Posted by: matheuscmss | September 25, 2008

Asaoka’s lecture on Verjovsky conjecture – part I

My third post on the series of discussions of lectures held at ICTP (Trieste, Italy) concerns Asaoka’s theorem recent solution of a conjecture of Verjovsky.

Let me start by recalling some basic definitions and facts.

Definition 1. A flow \phi^t on a compact manifold M is called an Anosov flow whenever there are a D\phi^t-invariant splitting TM= E^s\oplus T\phi\oplus E^u of the tangent bundle TM of M and a constant \lambda>0 such that

  • T\phi is the one-dimensional subbundle of TM of tangents to the orbits of \phi^t,
  • E^s is uniformly contracted: \|D\phi^t|_{E^s(p)}\|\leq e^{-\lambda\cdot t} for any p\in M and t>0,
  • E^u is uniformly expanded: \|D\phi^{-t}|_{E^u(p)}\|\leq e^{-\lambda\cdot t} for any p\in M and t>0.

We say that \phi^t is a codimension-one Anosov flow if \phi^t is an Anosov flow with \textrm{dim}(E^u)=1.

For sake of comparision, let me remind you the definition of Anosov diffeomorphisms:

Definition 2. A C^rdiffeomorphism f:M\to M is Anosov if there are a $Df$-invarinat splitting TM=E^s\oplus E^u and some constants C>0, \lambda<1 such that

  • E^s is uniformly contracted: \|Df^n|_{E^s(p)}\|\leq e^{-\lambda\cdot n} for any p\in M and n\in\mathbb{N},
  • E^u is uniformly expanded: \|Df^{-n}|_{E^s(p)}\|\leq e^{-\lambda\cdot n} for any p\in M and n\in\mathbb{N}.

Remark 1. As one can expect, there are some “dynamical” differences between these two definitions: Anosov diffeomorphism are discrete-time systems while Anosov flows are continuous-time systems. At a first glance this seems to be of little relevance, but there are fundamental distinct dynamical behaviors hidden here. In fact, given an Anosov flow \phi^t, one can infer some of its properties by the so-called time-one diffeomorphism \phi^1 associated to this flow. It turns out that \phi^1 is not an Anosov diffeomorphism: indeed, although we can decompose the tangent bundle TM=E^s\oplus T\phi\oplus E^u into three D\phi^1-invariant subbundles, since \phi^1 is an isometry along the direction T\phi (exercise), \phi^1 can’t be Anosov due to the presence of this ”neutral” (or central) direction. We will come back to this point later in the discussion.

Now, let us see some basic examples of Anosov systems:

Example 1 (Hyperbolic linear tori automorphisms). Take A a n\times n matrix with integer coefficients and determinant 1, i.e., A\in SL(n,\mathbb{Z}). It is not hard to see that A induces a diffeomorphism f_A on the torus \mathbb{T}^n:=\mathbb{R}^n/\mathbb{Z}^n. Moreover, one can show that f_A is an Anosov diffeomorphism whenever the spectrum \sigma(A) of A doesn’t intersect the unit circle S^1 (i.e., there are no eigenvalues with modulus 1): indeed, E^s is the sum of the eigenspaces associated to eigenvalues of modulus <1 and E^u is the sum of the eigenspaces associated to eigevalues of modulus >1. In this situation, f_A is called a hyperbolic toral automorphism.

Example 2 (Suspension of an Anosov diffeomorphism). Given a map T:M\to M and a “roof” function r:M\to\mathbb{R}^+, one can define a suspension flow associated to (T,r) in the following way: consider the space

\widetilde{M}=\{(x,s)\in M\times\mathbb{R}: 0\leq s<r(x)\}

and the flow

T^t(x,s)=(T^n(x), s+t-r^{(n)}(x))

where n\in\mathbb{N} is the unique integer such that

r^{(n)}(x):=\sum\limits_{j=1}^n r(T^j(x))\leq s+t<\sum\limits_{j=1}^{n+1} r(T^j(x)):=r^{(n+1)}(x).

An interesting feature of this construction is the following fact: the suspension flow of an Anosov diffeomorphism is an Anosov flow (exercise). In particular, we see that the suspension flow of an Anosov diffeomorphism shares the good dynamical properties of original diffeomorphism.

From the example 2, we see that the study of Anosov flows would be significantly more easy provided that a general Anosov flow possesses the dynamical structure of a suspension of an Anosov diffeomorphism. More precisely, our life would be easier if every Anosov flow is topologically conjugated to a suspension of an Anosov diffeomorphism since the dynamics of Anosov diffeomorphisms are well-studied, e.g.:

Theorem (Franks and Newhouse). A codimension-one Anosov diffeomorphism is always topologically conjugated to a hyperbolic toral automorphism.

However, our life is not always easy: Franks and Williams constructed an Anosov flow in a 3-manifold whose non-wandering set is not the whole manifold. In particular, this flow is not conjugated to a suspension flow. Indeed, since our manifold is 3-dimensional, the Anosov flow associated to the suspension is forced to be of codimension one. By the result of Newhouse, the diffeomorphism and, a fortiori, the suspension flow is transitive and, therefore, it can’t be topologically conjugated to a non-transitive system.

Roughly speaking, the idea of Franks and Williams is the following: Anosov flows possesses a central direction where the dynamics is isometric; taking advantage of this neutral direction, they construct a Reeb component preserved by the flow; since the 2-torus of the Reeb foliation separates the 3-torus into two distinct connected components, the flow isn’t transitive.

On the other hand, Verjovsky showed that this phenomenon is purely 3-dimensional: any codimension-one Anosov flow on a n-manifold with n\geq 4 is transitive. Motivated by this result, he made the following conjecture:

Verjovsky conjecture. Any codimension-one Anosov flow on a compact manifold M^n of dimension n\geq 4 is topologically equivalent to a suspension flow over an Anosov diffeomorphism of the torus.

As we already mentioned, Asaoka solved this conjecture based on a preprint of Simic (which is still under revision). More precisely, Asaoka’s strategy is the following:

  • by the theorem of Verjovsky, we know that any codimension-one Anosov flow is transitive (when n\geq 4);
  • by the results of Simic, a codimension-one volume-preserving Anosov is topologically equivalent to the suspension of a hyperbolic toral automorphism;

Thus, the proof of the Verjovsky conjecture is complete once we show the following theorem:

Theorem (Asaoka). Any transitive codimension-one Anosov flow is topologically conjugated to a smooth codimension-one volume-preserving Anosov flow.

Of course, the details of the proof of this result deserves another entire post, so that this is probably the best place to close the current discussion! Ja ne!

Edited 14/Feb/2011: As pointed out by M. N. below, it was found a gap in Simic’s preprint (which wasn’t filled up to today [to the best of my knowledge]), so that one can’t conclude Verjovsky conjecture from Asaoka’s theorem.


  1. Hi Matheus,

    You may notice the erratum of Asaoka’s paper.
    DOI: 10.1007/s00222-009-0211-9

    M. N.

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