On infinitely cohomologous to zero observables

TL;DR

This paper proves that for a broad class of piecewise expanding maps, the only bounded p-variation observables infinitely cohomologous to zero are constant, extending previous results to more general maps and observables.

Contribution

It introduces a new method using a Hilbert basis for L^2(hm) to analyze the action of transfer operators, generalizing earlier results to wider classes of maps and observables.

Findings

Bounded p-variation observables infinitely cohomologous to zero are constant.

The method employs a Hilbert basis to understand transfer operators.

Generalizes earlier results to broader classes of maps and observables.

Abstract

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.