Proceedings of Symposia in Pure Mathematics

An Example of a Smooth Hyperbolic Measure with Countably Many Ergodic

D.Dolgopyat, H. Hu, and Ya. Pesin

0.1. Introduction. We construct an example of a diffeomorphism with nonzero Lyapunov exponents with respect to a smooth invariant measure which has countably many ergodic components. More precisely we will prove the following result.

Theorem 0.1. There exists a C^{∞} diffeomorphism f of the three dimensional torus 𝕋^{3} such that

1. f preserves the Riemannian volume μ on 𝕋^{3};

2. μ is a hyperbolic measure;

3. f has countably many ergodic components which are open (mod 0).

0.2. Construction of the Diffeomorphism f. Let A: 𝕋^{2} → 𝕋^{2} be a linear hyperbolic automorphism. Passing if necessary to a power of A we may assume that A has at least two fixed points p and p′. Consider the map F = A × Id of the three dimensional torus 𝕋^{3} = 𝕋^{2} × 𝕊^{1}. We will perturb F to obtain the desired map f.

Consider a countable collection of intervals {I_{n}}_{n=1}^{∞} on the circle 𝕊^{1}, where

I_{2n} = [(n + 2)^{−1}, (n + 1)^{−1}], I_{2n−1} = [1 − (n + 1)^{−1}, 1 − (n + 2)^{−1}].

Clearly, ⋃_{n=1}^{∞} I_{n} = (0, 1) and int I_{n} are pairwise disjoint.

By Proposition 0.2 below, for each n one can construct a C^{∞} volume preserving ergodic diffeomorphism f_{n}: 𝕋^{2}×[0, 1] → 𝕋^{2}×[0, 1] which satisfies:

\begin{matrix}c \\ ◯ \end{matrix}2000 American Mathematical Society

2000 Mathematics Subject Classification. Primary: 37D25, 37C40.

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

D. Dolgopyat was partially supported by the Sloan Foundation. H. Hu was partially supported by the National Science Foundation grant #DMS-9970646. Ya. Pesin was partially supported by the National Science Foundation grant #DMS-9704564 and by the NATO grant CRG970161. D. Dolgopyat and Ya. Pesin wish to thank the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) for hospitality and support during our stay there in June-July 2000.

1. ∥F − f_{n}∥_{C^{n}} ≤ e^{−n^{2}};

2. for all 0 ≤ m < ∞, D^{m}f_{n}|𝕋^{2} × {z} = D^{m}F |𝕋^{2} × {z} for z = 0 or 1;

3. f_{n} has nonzero Lyapunov exponents μ-almost everywhere.

Let L_{n}: I_{n} → [0, 1] be the affine map and π_{n} = (Id, L_{n}): 𝕋^{2} × I_{n} → 𝕋^{2} × [0, 1]. We define the map f by setting f|𝕋^{2} × I_{n} = π_{n}^{−1} ∘ f_{n} ∘ π_{n} for all n and f|𝕋^{2} × {0} = F |𝕋^{2} × {0}. Note that for every n > 0 and 0 ≤ m ≤ n we have

\begin{aligned}∥D^{m}F |𝕋^{2} × I_{n} − π_{n}^{−1} ∘ D^{m}f_{n} ∘ π_{n}∥_{C^{n}} & ≤ ∥π_{n}^{−1} ∘ (D^{m}F − D^{m}f_{n}) ∘ π_{n}∥_{C^{n}} \\ & ≤ e^{−n^{2}} · n^{n} → 0 \end{aligned}

as n → ∞. It follows that f is C^{∞} on M and has the required properties.

0.3. Main Proposition. The goal of this section is to prove the following statement. Set I = [0, 1].

Proposition 0.2. For any k ≥ 2 and δ > 0, there exists a map g of the three dimensional manifold M = 𝕋^{2} × I such that:

1. g is a C^{∞} volume preserving diffeomorphism of M;

2. ∥F − g∥_{C^{k}} ≤ δ;

3. for all 0 ≤ m < ∞, D^{m}g|𝕋^{2} × {z} = D^{m}F |𝕋^{2} × {z} for z = 0 and 1;

4. g is ergodic with respect to the Riemannian volume and has nonzero Lyapunov exponents almost everywhere.

Before giving the formal proof let us outline the main idea. The result will be achieved in two steps. First applying a method of [SW] we construct a perturbation map which has nonzero average central exponent ∫_{M} χ^{c}(x) dμ(x) ≠ 0, where χ^{c}(x) denotes the Lyapunov exponent of x along the neutral subspace E^{c}(x). We then further perturb this diffeomorphism using a method of [NT] to ensure that it has the accessibility property and therefore, is ergodic (see Section B.4 for details).

We believe that this approach works in a more general setting. Namely, we conjecture that the following statement holds.

