝ(x, t), t) = G̃(Ĝ(x, 1), 1), t ∈ (1 − ε, 1].

Hence, ĝ^{t}∘g̃^{t} is area-preserving for any t ∈ [0, ε)∪(1−ε, 1]. Therefore, by Lemma 6, Ω_{1}^{t} = Ω_{0}^{t} and Ḡ(·, t) = id for all t ∈ [0, ε) ∪ (1 − ε, 1], because in this case Ω^{t} = 0, ρ^{t} = 0, a^{t} = b^{t} = 0 and V_{s}^{t} = 0 for every s. □

We also need Brin’s construction from [B2]. Given n ≥ 5, let A: 𝒯^{n−3} → 𝒯^{n−3} be a hyperbolic automorphism of the (n−3)-dimensional torus and h^{t}: L → L the suspension flow over A with the roof function H = 1, y ∈ 𝒯^{n−3}. The suspension manifold L is diffeomorphic to 𝒯^{n−3} × [0, 1]/ ∼, where ∼ is the identification (y, 1) = (Ay, 0). The flow h^{t} preserves volume on L and one can choose A so that L can be embedded into ℛ^{n−1} × 𝒮^{1} with trivial normal bundle.

3. Proof of the theorem: the case dimM ≥ 5

Consider the map

R = g × A : 𝒟^{2} × 𝒯^{n−3} → 𝒟^{2} × 𝒯^{n−3},

where g: 𝒟^{2} → 𝒟^{2} is Katok’s diffeomorphism and A: 𝒯^{n−3} → 𝒯^{n−3} is the automorphism from Brin’s construction. Consider the suspension flow over R with roof function H = 1 and the suspension manifold K = 𝒟^{2} × 𝒯^{n−3} × [0, 1]/ ∼, where ∼ is the identification (x, y, 1) = (g(x), A(y), 0). In other words, K is the manifold 𝒟^{2} × 𝒯^{n−3} × [0, 1] where the points (x, y, 1) are identified with the points (g(x), A(y), 0). Denote by Z the vector field of the suspension flow and by φ_{Z}^{t} the suspension flow itself.

For each point (x, y, t) ∈ K consider the coordinate system

(3.2)  (x_{1}, x_{2}, y_{1}, …, y_{n−3}, t) = (x, y, t)

in its neighborhood where x = (x_{1}, x_{2}) ∈ 𝒟^{2}, y = (y_{1}, … y_{n−3}) ∈ 𝒯^{n−3} and t ∈ [0, 1]. In this coordinate system, Z = (0, 0, 1).

Set N = 𝒟^{2} × L, where L is the suspension manifold in Brin’s construction (see the previous section). Write N = 𝒟^{2} × (𝒯^{n−3} × [0, 1]/ ∼) where ∼ is the identification (y, 1) = (A(y), 0) for all y ∈ 𝒯^{n−3}.

Consider the map F: K → N given by

F (x, y, t) = (G(x, t), y, t),

where G : 𝒟^{2} × [0, 1] → 𝒟^{2} is the diffeotopy constructed in Proposition 4. We have

F (x, y, 1) = (g(x), y, 1) = (g(x), A(y), 0) = F (g(x), A(y), 0).

Therefore, the map F is well-defined. It is easy to see that F preserves volume, is one-to-one and continuous. Hence, it is a homeomorphism. Formal differentiation yields

(3.3)  dF (x, y, t) = ( \begin{matrix}G_{x}(x, t) & 0 & G_{t}(x, t) \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) .

It follows from Statement 5 of Proposition 4 that for every k ≥ 1,

d^{k}F (x, y, 1) = d^{k}F (g(x), Ay, 0)

and hence, F is a C^{∞} diffeomorphism.

Consider the vector field Y = dF Z on N and let φ_{Y}^{t} be the corresponding flow. In the coordinate system (3.2), we have

Y (G(x, t), y, t) = (\frac{∂G}{∂t}(x, t), 0, 1), (x, y, t) ∈ K. (3.4)

The vector field Y is divergence free since it is the image of the divergence free vector field Z under the volume-preserving map F.

