Starts 12 Oct 2022 14:00
Ends 12 Oct 2022 15:00
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Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems
hybrid seminar
Leonardo Building - Luigi Stasi Seminar Room
Venue: for in-person attendees (Leonardo da Vinci, Luigi Stasi seminar room).
For virtual attendees:
https://zoom.us/meeting/register/tJAvf-GsqTspHdyZx3rgfg1RyiuEKtUdnq_Q
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Abstract: We are interested in the limit behaviour of Birkhoff sums over an infinite sigma-finite measure space. If the observable is integrable then - by a classical theorem by Aaronson - there exists no sequence of real numbers such that the Birkhoff sum normed by this sequence converges almost surely to 1. Under strong mixing conditions on the underlying system we prove a generalized strong law of large numbers for integrable observables using a truncated sum adding a suitable number of terms depending on the point of evaluation. For f not integrable we give conditions on f such that the Birkhoff sum normed by a sequence of real numbers converges almost surely to 1. Finally, we look at the example of the backwards continued fractions where we have an observable which is non-integrable wrt the measure restrictet to a finite measure set. We give a proper truncation also for this system. This is joint work with Claudio Bonanno.
For virtual attendees:
https://zoom.us/meeting/register/tJAvf-GsqTspHdyZx3rgfg1RyiuEKtUdnq_Q
After registering, you will receive a confirmation email containing information about joining the meeting.
Abstract: We are interested in the limit behaviour of Birkhoff sums over an infinite sigma-finite measure space. If the observable is integrable then - by a classical theorem by Aaronson - there exists no sequence of real numbers such that the Birkhoff sum normed by this sequence converges almost surely to 1. Under strong mixing conditions on the underlying system we prove a generalized strong law of large numbers for integrable observables using a truncated sum adding a suitable number of terms depending on the point of evaluation. For f not integrable we give conditions on f such that the Birkhoff sum normed by a sequence of real numbers converges almost surely to 1. Finally, we look at the example of the backwards continued fractions where we have an observable which is non-integrable wrt the measure restrictet to a finite measure set. We give a proper truncation also for this system. This is joint work with Claudio Bonanno.