seminar: A Birkhoff Theorem for global observables
speaker: Marco Lenci (Università di Bologna)abstract: It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in infinite ergodic theory is trivial: for any infinite-measure-preserving ergodic system the Birkhoff average of every integrable function is almost everywhere zero. Nor does it exist a different rescaling of the Birkhoff sum that leads to a non-degenerate pointwise limit. We give an analogue of Birkhoff's theorem for conservative, ergodic, infinite-measure-preserving dynamical systems where instead of integrable functions we use certain elements of $L\infty$, 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. Joint work with Sara Munday
timetable:
Wed 7 Mar, 16:30 - 17:30, Sala Conferenze Centro De Giorgi
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