Let us choose a C^{∞} function α : 𝒟^{2} → [0, 1] such that

(A1) α and all its partial derivatives of any order are equal to zero on ∂𝒟^{2}; (A2) α(x) > 0 for x ∈ int𝒟^{2}, α(x) = 1 for x ∈ 𝒟^{2} \ U. Define the vector field X on N by

X(G(x, t), y, t) = (\frac{∂G}{∂t}(x, t), 0, α(G(x, t))), (x, y, t) ∈ K. (3.5)

Note that by Statement 3 of Proposition 4, \frac{∂G}{∂t}(x, t) = V (G(x, t)) for x ∈ U. Therefore, Equalities (3.4) and (3.5) imply that for (x, y, t) ∈ N with x ∈ U,

Y (x, y, t) = (V (x), 0, 1), X(x, y, t) = (V (x), 0, α(x)). (3.6)

Let φ^{t} = φ_{X}^{t} be the flow on N generated by the vector field X. It is easy to see that X is divergence free and hence, φ^{t} is volume-preserving.

Lemma 7. All but one Lyapunov exponents of the flow φ_{X}^{t} are non-zero almost everywhere.

Proof. We begin with a construction of a map which is similar to Katok’s map: it has the same topological and hyperbolic properties but does not preserve area. It is better adopted to the flow. We will also use this map in the proof of the ergodicity of the flow.

We assume that the function α(x) is chosen such that the following additional condition holds:

(A3) α(x)^{−1}V (x) → 0 as x → ∂𝒟^{2}, where V is the vector field defined in Statement 3 of

Proposition 3. Consider the map g^{∗} : 𝒟^{2} → 𝒟^{2} such that g^{∗} = g on 𝒟^{2}\U and g^{∗} is the time-1

map of the flow g^{∗}^{t} generated by the vector field V^{∗} = α^{−1}V on U_{1} (see Statement 3 of Proposition 3). By (A1) − (A3), the map g^{∗} is well defined and is a diffeomorphism. It also satisfies Statements 2 – 4 of Propositions 2 and 3. Note that the map g^{∗} preserves a measure μ^{∗} which is absolutely continuous with respect to area with positive density ρ^{∗}(x); the latter is unbounded as x approaches ∂𝒟^{2}.

We now proceed as before replacing g by g^{∗}. Namely, define G^{∗} : 𝒟^{2} × [0, 1] → 𝒟^{2} by G^{∗}(x, t) = G(x, t) if x ∈ X \ U, and G^{∗}(x, t) = g^{∗}^{t}(x) otherwise. Let ϕ_{Z^{∗}}^{t} be the suspension flow over g^{∗} × A with the suspension manifold K^{∗} = X × 𝒯^{n−3} × [0, 1]/ ∼, where ∼ is the identification (x, y, 1) = (g^{∗}(x), A(y), 0) and Z^{∗} is the vector field of the suspension flow. Define the map F^{∗} : K^{∗} → N by

F^{∗}(x, y, t) = (G^{∗}(x, t), y, t)

and let Y^{∗} = dF^{∗}Z^{∗}. We have for x ∈ U,

\frac{∂G^{∗}}{∂t} (x, t) = α(g^{t}(x))^{−1}V (g^{t}(x))

and hence,

Y^{∗}(G^{∗}(x, t), y, t)) = (α(g^{∗}^{t}(x))^{−1}V (g^{∗}^{t}(x)), 0, t)

or equivalently,

Y^{∗}(x, y, t) = (α(x)^{−1}V (x), 0, t).

Since Y^{∗}(x, y, t) = Y (x, y, t) for x ∉ U, we obtain by (3.6) that X = αY^{∗}.

Define the vector field Z̃ on K^{∗} by

Z̃(x, y, t) = (dF^{∗})^{−1}X(F^{∗}(x, y, t)) (x, y, t) ∈ K^{∗}.

It is easy to see that ϕ_{X}^{t} = F^{∗} ∘ ϕt ; Z̃ ∘ F^{∗}^{−1} and the vector fields Z^{∗} and Z̃ have the same orbits. In other words, there is a function β: K^{∗} → ℛ^{+} such that for every (x, y, t) ∈ K^{∗},

(B1) Z̃(x, y, t) = β(x, y, t)Z^{∗}(x, y, t); (B2) β(x, y, t) = 0 if x ∈ ∂𝒟^{2}; (B3) β(x, y, t) = 1 if x ∉ U.

