Comments on the article "Hyperbolic systems on nilpotent covers"

Conservativity and ergodicity of the stable foliation on covers.

Conservativity of the stable foliation is easier to achieve than ergodicity. For instance, If the covering is trivial, the weak stable foliation is still conservative, whereas it cannot be ergodic. More can be said about conservativity.

We consider the case of a discrete group G. For an arbitrary group G, the proof of the main theorem shows that functions invariant by the strong stable foliation are invariant by elements of RxG of the form (l(t),[t]), t periodic orbit, as soon as these elements belong to the center of RxG. This is the case if G is abelian, and if G is nilpotent, one can still reduce the problem to the abelian case, by making a recurrence on the central series of the group.

However, the elements of the form (l(t),[t]), with [t]=0, belong to the center of RxG. Assuming the G-action is free, and the foliation is locally trivial, it shows that the strong stable foliation is conservative if <(l(t)| [t]=0)> is dense in R, that is, if the lengths of the periodic orbits on the cover, generate a dense subgroup of R. This condition is not necessary to get the conservativity, and does not imply ergodicity even if G is abelian. Yet it is satisfied in some cases.

For the geodesic flow on a negatively curved manifold, it is conjectured to hold as soon as the covering manifold is non elementary, and known to be true, for example, if there is a cusp, or if the non-wandering set of the flow on the cover is the whole manifold. It is not satisfied for the universal cover.

On the other hand, if the group G is not amenable, the stable foliation cannot be ergodic on the cover. In fact, there are no ergodic cocycle taking values in a non-amenable group. This result is due to :

Zimmer, Robert J.
Amenable ergodic group actions and an application to Poisson boundaries of random walks.
J. Functional Analysis 27 (1978), no. 3, 350--372.

This shows that, in general, conservativity does not imply ergodicity of the stable foliation on covers, even under strong connecting assumptions on the fibers.

The homologically full assumption and the transitivity of the strong stable foliation.

R. Sharp showed that the homologically full assumption : {[t]}=H_1 for an Anosov flow implies the condition : <(l(t),[t])> dense in RxH_1 , which ensures the transitivity of the strong stable foliation on the homology cover.

This is again true for hyperbolic flows on compact manifolds. Note however that these flows are usually not homologically full. Suspensions over subshifts of finite type are never homologically full, since they admit a global topological cross-section (see Schwartzman, Asymptotic cycles, Ann. of Math. (2) 66 (1957), 270--284 ; this article has a follow-up : Asymptotic cycles on non-compact spaces. Bull. London Math. Soc. 29 (1997), no. 3, 350--352.).

The homologically full assumption is an open condition, and thus gives an open set of mixing Anosov flows, in the C^k topology, for all k. Any positive continuous reparametrization of an homologically full Anosov flow is again homologically full, hence mixing. On the other hand, a flow with a global section always admits a continuous reparametrization which is not mixing.

The homologically full assumption implies the transitivity of W^ss on any connected abelian cover. Note however there are abelian covers, with deck transformation G, such that the implication {[t]}=G -> <(l(t),[t])> dense in RxG is false. A counterexample is given at the end of section 5 of the paper.

Let us explain why the argument given by R. Sharp does not work on this example. The function F which appear in the argument is defined on the suspension by

F(x,t) = 1 if x begins by 0
F(x,t) = alpha^(-1) if x begins by 1

for the example of section 5. The function H may be taken equal to

H = ( 1 + alpha F )/( 1 + alpha )

The function psi then obtained by the Livschits theorem is given by :

psi(x,t) = exp(2pi*i*t) if x begins by 0
psi(x,t) = 0 if x begins by 1

So the evaluation of the asymptotic cycle of any measure of full support, on the homology class associated to psi, is strictly positive, and one cannot conclude.

This product was zero in the homologically full case, because the asymptotic cycle of the measure was zero. But now, the condition {[t]}=G only insures that the asymptotic cycle of the measure is zero, when evaluated on elements of G. In the example of section 5, the homology class associated to psi does not belong to G.

The length of periodic orbits and the topological weak mixing property.

