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On infinitely cohomologous to zero observables

  • AMANDA DE LIMA (a1) and DANIEL SMANIA (a1)

Abstract

We show that for a large class of piecewise expanding maps T, the bounded p-variation observables u0 that admit an infinite sequence of bounded p-variation observables ui satisfying

\[ u_{i}= u_{i+1}\circ T-u_{i+1} \]
are constant. The method of the proof consists of finding a suitable Hilbert basis for L2(hm), where hm is the unique absolutely continuous invariant probability of T. On this basis, the action of the Perron–Frobenius and the Koopman operator on L2(hm) can be easily understood. This result generalizes earlier results by Bamón, Kiwi, Rivera-Letelier and Urzúa for the case T(x)=ℓx mod   1 , ∈ℕ∖ {0,1} and Lipschitzian observables u0.

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