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On infinitely cohomologous to zero observables

Published online by Cambridge University Press:  08 February 2012

AMANDA DE LIMA  and
DANIEL SMANIA
AMANDA DE LIMA
Affiliation:
Departamento de Matemática, ICMC-USP, Caixa Postal 668, São Carlos-SP, CEP 13560-970 São Carlos-SP, Brazil (email: [email protected], [email protected])
DANIEL SMANIA
Affiliation:
Departamento de Matemática, ICMC-USP, Caixa Postal 668, São Carlos-SP, CEP 13560-970 São Carlos-SP, Brazil (email: [email protected], [email protected])
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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 , ℓ∈ℕ∖ {0,1} and Lipschitzian observables u0.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 2 , April 2013 , pp. 375 - 399
Copyright
©2012 Cambridge University Press

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