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.

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.

{[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