Proposition 0.3. Consider a one parameter family g_{ε} with g_{0} = F. Then for sufficiently small ε, g_{ε} satisfies the conditions of Proposition 0.2 except for a positive codimension submanifold in the space of one parameter families.

Proof. We proceed with the proof of Main Proposition. Consider the linear hyperbolic map A of the torus 𝕋^{2}. We may assume that its eigenvalues are η and η^{−1}, where η > 1. Let p and p′ be fixed points of A. Choose a number ε_{0} > 0 such that d(p, p′) ≥ 3ε_{0}. Consider the local stable and unstable one-dimensional manifolds for A at points p and p′ of “size” ε_{0} and denote them respectively by V^{s}(p), V^{u}(p), V^{s}(p′), and V^{u}(p′).

Let us choose the smallest positive number n_{1} such that the intersection A^{−n_{1}}(V^{s}(p′)) ∩ V^{u}(p) ∩ B(p, ε_{0}) consists of a single point which we denote by q_{1} (here B(p, ε_{0}) is the ball in 𝕋^{2} of radius ε_{0} centered at p). Similarly, we choose the smallest positive number n_{2} such that the intersection A^{n_{2}}(V^{u}(p′)) ∩ V^{s}(p) ∩ B(p, ε_{0}) consists of a single point which we denote by q_{2}.

Given a sufficiently small number ε ∈ (0, ε_{0}),

ε ≤ \frac{1}{2} min\{d(p, q_{1}), d(p, q_{2})\},

there is ℓ ≥ 2 such that

A^{−ℓ}(q_{1}) ∉ B(p, ε), A^{−ℓ−1}(q_{1}) ∈ B(p, ε).  (0.1)

We now choose ε′ ∈ (0, ε) such that A^{−ℓ−1}(q_{1}) ∈ B(p, ε′).

Finally, we assume ε to be so small that for some q ∈ 𝕋^{2} we have

B(p, ε) ∩ (A^{−n_{1}}(V^{s}(p′)) ∪ A^{n_{2}}(V^{u}(p′))) = ∅,
A^{i}(B(q, ε)) ∩ B(q, ε) = ∅, A^{i}(B(q, ε)) ∩ B(p, ε) = ∅

for i = 1, …, N, where N > 0 will be determined later, and ε = ε(N).

Set Ω_{1} = B(p, ε_{0})×I and Ω_{2} = B^{uc}(q̄, ε_{0})×B^{s}(q̄, ε_{0}), where q̄ = (q, 1/2) and B^{uc}(q̄, ε_{0}) ⊂ V^{u}(q) × I and B^{s}(q̄, ε_{0}) ⊂ V^{s}(q) are balls of radius ε_{0} about q̄.

After this preliminary considerations we describe the construction of the map g.

Consider the coordinate system in Ω_{1} originated at (p, 0) ∈ M with x, y, and z-axes to be unstable, stable, and neutral directions respectively for the map F. If a point w = (x, y, z) ∈ Ω_{1} and F (w) ∈ Ω_{1} then F (w) = (ηx, η^{−1}y, z).

Choose a C^{∞} function ξ: I → ℝ^{+} satisfying:

1. ξ(z) > 0 on (0, 1);
2. ξ^{(i)}(0) = ξ^{(i)}(1) = 0 for i = 0, 1, …,k;
3. ∥ξ∥_{C^{k}} ≤ δ.

We also choose two C^{∞} functions φ = φ(x) and ψ = ψ(y) which are defined on the interval (−ε_{0}, ε_{0}) and satisfy

4. φ(x) = φ_{0} if x ∈ (−ε′, ε′) and ψ(y) = ψ_{0} if y ∈ (−ε′, ε′), where φ_{0} and ψ_{0} are positive constants;

5. φ(x) = 0 if |x| ≥ ε; ψ(y) ≥ 0 for any y and ψ(y) = 0 if |y| ≥ ε;

6. ∥φ∥_{C^{k}} ≤ δ, ∥ψ∥_{C^{k}} ≤ δ;

7. ∫_{0}^{±ε} φ(s) ds = 0. We now define the vector field X on Ω_{1} by

X(x, y, z) = (−ψ(y)ξ′(z) ∫_{0}^{x} φ(s)ds, 0, ψ(y)ξ(z)φ(x)) .

It is easy to check that X is a divergence free vector field supported on (−ε, ε) × (−ε, ε) × I.

We define the map h_{t} on Ω_{1} to be the time t map of the flow generated by X and we set h_{t} = Id on the complement of Ω_{1}. It is easy to see that h_{t} is a C^{∞} volume preserving diffeomorphism of M which preserves the y coordinate (the stable direction for the map F).

Consider now the coordinate system in Ω_{2} originated at (q, 1/2) with x, y, and z-axes to be unstable, stable, and neutral directions respectively. We then switch to the cylindrical coordinate system (r, θ, y), where x = r cos θ, y = y, and z = r sin θ.

