We show that for a large class of piecewise expanding maps T, the bounded
p-variation observables u_0 that admits an infinite sequence of bounded
p-variation observables u_i satisfying u_i(x)= u_{i+1}(Tx) -u_{i+1}(x) are
constant. The method of the proof consists in to find a suitable Hilbert basis
for L^2(hm), where hm is the unique absolutely continuous invariant probability
of T. In terms of this basis, the action of the Perron-Frobenious and the
Koopan operator on L^2(hm) can be easily understood. This result generalizes
earlier results by Bamon, Kiwi, Rivera-Letelier and Urzua in the case T(x)= n x
mod 1, n in N-{0,1} and Lipchitizian observables u_0.Comment: 24 pages. We included new results by A. Avila. He kindly agreed to
include them in this new version. We also fixed some typo
