Comments on the article "Cocycles and stable foliations of Axiom A flows"

Computation of the essential values for the strong stable relation.

Let T be the subgroup of \tilde{G} obtained by taking the closure of the subgroup generated by the weights of the periodic orbits : T=<[t]> .

In the article, it is shown that the essential values of the cocycle
c(x,y) = f^m(x)-f^n(y) over the relation
x~y <-> there exist m,n such that s^m(x)=s^n(y) ,
contains T. In fact, Theorem 2 shows that we have

E(c)= T.

Indeed, if this is not the case, one can consider the system obtained by quotienting \tilde{G} by T. The essential values of the quotient cocycle contain E(c)/T. All the weights are now zero, so that the quotient function obtained from f : Sigma -> \tilde{G} / T is cohomologous to zero via a Holder continuous function (Livsic theorem ; since Sigma is totally discontinuous, the coboundary may even be continuously lifted from Sigma to \tilde{G} ). So the system is topologically and measurably isomorphic to the system associated to the zero function. This system has no other essential values than 0. This shows that E(c)=T.

Note that this also shows that the function f: Sigma -> \tilde{G} is Holder cohomologous to a function which takes values in the group of essential values of the cocycle. Other deformation results are studied by V. Kaimanovich and K. Schmidt, in the preprint "Ergodicity of cocycles : general theory".

Now what are the essential values of the strong equivalence relation, given by the cocycle studied in Theorem 3 :
c'(x,y)= f^m(x)-f^m(y) , defined on the relation
x~y <-> there exists m such that s^m(x)=s^m(y) ?

As in the proof of Theorem 3, one reduces the problem to the previous one by considering (1,f) instead of f. Let proj be the projection from Zx\tilde{G} to \tilde{G}, and (l(t),[t]) the element of Zx\tilde{G} given by the period p and the weight f^p(t) of the periodic point t on Sigma.

E(c')= proj( {0}x\tilde{G} inter <(l(t),[t])> ).

From the final expression, there seems to be no other way to compute these essential values, than to come back to the weak stable relation.

The non arithmeticity condition.

There are other conditions which are equivalent to the transitivity of the relation given in Theorem 2, namely, the transitivity of the weak (resp. strong) stable foliation of the transformation
T(x,g)=(sigma(x),f(x)g) on Sigma x \tilde{G}.

One of them is the non-arithmeticity condition : There does not exist function g:Sigma -> S^1, and character Xi: \tilde{G} -> S^1, (resp. Xi: Zx\tilde{G} ->S^1) such that Xi o f = (g o sigma) / g. (resp. Xi o (1,f) = (g o sigma) / g )

On the one hand, if there exists such a Xi and g, the subgroup <[t]> generated by weights of periodic orbits is included in the kernel of Xi. On the other hand , if <[t]> is not dense, then one takes a cocycle on \tilde{G} which is zero on <[t]>, and apply the Lifschitz theorem to Xi o f to build g.

The cohomology of symbolic systems and the homologically full assumption.

R. Sharp [Sh93] gives a condition for Anosov flows that implies the transitivity of the strong stable foliation. He shows that the condition :
{[t]}=H_1 (H_1 the first homology group of the manifold) implies the condition :
<(l(t),[t])> dense in RxH_1 .

The condition {[t]}=H_1 is never satisfied for a suspension over a subshift of finite type. Indeed these systems admit a global topological cross section (cf Schwartzman, Asymptotic cycles). One can also compute the cohomology of a suspension, and the asymptotic cycle of an invariant measure, to convince oneself that the condition {[t]}=H_1 cannot be satisfied.

The Cech (or sheaf or SWAK) cohomology of a suspension over a subshift of finite type can be explicitly computed. One can also give a formula for the asymptotic cycle of a suspended measure.

Let S be a subshift of finite type, sigma the shift on S, r: S -> R a positive continuous function on S, and S_r the suspension

S_r=SxR/ (x,t)~(sigma(x),t-r(x))

In order to compute the first cohomology group, one applies the Mayer-Vietoris sequence to the cover U={0 < t < r(x)} and V={t < inf r/2 or t > inf r/2} This gives the exact sequence :

H^0(U)+H^0(V) -> H^0(U inter V) -> H^1(S_r) ->0

with the first arrow given by :

(f,g) -> ( f - g , f - (g o sigma) )

The 0th cohomology group is composed of locally constant functions. Taking for coefficient group Z, and noting that locally constant functions on S with values in Z is the same as Z-valued continuous functions C(S,Z), we see that :

H^1(S_r,Z) = C(S,Z)/ ( (f o sigma) -f )

that is, Z-valued functions depending on a finite number of coordinates, up to coboundaries. This computation is valid for any suspension over a topological paracompact space with H^1=0.

The first cohomology group is isomorphic to the Bruschlinsky group, which consists of functions on S_r with value in R/Z , up to functions admitting a continuous lifting to R. The isomorphism can be given explicitely :

Let psi : S_r -> R/Z an element of the Bruschlinsky group. The function t -> psi(x,t) may be lifted to a function phi_x(t) from SxR to R. Since psi is defined on S_r, the function (x,t) -> phi_x(t) - phi_(sigma(x))(t-r(x)) is a continuous function taking integer values. So, it does not depend on t, and defines a cohomology class, as explained earlier.

Now let mu be an invariant measure on S, and mu_r its suspension on S_r. Using the previous isomorphism, and the formula :

<[mu_r],psi> = 1/2pi*i int psi'/psi dmu_r,

one obtains :

<[mu_r],f> = int_S f dmu / int_S r dmu.

for any locally constant,integer valued function f on S.

As expected, <[mu_r],1> is greater than inf r. Hence there is no invariant measure with zero asymptotic cycle. In particular, zero is not realized as the homology class of a periodic orbit.

Yves Coudene 2/6/2002