By construction, the flow ϕ_{Z^{∗}}^{t} has non-zero Lyapunov exponents almost everywhere. Denote by E_{Z^{∗}}^{s}(x, y, t), E_{Z^{∗}}^{u}(x, y, t), E_{Z^{∗}}^{cs}(x, y, t) and E_{Z^{∗}}^{cu}(x, y, t) the stable, unstable, center-stable and center-unstable invariant subspaces at the point (x, y, t) ∈ K^{∗} repectively. Observe that the subspaces E_{Z^{∗}}^{cs}(x, y, t) and E_{Z^{∗}}^{cu}(x, y, t) are also invariant under the flow ϕt ; Z̃. Chose a point (x_{0}, y_{0}, t_{0}) and a vector v ∈ E_{Z^{∗}}^{u}(x_{0}, y_{0}, t_{0}). Note that for almost every (x_{0}, y_{0}, t_{0}) (with respect to the Riemannian volume) the proportion of time the trajectory {ϕt ; Z̃(x_{0}, y_{0}, t_{0})} spends in the set {(x, y, t): x ∉ U} is strictly positive. It follows that the Lyapunov exponent χ(v) at (x_{0}, y_{0}, t_{0}) with respect to the flow ϕt ; Z̃ is positive. □

Almost every point (x_{0}, y_{0}, t_{0}) ∈ N has stable, unstable, center-stable and center-unstable global manifolds W_{X}^{s}(x_{0}, y_{0}, t_{0}), W_{X}^{u}(x_{0}, y_{0}, t_{0}), W_{X}^{cs}(x_{0}, y_{0}, t_{0}) and W_{X}^{cu}(x_{0}, y_{0}, t_{0}) with respect to the flow φ_{X}^{t}. Similarly, almost every point (x_{0}, y_{0}, t_{0}) ∈ K^{∗} has stable, unstable, center-stable and center-unstable global manifolds W s ; Z̃(x_{0}, y_{0}, t_{0}), W u ; Z̃(x_{0}, y_{0}, t_{0}), W cs ; Z̃(x_{0}, y_{0}, t_{0}) and W cu ; Z̃(x_{0}, y_{0}, t_{0}) with respect to the flow φt ; Z̃. By Proposition 2 these foliations can be extended to foliations which are continuous everywhere except for the discontinuity set Q × 𝒯^{n−3} × [0, 1]/ ∼. We will use the same notations for these foliations. Observe that

(3.7)  π_{x}(W cs ; Z̃(x_{0}, y_{0}, t_{0})) = W_{g^{∗}}^{s}(x_{0}), π_{x}(W cu ; Z̃(x_{0}, y_{0}, t_{0})) = W_{g^{∗}}^{u}(x_{0})

and

(3.8)  π_{y}(W cs ; Z̃(x_{0}, y_{0}, t_{0})) = W_{A}^{s}(y_{0}), π_{y}(W cu ; Z̃(x_{0}, y_{0}, t_{0})) = W_{A}^{u}(y_{0}),

where π_{x}: K^{∗} → 𝒟^{2} and π_{y}: K^{∗} → 𝒯^{n−3} are natural projections.

We say that two points z, z′ ∈ N are accessible if there are points z = z_{0}, z_{1}, …, z_{ℓ−1}, z_{ℓ} = z′, z_{i} ∈ N such that z_{i} ∈ W_{X}^{u}(z_{i−1}) or z_{i} ∈ W_{X}^{s}(z_{i−1}) for i = 1, …, ℓ. The collection of points z_{0}, z_{1}, …, z_{ℓ} is called a path connecting z and z′ and is denoted by [z, z′] = [z_{0}, z_{1}, …, z_{ℓ}]. Accessibility is an equivalence relation. We say that the time-t map of the flow φ_{X}^{t} has accessibility property if the partition into accessibility classes is trivial (i.e. any two points z, z′ are accessible) and to have essential accessibility property if the partition into accessibility classes is ergodic (i.e. a measurable union of equivalence classes must have zero or full measure). Similarly, one defines (essential) accessibility of the time-t map of the flow φt ; Z̃ using its global stable and unstable foliations.

Lemma 8. For every t the time-t map of the flow φ_{X}^{t} has essentially accessibility property. Moreover, for any set ℰ of zero measure and almost any two points z, z′ ∉ ℰ one can find a path [z, z′] = [z_{0}, z_{1}, …, z_{ℓ}] such that each z_{i} ∉ ℰ.