Consider a C^{∞} function ρ: (−ε_{0}, ε_{0}) → ℝ^{+} satisfying:

8. ρ(r) > 0 if 0.2ε′ ≤ r ≤ 0.9ε and ρ(r) = 0 if r ≤ 0.1ε′ or r ≥ ε;
9. ∥ρ∥_{C^{k}} ≤ δ.

We define now the map h̃_{τ} on Ω_{2} by

h̃_{τ} (r, θ, y) = (r, θ + τψ(y)ρ(r), y).  (0.2)

and we set h̃_{τ} = Id on M \ Ω_{2}. It is easy to see that for every τ the map h̃_{τ} is a C^{∞} volume preserving diffeomorphism of M.

Let us set g = g_{tτ} = h_{t} ∘F ∘h̃_{τ}. For all sufficiently small t > 0 and τ, the map g_{tτ} is C^{k} close to F and hence, is a partially hyperbolic (in the narrow sense) C^{∞} diffeomorphism of M. It preserves the Riemannian volume in M and is ergodic by Lemma 0.4. It remains to show that g_{tτ} has nonzero Lyapunov exponents almost everywhere.

Denote by E_{tτ}^{s} (w), E_{tτ}^{u} (w), and E_{tτ}^{c} (w) the stable, unstable, and neutral subspaces at a point w ∈ M for the map g_{tτ}. It suffices to show that for almost everywhere point w ∈ M and every vector v ∈ E_{τ}^{c}(w), the Lyapunov exponent χ(w, v) ≠ 0.

Set κ_{tτ} (w) = Dg_{tτ} |E_{tτ}^{u} (w), w ∈ M. By Lemma 0.7, for all sufficiently small τ > 0,

∫_{M} log κ_{0τ} (w) dw < log η.

The subspace E_{tτ}^{u} (w) depends continuously on t and τ (for a fixed w; for details see the paper by Burns, Pugh, Shub, and Wilkinson in this volume) and hence, so does κ_{tτ}. It follows that for all sufficiently small τ > 0, there is t > 0 such that

∫_{M} log κ_{tτ} (w) dw < log η.

Denote by χ_{tτ}^{s} (w), χ_{tτ}^{u} (w), and χ_{tτ}^{c} (w) the Lyapunov exponents of g_{tτ} at the point w ∈ M in the stable, unstable, and neutral directions respectively (since these directions are on-dimensional the Lyapunov exponents do not depend on the vector). By the ergodicity of g_{tτ}, we have that for almost every w ∈ M,

χ_{tτ}^{u} (w) = lim_{n→∞} \frac{1}{n} log ∏_{i=0}^{n−1} κ_{tτ} (g_{tτ}^{i} (w)).

By the Birkhoff ergodic theorem, we get

χ_{tτ}^{u} (w) = ∫_{M} log κ_{tτ} (w) dw < log η.

Since E_{tτ}^{s} (w) = E_{00}^{s}(w) = E_{F}^{s} (w) for every t and τ, we conclude that χ_{tτ}^{s} (w) = − log η for almost every w ∈ M. Since g_{tτ} is volume preserving,

χ_{tτ}^{s} (w) + χ_{tτ}^{u} (w) + χ_{tτ}^{c} (w) = 0

for almost every w ∈ M. It follows that χ_{tτ}^{c} (w) ≠ 0 for almost every w ∈ M and hence, g_{tτ} has nonzero Lyapunov exponents almost everywhere. This completes the proof of the proposition. □

0.4. Ergodicity of the Map g_{tτ}.

Lemma 0.4. For every sufficiently small t > 0 and τ ≥ 0 the map g_{tτ} is ergodic.

Proof. Consider a partially hyperbolic (in the narrow sense) diffeomorphism f of a compact Riemannian manifold M preserving the Riemannian volume. Two points x, y ∈ M are called accessible (with respect to f if they can be joined by a piecewise differentiable piecewise nonsingular path which consists of segments tangent to either E^{u} or E^{s}. The diffeomorphism f satisfies the essential accessibility property if almost any two points in M (with respect to the Riemannian volume) are accessible. We will show that the map g_{tτ} has the essential accessibility property. The ergodicity of the map will then follow from the result by Pugh and Shub (see [PS]; see also the paper by Burns, Pugh, Shub, and Wilkinson in this volume).

Given a point w ∈ M, denote by A(w) the set of points q ∈ M such that w and q are accessible. Set I_{p} = {p} × (0, 1).

