Smooth Rigidity of Uniformly Quasiconformal Anosov Flows

Yong Fang

Laboratoire de Mathématique d’Orsay, U.M.R. 8628 du C.N.R.S, Université Paris-sud, France (e-mail: fangyong1@yahoo.fr)

Abstract — We classify the C^{∞} volume-preserving uniformly quasiconformal Anosov flows, such that E^{+} ⊕ E^{−} is C^{∞} and the dimensions of E^{+} and E^{−} are at least two.

1. Introduction Our purpose is to prove the following theorem. For the terminology and

the motivations, see our previous preprint, [Fa1].

Theorem 1. Let ϕ_{t} be a C^{∞} volume-preserving uniformly quasiconformal Anosov flow on a closed manifold M. If E^{+} ⊕ E^{−} is C^{∞} and the dimensions of E^{+} and E^{−} are at least 2, then up to a constant change of time scale and finite covers, ϕ_{t} is C^{∞} flow equivalent either to the suspension of a hyperbolic automorphism of a torus, or to a canonical perturbation of the geodesic flow of a hyperbolic manifold.

For each vector field X on M, a canonical perturbation of the flow of X is by definition the flow of \frac{X}{1+α(X)}, where α is a smooth closed 1-form on M such that 1 + α(X) > 0.

2. A continuous invariant connection.

Suppose that ϕ_{t} satisfies the conditions in Theorem 1. Then by Lemma 2.1.2. of [Fa1], E^{+} and E^{−} are both C^{∞}. By Subsection 2.1. of [Fa1], we have the linearizations of ϕ_{t}, {h_{x}^{±}}_{x∈M}. In addition, we have two continuous connections ∇^{+} and ∇^{−} along F^{+} and F^{−} which are smooth along the leaves. Denote by ν the ϕ_{t}-invariant volume form. Since the flow is uniformly quasiconformal, then ϕ_{t} has only three Lyapunov exponents with respect to ν, {a^{−}, 0, a^{+}}. We can construct as follows a continuous ϕ_{t}-invariant connection ∇, such that

∇X = 0, ∇E^{±} ⊆ E^{±},
∇_{Y^{±}}Z^{∓} = P^{∓}[Y^{±}, Z^{∓}], ∇_{Y^{±}}Z^{±} = (∇^{±})_{Y^{±}}Z^{±},
∇_{X}Y^{±} = [X, Y^{±}] + a^{±}Y^{±}, ∀ Y^{±}, Z^{±} ⊆ E^{±},

where P^{±} denote the projections of T M onto E^{±} with respect to the Anosov splitting.

3. A time change.

Let ϕ_{t} be as in the previous section. Fix a Lyapunov metric g on M. Then ∃ b > 0, such that

∥ Dϕ_{∓t}(u^{±}) ∥≤ e^{−bt} ∥ u^{±} ∥, ∀ t > 0, ∀ u^{±} ∈ E^{±}.  (∗)

Up to finite covers, we suppose that E^{+} and E^{−} are both orientable. Denote by n and m the dimensions of E^{+} and E^{−} and by ν^{±} the volume forms of g |_{E^{±}} on E^{±}. For ∀ x ∈ M and ∀ t ∈ ℝ, define

det(Dϕ_{t}) |_{E_{x}^{±}}:= \frac{(ϕ_{t}^{∗}ν^{±})_{x}}{ν_{x}^{±}} .

Then by (∗), we get for ∀ t > 0,

det(Dϕ_{t}) |_{E_{x}^{+}}≥ e^{nbt}, det(Dϕ_{t}) |_{E_{x}^{−}}≤ e^{−mbt}.  (∗∗)

For ∀ x ∈ M, we define

ϕ^{±}(x) := \frac{∂}{∂t} |_{t=0} log(det(Dϕ_{t}) |_{E_{x}^{±}}).

Since E^{±} are both C^{∞}, then ϕ^{±} are both smooth. In addition by (∗∗), we get

ϕ^{+} ≥ nb > 0, ϕ^{−} ≤ −mb < 0.

Thus we can define a C^{∞} time change Y := \frac{X}{ϕ^{+}} . Denote its flow by ϕ_{t}^{Y}. Now we want to see that ϕ_{t}^{Y} satisfies the conditions in Theorem 1. of [Fa1].

Certainly, ϕ_{t}^{Y} is also a C^{∞} volume-preserving Anosov flow. By [Sa], it is also uniformly quasiconformal. In addition by [Par], the Bowen-Margulis measure of ϕ_{t}^{Y} is in the Lebesgue measure class. Denote by E_{Y}^{+} and E_{Y}^{−} the unstable and stable distributions of ϕ_{t}^{Y} .