Proof. It suffices to establish essential accessibility property of the time-t map of the flow φt ; Z̃. Note that the map g^{∗} has essential accessibility property, indeed, any two points outside of the discontinuity set Q are accessible. Note also that the automorphism A in Brin’s construction is accessible. Denote by Q^{∗} = {(x, y, t) ∈ K^{∗} : x ∈ Q, y ∈ 𝒯^{n−3}, t ∈ [0, 1]}. Fix t and consider the time-t map of the flow φt ; Z̃. Relations (3.7) and (3.8) imply that for any two points (x, y) and (x′, y′) ∈ 𝒟^{2} × 𝒯^{n−3} and any t ∈ [0, 1] the point (x, y, t) is accessible to a point (x′, y′, t′). In particular, any point (x, y, t) ∈ K^{∗} \ Q^{∗} is accessible to a point in Π_{p,q} = {(p, q, t′) ∈ K^{∗} : t′ ∈ [0, 1]} for some p ∈ 𝒟^{2} and q ∈ 𝒯^{n−3}. It remains to show that any two points in Π_{p,q} are accessible.

Let q ∈ 𝒯^{n−3} be a periodic point of the automorphism A in Brin’s construction and p, p′ be two periodic points of g^{∗} such that the orbit of p and the orbit of p′ except for p′ itself (under g^{∗}) lie outside the neighborhood U. Consider local stable and unstable leaves (curves) at p and p′, V_{g^{∗}}^{s}(p), V_{g^{∗}}^{u}(p), V_{g^{∗}}^{s}(p′), and V_{g^{∗}}^{u}(p′). One can choose points p and p′ such that the local unstable leaf from p to V_{g^{∗}}^{u}(p)∩V_{g^{∗}}^{s}(p′), and the local stable and unstable leaves from p to V_{g^{∗}}^{s}(p) ∩ V_{g^{∗}}^{u}(p′) also lie outside the neighborhood U.

For a point (p, q, t) ∈ Π_{p,q}, let γ_{p}^{u} ⊂ W u ; Z̃(p, q, t) and γ_{p}^{s} ⊂ W s ; Z̃(p, q, t) be the curves such that π_{x}(γ_{p}^{u}) ⊂ V_{g^{∗}}^{u}(p), π_{y}(γ_{p}^{u}) = q and π_{x}(γ_{p}^{s}) ⊂ V_{g^{∗}}^{s}(p), π_{y}(γ_{p}^{s}) = q. Since γ^{u} and γ^{s} lie outside U and the point (p, q) is periodic, the t coordinate of a point remains unchanged when this point moves along γ_{p}^{u} from (p, q, t) toward γ_{p}^{u} ∩ π_{x}^{−1}(V_{g^{∗}}^{u}(p) ∩ V_{g^{∗}}^{s}(p′)) ∩ π_{y}^{−1}(q). So does the t coordinate of a point moving along γ_{p}^{s} from (p, q, t) toward γ_{p}^{s} ∩ π_{x}^{−1}(V_{g^{∗}}^{u}(p′) ∩ V_{g^{∗}}^{s}(p)) ∩ π_{y}^{−1}(q). On the other hand, for a point (p′, q, τ) ∈ Π_{p′,q}, if we choose γ_{p′}^{u} and γ_{p′}^{s} in a similar way, then the third coordinate τ increases when a point moves along γ_{p′}^{u} from (p, q, τ) toward γ_{p}^{u} ∩ π_{x}^{−1}(V_{g^{∗}}^{u}(p′) ∩ V_{g^{∗}}^{s}(p)) ∩ π_{y}^{−1}(q) and the third coordinate τ decreases when a point moves along γ_{p′}^{s} from (p, q, τ) toward γ_{p}^{u} ∩ π_{x}^{−1}(V_{g^{∗}}^{u}(p) ∩ V_{g^{∗}}^{s}(p′)) ∩ π_{y}^{−1}(q) (we assume that α(p′) < 1). We can now use the argument in [DHP] (see the proof of Lemma B.4) to obtain that the point (p, q, t) is accessible to some point (p, q, t′) ∈ Π_{p} with t′ < t, and then any two points in Π_{p} are accessible. This show that for every t the time-t map of the flow φ_{X}^{t} has essentially accessibility property.