Lemma 0.5. For every z ∈ (0, 1),

A(p, z) ⊃ I_{p}.  (0.3)

Proof of Lemma 0.5. We use the coordinate system (x, y, z) in Ω_{1} described above. Since the map h_{t} preserves the center leaf I_{p}, we have that

h_{t}(0, 0, z) = (h_{t}^{1}(0, 0, z), h_{t}^{2}(0, 0, z), h_{t}^{3}(0, 0, z)) = (0, 0, h_{t}^{3}(0, 0, z))

for z ∈ (0, 1). It suffices to show that for every z ∈ (0, 1),

A(p, z) ⊃ {(p, a) : a ∈ [(h_{t}^{−ℓ})^{3}(p, z), z]},  (0.4)

where ℓ is chosen by (0.1). In fact, since accessibility is a transitive relation and h_{t}^{−n}(p, z) → (p, 0) for any z ∈ (0, 1), (0.4) implies that A(p, z) ⊃ {(p, a) : a ∈ (0, z]}. Since this holds true for all z ∈ (0, 1) and accessibility is a reflexive relation, we obtain (0.3).

Now we proceed with the proof of (0.4).

Let q_{1} ∈ V_{tτ}^{u}(p) and q_{2} ∈ V_{tτ}^{s} (p) be two points constructed in Section 0.3. The intersection V_{tτ}^{s} (q_{1}) ∩ V_{tτ}^{u}(q_{2}) is not empty and consists of a single point q_{3}. We will prove that for any z_{0} ∈ (0, 1), there exist z_{i} ∈ (0, 1), i = 1, 2, 3, 4 such that

\begin{aligned}(q_{1}, z_{1}) & ∈ V_{tτ}^{u}((p, z_{0})), (q_{3}, z_{3}) ∈ V_{tτ}^{s} ((q_{1}, z_{1})), \\ (q_{2}, z_{2}) & ∈ V_{tτ}^{u}((q_{3}, z_{3})), (p, z_{4}) ∈ V_{tτ}^{s} ((q_{2}, z_{2})) \end{aligned}

and

z_{4} ≤ (h_{t}^{−ℓ})^{3}(p, z_{0}).  (0.5)

This means that (p, z_{4}) ∈ A(p, z_{0}). By continuity, we conclude that

{(p, a) : a ∈ [z_{4}, z_{0}]} ⊂ A(p, z_{0})

and (0.4) follows.

Since g_{tτ} preserves the xz-plane, we have that V_{tτ}^{uc}((p, z_{0})) = V_{F}^{uc}((p, z_{0})). Hence, there is a unique z_{1} ∈ (0, 1) such that (q_{1}, z_{1}) ∈ V_{tτ}^{u}((p, z_{0})). Notice that

g_{tτ}^{−n}(p, z_{0}) = (p, h_{t}^{−n}((p, z_{0})), g_{tτ}^{−n}(q_{1}, z_{1}) = (A^{−n}q_{1}, z_{1})

for n ≤ ℓ. This is true because the points A^{−n}q_{1}, n = 0, 1, …, ℓ lie outside the ε-neighborhood of I_{p}, where the perturbation map h_{t} = Id. Similarly, since the points A^{−n}q_{1}, n > ℓ lie inside the ε′-neighborhood of I_{p}, and the third component of h_{t} depends only on the z-coordinate, we have

g_{tτ}^{−n}(q_{1}, z_{1}) = (A^{−n}q_{1}, h_{t}^{−n+ℓ}z_{1}).

Since d(g_{tτ}^{−n}((p, z_{0})), g_{tτ}^{−n}((q_{1}, z_{1}))) → 0 as n → ∞, we have

d(h_{t}^{−n}((p, z_{0})), h_{t}^{−n+ℓ}((p, z_{1}))) → 0

as n → ∞. It follows that z_{1} = (h_{t}^{−ℓ})^{3}((p, z_{0})).

By the construction of the map h_{t} (that is h_{t} = Id outside Ω_{1}) the sets A^{−n_{1}}V_{tτ}^{s} (p′) and A^{n_{2}}V_{tτ}^{u}(p′) are pieces of horizontal lines. This means that z_{2} = z_{3} = z_{1}.

Since the third component of h_{t} is nondecreasing from (q_{2}, z_{2}) to (p, z_{4}) along V_{tτ}^{s} (p), we conclude that z_{4} ≤ z_{3} = z_{1} = (h_{t}^{−ℓ})^{3}(p, z_{0}) and thus (0.5) holds. □

The essential accessibility property follows from Lemma 0.5 and the following statement.

Lemma 0.6 (see [NT]). Assume that any two points in I_{p} are accessible. Then the map g_{tτ} satisfies the essential accessibility property.

Proof of Lemma 0.6. It is easy to see that for any two points x, y ∈ M which do not lie on the boundary of M one can find points x′, y′ ∈ I_{p} such that the pairs (x, x′) and (y, y′) are accessible. By Lemma 0.5 the points x′, y′ are accessible. Since accessibility is a transitive relation the result follows. □

□

0.5. Hyperbolicity of the Map g_{0τ}. In this section we show that for all sufficiently small τ, the map g_{0τ} has nonzero average Lyapunov exponent in the central direction. Since this map is ergodic this implies that g_{0τ} has nonzero Lyapunov exponents almost everywhere.

