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.

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.

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.

Remarks:

-- 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 .

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.

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

(x,y,t)->(x,y,t+s)

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).

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.

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