Let us note that for any zero measure set in the disk and almost any two points outside this set there exists a path for Katok’s map which consists of points lying outside this set. Similar statement holds for the automorphism in Brin’s construction.

To complete the proof consider a set ℰ of zero measure. For almost any two points z, z′ ∉ ℰ and almost every points w and w′ close to (p, q) one can find paths connecting the point z to (w, s) and z′ to (w′, s′) for some s and s′ such that every point in these paths lies outside ℰ. Note that the quadrilateral path from (p, q, t) to (p, q, t′) in the above argument can be replaced by a nearby paths from (w, s) to (w′, s′) such that both (w, s) and (w′, s′) and all other points in the path do not belong to ℰ. The desired result follows. □

Lemma 9. The flow φ_{X}^{t} on N is Bernoulli.

Proof. By results in [P3], a flow with non-zero Lyapunov exponents is Bernoulli if it is a K-flow, i.e., the Pinsker algebra 𝒫, (the largest subalgebra for which ϕ^{t}|𝒫 has zero entropy), is trivial.

Let ℬ be the σ-algebra of Borel subsets in N and 𝒜 ⊂ ℬ a subalgebra. Denote by Sat_{0}(𝒜) the saturation of 𝒜 by sets of measure zero, that is,

Sat_{0}(𝒜) = {B ∈ ℬ : there exists A ∈ 𝒜 such that μ(A △ B) = 0}

(where μ is volume). Let 𝒮^{s} (respectively, 𝒮^{u}) be the subalgebra of Borel sets that consist of whole stable (respectively, unstable) leaves, that is, for E ∈ 𝒮^{s} (respectively, E ∈ 𝒮^{u}) and x ∈ E we have W_{X}^{s}(x) ⊂ E (respectively, W_{X}^{u}(x) ⊂ E). By [P2] (see Theorem 2), the Pinsker algebra 𝒫 is contained in Sat_{0}(𝒮^{s}). Similarly, 𝒫 is contained in Sat_{0}(𝒮^{u}). Therefore, 𝒫 is contained in Sat_{0}(𝒮^{u})∩Sat_{0}(𝒮^{u}) and we wish to show that this intersection is the trivial algebra.

Let 𝒜 ⊂ Sat_{0}(𝒮^{u}) ∩ Sat_{0}(𝒮^{u}) with μ(𝒜) > 0. We shall show that μ(𝒜) = 1. Recall that a point x ∈ N is a density point of 𝒜 if

lim_{r→0} \frac{μ(B(x, r) ∩ 𝒜)}{μ(B(x, r))} = 1.

Denote by D(𝒜) the set of density points of 𝒜. By the Lebesgue-Vitali theorem D(𝒜) = 𝒜 (mod 0) and hence, it suffices to show that μ(D(𝒜)) = 1. Let ℰ_{1} the complement of D(𝒜) ∪ D(N\𝒜). Applying again the Lebesgue-Vitali theorem we obtain that μ(ℰ_{1}) = 0.

Recall that the diffeomorphism g^{∗}: 𝒟^{2} → 𝒟^{2} has non-zero Lyapunov exponents and results of smooth ergodic theory applies (see [BP] for relevant notions and details). Let ℛ^{ℓ} ⊂ 𝒟^{2} be the regular set (of level ℓ) for g^{∗}. The set ⋃_{ℓ} ℛ^{ℓ} has full measure in 𝒟^{2} and so does the set \overline{ℛ} = ⋃_{ℓ} D(ℛ^{ℓ}).

For x ∈ 𝒟^{2} and outside the discontinuity set let γ_{g^{∗}}^{s}(x, r) and γ_{g^{∗}}^{u}(x, r) be arcs in V_{g^{∗}}^{s}(x) and V_{g^{∗}}^{u}(x) centered at x of length r.