Lemma 0.7. For any sufficiently small τ > 0,

∫_{M} log κ_{0τ} (w)dw < log η.  (0.6)

Proof. Our approach is an elaboration of an argument in [SW].

For any w ∈ M, we introduce the coordinate system in T_{w}M associated with the splitting E_{F}^{u} (w) ⊕ E_{F}^{s} (w) ⊕ E_{F}^{c} (w). Given τ ≥ 0 and w ∈ M, there exists a unique number α_{τ} (w) such that the vector v_{τ} (w) = (1, 0, α_{τ} (w))^{t} lies in E_{0τ}^{u} (w) (where t denotes the transpose). Since the map h̃_{τ} preserves the y coordinate, by the definition of the function α_{τ} (w), one can write the vector Dg_{0τ} (w)v_{τ} (w) in the form

Dg_{0τ} (w)v_{τ} (w) = (κ̄_{τ} (w), 0, κ̄_{τ} (w)α_{τ} (g_{t0}(w)))^{t}  (0.7)

for some κ̄_{τ} (w) > 1. Since the expanding rate of Dg_{0τ} (w) along its unstable direction is κ_{0τ} (w) we obtain that

κ_{0τ} (w) = κ̄_{τ} (w)\begin{matrix}√ & \overline{1 + α_{τ} (g_{0τ} (w))^{2}} \\ √ & \overline{1 + α_{τ} (w)^{2}} \end{matrix} .

Since E_{0τ}^{u} (w) is close to E_{00}^{u}(w) the function α_{τ} (w) is uniformly bounded. Using the fact that the map g_{0τ} preserves the Riemannian volume we find that

L_{τ} = ∫_{M} log κ_{0τ} (w) dw = ∫_{M} log κ̄_{τ} (w) dw.  (0.8)

Consider the map h̃_{τ}. Since it preserves the y-coordinate using (0.2), we can write that

h̃_{τ} (x, y, z) = (r cos σ, y, r sin σ),

where σ = σ(τ, r, θ, y) = θ + τψ(y)ρ(r). Therefore, the differential

Dh̃_{τ}: E_{F}^{u} (w) ⊕ E_{F}^{c} (w) → E_{F}^{u} (g_{0τ} (w)) ⊕ E_{F}^{c} (g_{0τ} (w))

can be written in the matrix form

Dh̃_{τ} (w) = (\begin{matrix}A(τ, w) & B(τ, w) \\ C(τ, w) & D(τ, w) \end{matrix})

= (\begin{matrix}r_{x} cos σ − rσ_{x} sin σ & r_{y} cos σ − rσ_{y} sin σ \\ r_{x} sin σ + rσ_{x} cos σ & r_{y} sin σ + rσ_{y} cos σ \end{matrix}) ,

where

r_{x} = \frac{∂r}{∂x} = \frac{x}{r} = cos θ, r_{z} = \frac{∂r}{∂z} = \frac{y}{r} = sin θ,

σ_{x} = \frac{∂σ}{∂x} = \frac{−z}{r^{2}} + \frac{z}{r} τρ̃_{r}(y, r) = \frac{sin θ}{r} + τρ̃_{r}(y, r) cos θ,

σ_{z} = \frac{∂σ}{∂z} = \frac{x}{r^{2}} + \frac{x}{r}τρ̃_{r}(y, r) = \frac{cos θ}{r} + τρ̃_{r}(y, r) sin θ,

8 D.DOLGOPYAT, H. HU, AND YA. PESIN

and ρ̃(y, r) = ψ(y)ρ(r). It is easy to check that

A = A(τ, w) = 1 − τrρ̃_{r} sin θ cos θ − \frac{τ^{2}ρ^{2}}{2} − τ^{2}rρ̃ρ̃_{r} cos^{2} θ + O(τ^{3}),
B = B(τ, w) = −τρ̃− τrρ̃_{r} sin^{2} θ − τ^{2}rρ̃ρ̃_{r} sin θ cos θ + O(τ^{3}),
C = C(τ, w) = τρ̃+ τrρ̃_{r} cos^{2} θ − τ^{2}rρ̃ρ̃_{r} sin θ cos θ + O(τ^{3}),
D = D(τ, w) = 1 + τrρ̃_{r} sin θ cos θ − \frac{τ^{2}ρ^{2}}{2} − τ^{2}rρ̃ρ̃_{r} sin^{2} θ + O(τ^{3}).

(0.9)

By Lemma 0.8 below, we have

L_{τ} = log η − ∫_{M} log(D(τ, w) − ηB(τ, w)α_{τ} (g_{0τ} (w)))dw.