Lemma 3.1. Under the notations above, E_{Y}^{+} and E_{Y}^{−} are both C^{∞}.

Proof. Denote by ω^{±} the dual sections of ν^{±}. Thus ω^{+} and ω^{−} are nowhere-vanishing sections of ∧^{n}E^{+} and ∧^{m}E^{−}. Denote again by ∇ the induced connections on ∧^{n}E^{+} and ∧^{m}E^{−} of ∇. Define two 1-forms β^{+} and β^{−} on M, such that

∇_{X}ω^{±} = β^{±}(X)ω^{±}, β^{±}(Y^{±}) = 0,
∇_{Y^{∓}}ω^{±} = β^{±}(Y^{∓})ω^{±}, ∀ Y^{±} ⊆ E^{±}.

Although ∇ is just a continuous connection on M, but by definition, β^{+} and β^{−} are easily seen to be both C^{∞}.

By the condition (ii) of the linearizations (see Subsection 2.1 of [Fa1]), we get

h_{ϕ_{t}(x)}^{±} = Dϕ_{t} ∘ h_{x}^{±} ∘ ϕ_{−t}, ∀ x ∈ M, ∀ t ∈ ℝ.

Thus the induced connections of ∇ on the leaves of F^{±,0} are all C^{∞}. Now fix arbitrarily a leaf of F^{+,0}, W_{x}^{+,0}. Denote by ∇_{x}^{+,0} the restriction of ∇ on W_{x}^{+,0}. Then ∇_{x}^{+,0} is a C^{∞} linear connection on W_{x}^{+,0}. Denote by τ_{t} the ∇_{x}^{+,0}-parallel transport of E_{x}^{+} along the ϕ_{t}-orbit of x. Then by the definition of ∇, we get

τ_{t} = e^{−a^{+}t} · Dϕ_{t}.

Denote by Δ_{t} the determinant of τ_{t} with respect to ν^{+}. Then

Δ_{t} = e^{−na^{+}t} · det(Dϕ_{t}) |_{E_{x}^{+}} .

Thus by differentiating this relation, we get

−β^{+}(X)_{x} = ϕ^{+}(x) − n · a^{+}.

So

−β^{+}(X) = ϕ^{+} − n · a^{+}.  (∗)

Similarly, we get

−β^{−}(X) = ϕ^{−} + m · a^{−}.

For ∀ Y^{−} ⊆ E^{−}, define Ω^{+}(X, Y^{−}) such that

R(X, Y^{−})ω^{+} = Ω^{+}(X, Y^{−})ω^{+}.

Note that R(X, E^{−})E^{+} is well-defined, although the curvature tensor R of ∇ is not globally well-defined. By a simple calculation, we get

dβ^{+}(X, Y^{−}) = Ω^{+}(X, Y^{−}).

Since ∇ is ϕ_{t}-invariant, then

Ω^{+}(X, Y^{−}) ∘ ϕ_{−t} = Ω^{+}(ϕ_{t}X, ϕ_{t}Y^{−}), ∀ t ∈ ℝ.

View Ω^{+}(X, ·) as a smooth section of (E^{−})^{∗}. Thus by the Anosov property of ϕ_{t}, we get

dβ^{+}(X, E^{−}) = Ω^{+}(X, E^{−}) = 0.

Denote by λ the canonical 1-form of ϕ_{t} and define

α^{+} := na^{+}λ − β^{+}, f := \frac{1}{α^{+}(X)}.

Thus by (∗), α^{+}(X) = ϕ^{+} and fX = \frac{X}{ϕ^{+}} .

By Lemma 1.2. of [LMM], the stable distribution of fX, E_{Y}^{−}, is given as follows :

E_{Y}^{−} = {u^{−} + θ(u^{−}) · X | ∀ u^{−} ∈ E^{−}},

where θ is the unique section of (E^{−})^{∗} which satisfies the following relation,

L_{X}(f^{−1}θ) = f^{−2} · df |_{E^{−}} .  (∗∗)

Since λ is ϕ_{t}-invariant, then i_{X}dλ = 0. Thus we have

dα^{+}(X, Y^{−}) = 0, ∀ Y^{−} ∈ E^{−}.

Now using this fact, we can easily verify that −\frac{α^{+}}{α^{+}(X)} |_{E^{−}} satisfies (∗∗). Thus

E_{Y}^{−} = \{u^{−} − \frac{α^{+}(u^{−})}{α^{+}(X)} · X | ∀ u^{−} ∈ E^{−}\}.

So E_{Y}^{−} is smooth.