Choose two points x and x′ outside the discontinuity set which are sufficiently close to each other. Consider the holonomy map θ : V_{g^{∗}}^{s}(x) → W_{g^{∗}}^{s}(x′) generated by the family of local leaves V_{g^{∗}}^{u}(y), y ∈ V_{g^{∗}}^{s}(x). Since the foliation W_{g^{∗}}^{s} is absolutely continuous the map θ moves the conditional measure (length) μ_{x}^{s} on V_{g^{∗}}^{s}(x) to a measure on V_{g^{∗}}^{s}(x′) which is absolutely continuous with respect to μ_{x′}^{s}. The Jacobian Jac(θ)(y) is not bounded. It is however bounded if we allow y to run over the set V_{g^{∗}}^{s}(x) ∩ ℛ^{ℓ}. More precisely, for every x ∈ ℛ^{ℓ} there is J = J(ℓ) such that

(3.9)  μ_{x′}^{s}(θ(V_{g^{∗}}^{s}(x) ∩ ℛ^{ℓ})) ≤ Jμ_{x}^{s}(V_{g^{∗}}^{s}(x) ∩ ℛ^{ℓ}).

Since x′ ∈ D(ℛ^{ℓ}) without loss of generality we may assume that for sufficiently small r,

(3.10)  μ_{x′}^{s}(γ^{s}(x′, 3Jr) ∩ ℛ^{ℓ}) ≥ 0.9μ_{x′}^{s}(γ^{s}(x′, 3Jr)).

We claim that

(3.11)  θ(γ^{s}(x, r)) ⊂ γ^{s}(x′, 3Jr)

for all sufficiently small r. Indeed, if (3.11) does not hold then μ_{x′}^{s}(γ′) ≥ 3Jr where γ′ = θ(γ) is the longer component of θ(γ^{s}(x, r)) \ {0} and γ = θ^{−1}(γ′) ⊂ γ^{s}(x, r). By (3.10), we have

μ_{x′}^{s}(γ^{s}(x′, 3Jr) ∩ γ′ ∩ (N \ ℛ^{ℓ})) ≤ 0.2μ_{x′}^{s}(γ^{s}(x′, 3Jr) ∩ γ′).

It follows that

μ_{x′}^{s}(θ(γ^{s}(x, r)) ∩ ℛ^{ℓ}) ≥ μ_{x′}^{s}(γ^{s}(x′, 3Jr) ∩ γ′ ∩ ℛ^{ℓ}) ≥ 0.8μ_{x′}^{s}(γ^{s}(x′, 3Jr) ∩ γ′)
≥0.8 · 0.5 μ_{x′}^{s}(γ^{s}(x′, 3Jr)) ≥ 0.4 · 3Jμ_{x′}^{s}(γ^{s}(x′, 3Jr)) > Jμ_{x}^{s}(γ^{s}(x, r)),

contradicting to (3.9).

Let ℰ_{2} be the set of (x, y, t) ∈ N, where x ∈ 𝒟^{2} with x ∉ D(ℛ^{ℓ}) for any ℓ > 0, y ∈ 𝒯^{n−3} and t ∈ ℝ. Clearly, μ(ℰ_{2}) = 0. Let ℰ = ℰ_{1} ∪ ℰ_{2}. We have μ(ℰ) = 0.

Choose a point z ∈ D(𝒜) \ ℰ and a point z′ ∈ V_{X}^{u}(z) \ ℰ (recall that V_{X}^{u}(z) is a local unstable manifold at z for the flow ϕ_{X}^{t}). Consider the holonomy map θ : V_{X}^{cs}(z) → W_{X}^{cs}(z′) generated by the family of local unstable manifolds of ϕ_{X}^{t} (here V_{X}^{cs}(z) is a local centerstable manifold at z and W_{X}^{cs}(z′) is the global center-stable manifold at z′ for the flow ϕ_{X}^{t}). Consider a small ball B^{cs}(z, r) ⊂ V_{X}^{cs}(z) centered at z. By (3.11), the size of θ(B^{cs}(z, r)) in the stable direction E_{g^{∗}}^{s}(z) of the disc is bounded by C_{1}r for some C_{1} > 3J independent of r. Since the stable foliation W_{A}^{s} of the torus is smooth, the size of θ(B^{cs}(z, r)) in the stable direction E_{A}^{s}(z) is bounded by C_{2}r for some C_{2} > 0. Also, the size of θ(B^{cs}(z, r)) in the central direction is bounded by C_{3}r for some C_{3} > 0. Hence, the point z′ cannot belong to D(N \ 𝒜). Note that z ∉ ℰ implies z′ ∈ D(𝒜) ∪ D(N \ 𝒜) and hence, z′ ∈ D(𝒜).