By Lemma 0.9, we have

\frac{dL_{τ}}{dτ} |_{τ=0} = 0, \frac{d^{2}L_{τ}}{dτ^{2}} |_{τ=0} < 0.

So we can choose τ so small that L_{τ} ≠ log η. □

Lemma 0.8.

L_{τ} = log η − ∫_{M} log(D(τ, w) − ηB(τ, w)α_{τ} (g_{0τ} (w)))dw. Proof of Lemma 0.8. Since g_{0τ} = h_{0} ∘ F ∘ h̃_{τ} = F ∘ h̃_{τ}, we have that

D_{τ} (w) = Dg_{0τ} (w)|E_{0τ}^{u} (w) ⊕ E_{0τ}^{c} (w) = (\begin{matrix}ηA(τ, w) & ηB(τ, w) \\ C(τ, w) & D(τ, w) \end{matrix}) .

By (0.7),

D_{τ} (w) (\begin{matrix}1 \\ α_{τ} (w) \end{matrix}) = (\begin{matrix}ηA(τ, w) + ηB(τ, w) α_{τ} (w) \\ C(τ, w) + D(τ, w) α_{τ} (w) \end{matrix}) = (\begin{matrix}κ_{τ} (w) \\ κ_{τ} (w) α_{τ} (g_{0τ} (w)) \end{matrix}) .

(0.10)

Since h̃_{τ} is volume preserving, AD − BC = 1 and therefore,

A + Bα = \frac{1}{D} + \frac{B}{D} (C + Dα).

Comparing the components in (0.10), we obtain

\begin{matrix}κ_{τ} (w) = & η(A(τ, w) + B(τ, w) α_{τ} (w)) \\ = & η (\frac{1}{D(τ, w)} + \frac{B(τ, w)}{D(τ, w)}(C(τ, w) + D(τ, w) α_{τ} (w))) \\ = & η (\frac{1}{D(τ, w)} + \frac{B(τ, w)}{D(τ, w)}(κ_{τ} (w) α_{τ} (g_{0τ} (w)))) . \end{matrix}

Solving for κ_{τ} (w), we get

κ_{τ} (w) = \frac{η}{D(τ, w) − ηB(τ, w) α_{τ} (g_{0τ} (w))}.

The desired result follows from (0.8). □

Lemma 0.9.

\frac{dL_{τ}}{dτ} |_{τ=0} = 0,  \frac{d^{2}L_{τ}}{dτ^{2}} |_{τ=0} < 0.  (0.11)

Proof of Lemma 0.9. In order to simplify notations we set D′_{τ} = \frac{∂D}{∂τ}, B′_{τ} = \frac{∂B}{∂τ}, C′_{τ} = \frac{∂C}{∂τ}, D′′_{ττ} = \frac{∂^{2}D}{∂τ^{2}} , and B′′_{ττ} = \frac{∂^{2}B}{∂τ^{2}} . Since the function α_{τ} (w) is differentiable over τ (see the paper by Burns, Pugh, Shub, and Wilkinson in this volume) by Lemma 0.8, we find

\frac{dL_{τ}}{dτ} = − ∫_{M} \frac{D′_{τ} − ηB′_{τ} α(g_{0τ} (w)) − ηB \frac{∂α_{τ} (w)}{∂τ}(g_{0τ} (w))}{D(τ, w) − ηB(τ, w)α_{τ} (w)(g_{0τ} (w))} dw

and therefore,

\begin{aligned}\frac{d^{2}L_{τ}}{dτ^{2}} = & ∫_{M} (\frac{D′_{τ} − ηB′_{τ} α(g_{0τ} (w)) − ηB(τ, w)\frac{∂α_{τ} (w)}{∂τ}(g_{0τ} (w))}{D(τ, w) − ηB(τ, w)α_{s}(g_{0τ} (w))})^{2} dw \\ & − ∫_{M} \frac{E(τ, w)}{D(τ, w) − ηB(τ, w) α_{τ} (g_{0τ} (w))}dw, \end{aligned}

where

\begin{matrix}E(τ, w) = & D′′_{ττ} − ηB′′_{ττ} α(g_{0τ} (w)) \\ − ηB(τ, w)\frac{∂^{2}α_{τ} (w)}{∂τ^{2}}(g_{0τ} (w)) − 2ηB′_{τ} \frac{∂α_{τ} (w)}{∂τ}(g_{0τ} (w)). \end{matrix}

Note that for all w ∉ Ω_{2},

A(τ, w) = D(τ, w) = 1, C(τ, w) = B(τ, w) = 0

and for all w ∈ M,

A(0, w) = D(0, w) = 1, C(0, w) = B(0, w) = 0, α_{0}(w) = 0.