Theorem :
Let phi_s be a transitive flow on a metric space, which satisfies the closing lemma (no uniformity assumptions are needed on the delta,epsilon). Then the condition on the length of periodic orbits
< {l(t)| t periodic orbit} > is dense in R , is equivalent to the property :
There exists no S^1-valued continuous function and real a such that : F o phi_s = exp(2*pi*i*s*a) F .

Proof :
If s,F exist, then, applying the relation to periodic points, one obtains that l(t) is in a^(-1) Z. If < l(t) > is contained in a^(-1)Z, one builds F as in the Livsic theorem (A. N. Livsic, homology properties of Y-systems, [Math. Notes 10 (1971), 758--763]). The relation allows to define F on a transitive orbit ; one then checks using the closing lemma that F is locally uniformly continuous, hence can be extended to the whole space. Note that if the delta, epsilon in the closing lemma are valid on the whole manifold, and not just locally, then the function F is uniformly continuous.

-- In this theorem, the fixed points of the flow should be considered as periodic points with arbitrary periods. If the flow admits a fixed point, both sides of the equivalence are satisfied. These results are interesting only for flows with no fixed points.
-- The closing lemma is not only satisfied by hyperbolic systems, but also by the geodesic flow on manifolds of higher rank. cf Eberlein, geometry of non-positively curved manifolds.

When the flow is defined on a cover, one proves in a similar way :

Theorem :
Let phi_s be a transitive flow defined on a metric space X, phi'_s some covering map on a G-cover X' of X , and satisfying the closing lemma. Let us assume that G is abelian. Then the condition :
< {(l(t),[t]) | t periodic orbit of phi_s} > is dense in RxG
is equivalent to the property :
There exists no S^1-valued continuous function on X' and character Xi : RxG-> S^1 such that : F o g o phi'_s = Xi(s,g) F .

The length of periodic orbits and the transitivity of the strong stable foliation.

Theorem :
We consider a flow phi_s on a metric space, which admits a local product structure (in the neighborhood of periodic orbits ; epsilon and delta in the definition of the product structure may depend on the orbit). Then, the condition : <{l(t)}> dense in R
implies the transitivity of the strong stable leaf of every positively transitive (with respect to the flow) points.

For a geodesic flow on a non-elementary negatively curved manifold, this was proven by F. Dalbo.

Proof :
One proves that if x is a transitive point, and s is some arbitrary real number, then W^ss(x) comes arbirtrary close to phi_s(x). Let t_i be a finite set of periodic orbits and n_i integers such that the sum of n_i*l(t_i) is close to s. Then the orbit of x comes very close to each of these periodic orbits. Using the same procedure as in the article, one may shrink or expand the orbit of x , in order to obtain a point y whose trajectory turns n_i times more (or less if n_i is negative) around t_i. This point is on the stable leaf of x, close to the trajectory of x, and the time shift is close to s.

Remarks :
-- Transitivity does not always imply positive transitivity. Let us consider a flow on a compact set made of two periodic orbits and a third orbit connecting the two periodic orbits. A point on the third orbit is transitive, but not positively transitive nor recurrent.
-- Let us assume that the metric space has the Baire property. If there exists a transitive point whose orbit has an empty interior (this is the case for example if the periodic points are dense and the space is not reduced to a single orbit), then the flow is positively transitive. Indeed this implies that either the alpha limit and omega limit set of the transitive point is everything ; hence, the point is non-wandering and the non-wandering set is everything. From this, one easily shows that two open sets are positively connected (Walters, Springer GTM 79, th. 5.10).
-- The condition < l(t) > dense in R is not necessary to obtain the transitivity of the strong stable foliation ; an example of a dissipative flow with a transitive foliation and no periodic orbit is constructed in the next section.

Converse :
There is a converse to the theorem (assuming the closing lemma): if the flow admits a point with a non-empty omega limit set, and with dense strong stable leaf, then it is topologically weak mixing, and the previous section shows that < l(t) > = R.