We shall now show that μ(𝒜) = 1. By Lemma 8, for almost every point z′ ∈ N \ ℰ one can find a point z ∈ D(A) \ ℰ such that z and z′ are accessible through a path z_{0}, …, z_{ℓ} such that z_{i} ∉ ℰ. Repeating the above arguments we obtain that z_{1}, …, z_{ℓ} = z′ ∈ D(𝒜). Hence, μ(D(𝒜)) = 1. The desired result follows. □

By identifying some boundary points, it is easy to see that the manifold N can be mapped onto the n-dimensional disc 𝒟^{n} via a map ϕ : N → 𝒟^{n} such that ϕ(N) = 𝒟^{n} and ϕ|int(N) is a diffeomorphism. Since X|∂N = 0, dϕ(X) is smooth on 𝒟^{n}. There is also a mapping ψ : 𝒟^{n} → M (see [K]), and the vector field dψdϕ(X) generates the flow with the desired properties.

4. Proof of the theorem: the case dimM = 3 and 4

In the case dimM = 3, the proof is essentially the same. Consider the suspension flow over g with roof function 1. The suspension manifold K = 𝒟^{2} × [0, 1]/ ∼ (where ∼ is the identification (x, 1) = (g(x), 0)) is diffeomorphic to N = 𝒟^{2} × 𝒮^{1} = {(x, t) : x ∈ M, t ∈ [0, 1]}/ ∼ (where ∼ is the identification (x, 0) = (x, 1)). Let Z be the vector field of the suspension flow. For each (x, t) ∈ K we have Z = (0, 1).

Let F : K → N be given by F (x, t) = (G(x, t), t) (see Proposition 4). We have

dF (x, t) = ( \begin{matrix}G_{x}(x, t) & G_{t}(x, t) \\ 0 & 1 \end{matrix} ) .

Consider the vector field Y = dF Z. Note that

Y (G(x, t), t) = (\frac{∂G}{∂t}(x, t), 1).

Define the vector field X on N by

X(G(x, t), t) = (\frac{∂G}{∂t}(x, t), α(G(x, t))) ,

where α(x) is a C^{∞} function on 𝒟^{2} satisfying Conditions (A1) – (A3). The vector field X is divergence-free and the flow ϕ_{X}^{t} has all the desired properties.

In the case dimM = 4 we start with a Bernoulli map with non-zero Lyapunov exponents on a 3-manifold constructed in [DP]. More precisely, define

S = g × id : 𝒟^{2} × 𝒮^{1} → 𝒟^{2} × 𝒮^{1}.

Let T (x, y) = (g(x), T_{γ(x)}y) : 𝒟^{2} × 𝒮^{1} → 𝒟^{2} × 𝒮^{1}, where T_{γ(x)} is rotation by γ(x). Here γ: 𝒟^{2} → ℝ is a nonnegative C^{∞} function which is equal to zero in a small neighborhood of the discontinuity set \overline{Q} = \{q_{1}, q_{2}, q_{3}, ∂𝒟^{2}\} × 𝒮^{1} and is positive elsewhere.

It is shown in [DP] that the function γ can be chosen so that the map T is robustly accessible, i.e., any C^{1} perturbation R of T is accessible provided R coincides with T in a small neighborhood of the discontinuity set \overline{Q}. Moreover, there is a perturbation R of T which has non-zero Lyapunov exponents and is of the form R = ϕ ∘ T. Here the map ϕ differs from id in a small neighborhood of a point z_{0} ∈ 𝒟^{2} × 𝒮^{1} which lies outside a small neighborhood \overline{U} of the set \overline{Q}. To describe the perturbation ϕ consider the coordinate system ξ = {ξ_{1}, ξ_{2}, ξ_{3}} in an open disc B(z_{0}, ϵ) of asufficiently small radius ϵ centred at z_{0} such that

(1) dm = dξ;

(2) E_{T}^{c} (z_{0}) = ∂/∂ξ_{1}, E_{T}^{s} (z_{0}) = ∂/∂ξ_{2}, E_{T}^{u}(z_{0}) = ∂/∂ξ_{3}.