It follows that

\frac{dL_{τ}}{dτ} |_{τ=0} = ∫_{Ω_{2}} D′_{τ} dw,  (0.12)

and also that

\frac{d^{2}L_{τ}}{dτ^{2}} |_{τ=0} = ∫_{Ω_{2}} [(D′_{τ})^{2} − D′′_{ττ} + 2ηB′_{τ} \frac{∂α_{τ} (w)}{∂τ}(g_{0τ} (w))]_{τ=0} dw. (0.13)

By (0.9), we obtain that

D′_{τ} (0, w) = rρ̃_{r}(r) sin θ cos θ

and hence,

∫_{Ω_{2}} D′_{τ} dw = 0.

Therefore, (0.12) implies the equality in (0.11).

We now proceed with the inequality in (0.11). Applying Lemma 0.10 below we obtain that

\frac{∂α}{∂τ}(g_{0τ} (w))|_{τ=0} = \frac{C′_{τ} (0, w)}{η} + ∑_{n=1}^{∞} \frac{C′_{τ} (0, g_{00}^{−n}(w))}{η^{n+1}}.

It follows that

\begin{matrix}2ηB′_{τ} (0, w)\frac{∂α}{∂τ}(g_{0τ} (w))|_{τ=0} = & 2B′_{τ} (0, w)C′_{τ} (0, w) \\ + 2B′_{τ} (0, w) ∑_{n=1}^{∞} \frac{C′_{τ} (0, g_{00}^{−n}(w))}{η^{n}}. \end{matrix}

First, we evaluate the term

F(w) = D′_{τ} (0, w)^{2} − D′′_{ττ} (0, w) + 2B′_{τ} (0, w)C′_{τ} (0, w).

Using (0.9), we find that

F(w) =(rρ̃_{r} sin θ cos θ)^{2} + (ρ^{2} + 2rρ̃ρ̃_{r} sin^{2} θ)
− 2(ρ̃+ rρ̃_{r} sin^{2} θ)(ρ̃+ rρ̃_{r} cos^{2} θ)
= − ρ^{2} − (rρ̃_{r} sin θ cos θ)^{2} − 2rρ̃ρ̃_{r} cos^{2} θ.

(0.14)

Recall that Ω_{2} = B^{uc}(q̄, ε_{0}) × B^{s}(q̄, ε_{0}) and ρ̃(r) = 0 if r ≥ ε. We have

∫_{Ω_{2}} 2rρ̃ρ̃_{r} cos^{2} θ dw = ∫_{−ε_{0}}^{ε_{0}} dy ∫_{0}^{2π} 2 cos^{2} θ dθ ∫_{0}^{ε} r^{2}ρ̃ρ̃_{r} dr.  (0.15)

Since 0 = ρ̃(0) = ρ̃(ε) (by the definition of the function ρ), we find that

∫_{0}^{ε} r^{2}ρ̃ρ̃_{r} dr = \frac{1}{2}r^{2}ρ^{2}|_{0}^{ε} − ∫_{0}^{ε} rρ^{2} dr = − ∫_{0}^{ε} rρ^{2} dr.  (0.16)

We also have that

∫_{0}^{2π} 2 cos^{2} θ dθ = ∫_{0}^{2π} dθ.  (0.17)

It follows from (0.15)–(0.17) that

− ∫_{Ω_{2}} 2rρ̃ρ̃_{r} cos^{2} θ dw = ∫_{Ω_{2}} rρ^{2} dw ≤ ε ∫_{Ω_{2}} ρ^{2} dw.  (0.18)

Arguing similarly one can show that

− ∫_{Ω_{2}} rρ̃_{r} sin θ cos θ dw = −\frac{1}{8} ∫_{Ω_{2}}(rρ̃)^{2} dw  (0.19)

Thus we conclude using (0.14), (0.18), and (0.19) that

∫_{Ω_{2}} F(0, w) dw ≤ −(1 − ε) ∫_{Ω_{2}} ρ^{2} dw − \frac{1}{8} ∫_{Ω_{2}}(rρ̃)^{2} dw < 0.  (0.20)

We now evaluate the remaining term

G(0, w) = ∑_{n=1}^{∞} \frac{1}{η^{i}} ∫_{Ω_{2}} 2B′_{τ} (0, w)C′_{τ} (0, g_{00}^{−n}(w)) dw.

Since the map g_{00} = F preserves the Riemannian volume we obtain that

\begin{aligned}∫_{Ω_{2}} 2B′_{τ} (0, w)C′_{τ} (0, g_{00}^{−n}(w)) dw & ≤ ∫_{Ω_{2}} B′_{τ} (0, w)^{2} dw + ∫_{Ω_{2}} C′_{τ} (0, g_{00}^{−n}(w))^{2} dw \\ & = ∫_{Ω_{2}} B′_{τ} (0, w)^{2} dw + ∫_{Ω_{2}} C′_{τ} (0, w)^{2} dw \end{aligned}

Applying (0.9), we find that

∫_{Ω_{2}} B′_{τ} (0, w)^{2} dw + ∫_{Ω_{2}} C′_{τ} (0, w)^{2} dw
= ∫_{Ω_{2}}(ρ̃+ rρ̃_{r} sin^{2} θ)^{2} dw + ∫_{Ω_{2}}(ρ̃+ rρ̃_{r} cos^{2} θ)^{2} dw
≤ 4 (∫_{Ω_{2}} ρ^{2} dw + ∫_{Ω_{2}} r^{2}ρ_{r}^{2} dw) .