Proof of the weak mixing :
Suppose that there exists F locally uniformly continuous, and a such that F o phi_s = exp(2*pi*i*s*a) F . Let U be a neighborhood of a point z in the omega limit set of x, on which F is uniformly continuous. There exists some sequence s_i going to infinity such that phi_{s_i}(x) belongs to U and converges to z. Let y be a point on the strong stable foliation of x. If i is large, phi_{s_i}(y) is in U and close to phi_{s_i}(x), so |F(phi_{s_i}(y))-F(phi_{s_i}(x))| = |F(y)-F(x)| is close to zero. This shows that F is constant on the strong stable foliation of x, hence constant.

Remarks :
-- The omega limit set of a positively transitive point contains all the points which are not on the positive orbit of the point. This implies that the omega limit set of a positively transitive point is everything (the positive orbit of the point accumulates on a point on its backward orbit, hence on the whole orbit). In particular, a positively transitive point is recurrent.
-- The set of positively transitive points is a G_delta set. Let us assume that the strong stable foliation satisfies the following condition : for all open set U, the union of all leaves intersecting U form an open set ; this is a topological analog of the non-singularity of the foliation in the measure theoretic case, and is easily checked in practice. Under that assumption, the set of points with a dense strong stable leaf form a G_delta set. So, if the flow is positively transitive and admits a point with a dense leaf, then there is a (dense set of) positively transitive points with dense strong stable leaf.

On polycyclic covers.

Here is an example of an Anosov flow on a polycyclic cover for which the weak stable foliation is not ergodic, even if all levels are connected by the action.

Let r(x) be a Z^3 periodic function on R^3. Consider the following transformations on R^3 :
T_1 (x,y,t) = (x+1,y,t)
T_2 (x,y,t) = (x,y+1,t)
T_3 (x,y,t) = (2x+y,x+y,t-r(x))
and the flow on the third factor

The group generated by T_1, T_2, T_3 is polycyclic (hence solvable), but not nilpotent.

The flow on R^3/<{T_1,T_2,T_3}> is an Anosov flow. For generic r, it is mixing. On the cover R^3/<{T_1,T_2}>, the flow is totally dissipative, its non-wandering set is empty ; yet its weak stable foliation is ergodic, and for generic r, its strong stable foliation is ergodic. This follows from the main theorem of the paper.

On R^3, the stable leaves are graphs, so that the weak and strong stable foliations are not transitive. Yet, the Frobenius elements of the first periodic orbits are given by :

[(0,0,0)]^(-1) = T_3
[(1/5,2/5,0)]^(-1) = T_1^(-2) T_2^(-1) T_3^2
[(3/5,1/5,0)]^(-1) = T_1^(-3) T_2^(-2) T_3^2

This means that <[t]>=G.

Note that there are no non-constant bounded harmonic functions on the cover (Lyons & Sullivan).

Equivariant lifts.

Given some covering on a manifold M, and a hyperbolic diffeomorphism on M, there does not always exist an equivariant lifting of the diffeomorphism to the cover.

The main theorem of the paper is usually false without the assumption of commutation, between the lift and the deck transformation group of the cover.

For example, the hyperbolic transformation on the torus (R/Z)^2, given by the matrix :
2 1
1 1
admits no continuous lift to R^2 which commutes with Z^2. It admits a lift to R^2, given by the linear action of the matrix on R^2. The stable foliation of this lift is not transitive nor ergodic.
Another example is given by the action of the shift on (R/Z)^N, which cannot be lifted to R^N in a Z^N equivariant way.

Relaxing the hypothesis on the distance on G

Theorem 2 and 3 ask for the existence of a biinvariant metric d on G. This hypothesis is needed, in order to lift the product structure to the cover. It can be weakened to the following :

Consider the sequence d(f^k(x),f^k(y)). Then we ask that the sequence has a limit as k tends to infinity. This limit depends on x and y ; for fixed y (almost all y) , it should go to zero as x tends to y while staying on W^ss(y).

This is an easy check ; the existence of the limit is perhaps not necessary ; the fact that

limsup_k d(f^k(x),f^k(y))

tends to zero as x tends to y while staying on W^ss(y), is perhaps enough. This should be settled by a careful proofreading. Note that the set of points y satisfying that condition is invariant by the base transform, hence of measure 0 or 1. The condition is satisfied, for example if d is biinvariant and f holder, or if f is locally constant.

Yves Coudene 11/6/2002