Let ψ(t) be a C^{∞} function with support in (−ϵ, ϵ). Set τ = ∥ξ∥^{2}/γ^{2} and

ϕ^{−1}(ξ) = (ξ_{1} cos(ϵψ(τ)) + ξ_{2} sin(ϵψ(τ)), −ξ_{1} sin(ϵψ(τ)) + ξ_{2} cos(ϵψ(τ)), ξ_{3}).

Choose a function ψ̂(x, t) : ℝ × [0, 1] → ℝ with the following properties (1) ψ̂(x, 0) = 0 and ψ̂(x, 1) = ψ(x) for any x ∈ ℝ;

(2) ψ̂ is C^{∞} in x and t and ψ̂′_{t}(x, t) = 0 for t = 0, 1 and x ∈ ℝ;

(3) ∥ψ̂(x, t)∥_{r} < δ for all t ∈ [0, 1].

Define the map Ψ : 𝒟^{2} × 𝒮^{1} × [0, 1] → 𝒟^{2} × 𝒮^{1} by the formula

Ψ(x, y, t) = ϕ^{t}(x, y),

where

ϕ^{t}(ξ) = (ξ_{1} cos(ϵψ̂(τ, t)) + ξ_{2} sin(ϵψ̂(τ, t)), −ξ_{1} sin(ϵψ̂(τ, t)) + ξ_{2} cos(ϵψ̂(τ, t)), ξ_{3}).

Note that Ψ has the properties similar to those of the map G in Proposition 4. Namely, (1) Ψ is C^{∞} in (x, y, t);

(2) Ψ(·, 0) = id and Ψ(·, 1) = ϕ;

(3) for any t ∈ [0, 1] the map Ψ(·, t) : 𝒟^{2} × 𝒮^{1} → 𝒟^{2} × 𝒮^{1} is area-preserving;
(4) Ψ_{t}(x, y, 1) = Ψ_{t}(ϕ(x, y), 0);
(5) Ψ(x, y, t) = id for x ∈ \overline{U}.

Set

H = {(x, y, t) : x ∈ 𝒟^{2}, y ∈ 𝒮^{1}, t ∈ [0, 1]}/ ∼_{1}

with the identification ∼_{1}: (x, y, 1) = (T (x, y), 0) = (g(x), T_{γ(x)}(y), 0) and

K = {(x, y, t) : x ∈ 𝒟^{2}, y ∈ 𝒮^{1}, t ∈ [0, 1]}/ ∼_{2}

with the identification ∼_{2}: (x, y, 1) = (ϕ ∘ T (x, y), 0) = (ϕ(g(x), T_{γ(x)}(y)), 0). Consider the diffeomorphism G̃ : H → K given by

G̃(x, y, t) = (Ψ^{−1}(x, y, t), t),

where Ψ^{−1}(x, y, t) for each t is the inverse of Ψ. Note that T itself is diffeotopic to S. Let Ḡ be the diffeomorphism K → K′ where K′ is the suspension manifold of the suspension flow over S. The manifold K′ is diffeomorphic to N = 𝒟^{2} × S^{1} × S^{1}. If Ĝ : K′ → N then F = G̃ ∘ Ḡ ∘ Ĝ : H → N is a diffeomorphism.

Let Z be the vector field on H of the suspension flow over R . Obviously, Z = (0, 0, 1) is divergence free. Set Y = dF Z. Since F (x, y, t) = (G(x, t), y, t) for (x, y, t) in a neighborhood Ũ of the boundary of H then for (x, y, t) ∈ Ũ,

Y (F (x, y, t)) = Y (G(x, t), y, t) = (\frac{∂G}{∂t}(x, t), 0, 1) .

Let X be the vector field on N defined by the formula

X = (\frac{∂G}{∂t}(x, t), 0, α(x)) ,

where α(x) satisfies Conditions (A1) – (A3). Clearly, X is divergence free and the flow generated by X has all the desired properties.

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H. Hu, Department of Mathematics,Pennsylvania State University, University Park, PA 16802, USA

   E-mail address: hu@math.psu.edu

Current address: Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

   E-mail address: hu@math.msu.edu

Ya. Pesin, Department of Mathematics,Pennsylvania State University, University Park, PA 16802, USA

   E-mail address: pesin@math.psu.edu

A. Talitskaya, Department of Mathematics,Pennsylvania State University, University Park, PA 16802, USA

   E-mail address: anjuta@math.psu.edu