63 research outputs found
Pointwise convergence of Birkhoff averages for global observables
It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in
infinite ergodic theory is trivial; it states that for any
infinite-measure-preserving ergodic system the Birkhoff average of every
integrable function is almost everywhere zero. Nor does a different rescaling
of the Birkhoff sum that leads to a non-degenerate pointwise limit exist. In
this paper we give a version of Birkhoff's theorem for conservative, ergodic,
infinite-measure-preserving dynamical systems where instead of integrable
functions we use certain elements of , which we generically call
global observables. Our main theorem applies to general systems but requires an
hypothesis of "approximate partial averaging" on the observables. The idea
behind the result, however, applies to more general situations, as we show with
an example. Finally, by means of counterexamples and numerical simulations, we
discuss the question of finding the optimal class of observables for which a
Birkhoff theorem holds for infinite-measure-preserving systems.Comment: Final version. 33 pages, 10 figure
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