Since ϕ_{t} preserves a volume form, then ϕ^{+} and −ϕ^{−} are cohomologous, i.e. there exists a smooth function H, such that

ϕ^{+} + ϕ^{−} = X(H).

Recall that ϕ^{+} > 0 and −ϕ^{−} > 0. So we have

\frac{X}{ϕ^{+}} = \frac{X}{−ϕ^{−} + X(H)} = \frac{\frac{X}{−ϕ^{−}}}{1 + (\frac{X}{−ϕ^{−}})(H)}.

Then the flow of \frac{X}{−ϕ^{−}} is easily seen to be C^{∞} flow equivalent to ϕ_{t}^{Y} . Define

α^{−} := ma^{−}λ + β^{−}.

Denote by E+_{\frac{X}{−ϕ^{−}}} the unstable distribution of \frac{X}{−ϕ^{−}}. Then by similar arguments

as above, we get

E+_{\frac{X}{−ϕ^{−}}} = \{u^{+} − \frac{α^{−}(u^{+})}{α^{−}(X)} · X | ∀ u^{+} ∈ E^{+}\}.

Thus E+_{\frac{X}{−ϕ^{−}}} is smooth. We deduce that E_{Y}^{+} is C^{∞}. □

Then ϕ_{t}^{Y} satisfies the conditions in Theorem 1. of [Fa1]. So up to a constant change of time scale and finite covers, ϕ_{t}^{Y} is C^{∞} flow equivalent either to the suspension of a hyperbolic automorphism of a torus, or to the geodesic flow of a hyperbolic manifold.

Now we finish the proof of Theorem 1. as follows. Lift ϕ_{t} and ϕ_{t}^{Y} to the covering where ϕ_{t}^{Y} becomes an algebraic flow. Recall that ϕ_{t} is just a time change of ϕ_{t}^{Y}. If ϕ_{t}^{Y} is a suspension, then by [Fa2], up to a constant change of time scale, ϕ_{t} is C^{∞} flow equivalent to ϕ_{t}^{Y}. So Theorem 1. is true in this case.

If ϕ_{t}^{Y} is a geodesic flow, then by Lemma B of [Fa], there exists a C^{∞} 1-form α on M, such that

X = \frac{Y}{α(Y)}, α(Y) > 0,

and

L_{Y} dα = 0,

i.e. dα is ϕ_{t}^{Y} -invariant. Denote by λ_{Y} the canonical 1-form of ϕ_{t}^{Y}. Then by [Ham], ∃ a ∈ ℝ, such that

dα = a · dλ_{Y} .

Thus there exists a C^{∞} closed 1-form γ, such that

α = a · λ_{Y} + γ.

If a ≤ 0, then γ(Y) > 0. Thus we can find a closed global section of ϕ_{t}^{Y}, which is impossible for a contact flow. We deduce that a > 0. So up to a constant change of time scale, ϕ_{t} is C^{∞} flow equivalent to a canonical perturbation of the geodesic flow of a hyperbolic manifold. So Theorem 1. is true.

4. Remarks about diffeomorphisms.

We can draw from Theorem 1. the following

Corollary 4.1. Let ϕ be a C^{∞} volume-preserving uniformly quasiconformal Anosov diffeomorphism on a closed manifold N. If E^{+} and E^{−} are both of dimension at least 2, then up to finite covers, ϕ_{t} is C^{∞} conjugate to a hyperbolic automorphism of a torus.

Proof. Take the suspension of ϕ, (M, ϕ_{t}). Then ϕ_{t} satisfies the conditions in Theorem 1. Since N is a global section of ϕ_{t}, then ϕ_{t} can not be the time change of a geodesic flow. Thus by Theorem 1, up to a constant change of time scale, ϕ_{t} is finitely covered by the suspension of a hyperbolic automorphism of a torus. In particular, ϕ_{t} preserves a C^{∞} linear connection ∇. Then ϕ preserves the C^{∞} linear connection ∇ |_{N}. Since in addition, the unstable and stable distributions of ϕ are smooth (see [Sa]), then we deduce the corollary by [BL]. □

In [K-Sa], a similar proposition is proved for the topologically transitive case by assuming that the dimensions of E^{±} are at least three. Their idea, though quite elegant, meets some essential difficulties if we suppose that dimE^{+} = 2 and dimE^{−} ≥ 2.

Finally we pose the following question, of which we prefer a positive answer.

Question. Let ϕ be a C^{∞} volume-preserving uniformly quasiconformal Anosov diffeomorphism. Is it rigid if we suppose that dimE^{+} = 1 and dimE^{−} ≥ 2 ?

Acknowledgements. The author would like to thank P. Foulon and P. Pansu for discussions.

References

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