It follows that for sufficiently large N > 0 (which does not depend on ε)

∑_{i=N}^{∞} \frac{1}{η^{i}} ∫_{Ω_{2}} 2B′_{τ} (0, w)C′_{τ} (0, g_{00}^{−i}(w)) dw ≤ \frac{1}{10} (∫_{Ω_{2}} ρ^{2} dw + ∫_{Ω_{2}} r^{2}ρ_{r}^{2} dw) .

(0.21)

Note that if g_{00}^{−n}Ω_{2} ∩ Ω_{2} = ∅, then B′_{τ} (0, w)C′_{τ} (0, g_{00}^{−n}(w)) = 0 for all w. Hence,

∫_{Ω_{2}} 2B′_{τ} (0, w)C′_{τ} (0, g_{00}^{−n}(w)) dw = 0.

We may choose the point q and a small ε such that g_{00}^{−n}Ω_{2} ∩ Ω_{2} = F^{−n}Ω_{2} ∩ Ω_{2} = ∅ for all n = 1, 2, …, N. It follows from (0.13), (0.20), and (0.21) that

\begin{aligned}\frac{d^{2}L_{τ}}{dτ^{2}} |_{τ=0} & = ∫_{Ω_{2}} F(0, w) dw + ∫_{Ω_{2}} G(0, w) dw \\ & ≤ − (\frac{9}{10} − ε) ∫_{Ω_{2}} ρ^{2} dw − \frac{1}{40} ∫_{Ω_{2}} r^{2}ρ_{r}^{2} dw < 0. \end{aligned}

The desired result follows. □

Lemma 0.10.

\frac{∂α}{∂τ}(g_{0τ} (w))|_{τ=0} = ∑_{n=0}^{∞} \frac{C′_{τ} (0, g_{00}^{−n}(w))}{η^{n+1}}.

Proof of Lemma 0.10. Define

R(τ, w, α) = \frac{C(τ, w) + D(τ, w)α}{η(A(τ, w) + B(τ, w)α)}.

It follows from (0.7) that

α_{τ} (g_{0τ} (w)) = R(τ, w, α_{τ} (w)).  (0.22)

By (0.7) and (0.9), we have

\frac{∂R}{∂τ}|_{τ=0} = \frac{(C′_{τ} + D′_{τ} α)(A + Bα) + (C + Dα)(A′_{τ} + B′_{τ} α)}{η(A + Bα)^{2}} |_{τ=0} = \frac{C′_{τ} (0, w)}{η}.

Since A(0, w), B(0, w), C(0, w), and D(0, w) are constant functions over w = (x, y, z) we obtain that

\frac{∂H}{∂x} |_{τ=0} = \frac{∂H}{∂z} |_{τ=0} = 0

for H = A, B, C, D. This implies that

\frac{∂R}{∂x}|_{τ=0} = \frac{∂R}{∂z}|_{τ=0} = 0.

Since AD − BC = 1,

\frac{∂R}{∂α}|_{τ=0} = \frac{AD − BC}{η(A + Bα)^{2}} |_{τ=0} = \frac{1}{η}.

It follows from (0.22) that

\frac{∂α}{∂τ}(g_{0τ} (w))|_{τ=0} = \frac{C′_{τ} (0, w)}{η} + \frac{1}{η} · \frac{∂α}{∂τ}(w)|_{τ=0}.

Since this inequality holds for any w, replacing w with g_{0τ}^{−1}(w) we obtain

\frac{∂α}{∂τ}(w)|_{τ=0} = \frac{C′_{τ} (0, g_{0τ}^{−1}(w))}{η} + \frac{1}{η} · \frac{∂α}{∂τ}(g_{0τ}^{−1}(w))|_{τ=0}.

The result follows by induction. □

References

[NT] V. Niţică and A. Török, An open and dense set of stably ergodic diffeomorphisms in a neighborhood of a nonergodic one, Topology, to appear.

[PS] C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. 2 (2000), no. 1, 1–52.

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

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, U.S.A.

E-mail address: dolgop@math.psu.edu
URL: http://www.math.psu.edu/dolgop/

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, U.S.A.

E-mail address: hu@math.psu.edu
URL: http://www.math.psu.edu/hu/

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, U.S.A.

E-mail address: pesin@math.psu.edu
URL: http://www.math.psu.edu/pesin/