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Rigidity versus flexibility phenomena for periodic Hamiltonian flows

Luca Asselle
Schedule: 
Tuesday, April 13, 2021 - 15:00 to 16:00
Location: 
Online
Abstract: 
Periodic Hamiltonian flows are protagonist of fascinating rigidity and flexibility phenomena: Bertrand proved
in 1873 that the only central forces for which all bounded orbits are periodic are given by the harmonic and
the gravitational potential; on the other hand, Zoll showed in 1903 that on the two-sphere there is an
infinite dimensional space of Riemannian metrics of revolution whose all geodesics are periodic. The latter
result was extended by Guillemin in 1976 by explicitly determining the tangent space to the space of Zoll
metrics at the round metric. In this talk we will focus on periodic Hamiltonian flows in the class of magnetic
systems on surfaces where rigidity and flexibility phenomena coexist. We will first construct explicit
examples of magnetic systems which are periodic at a given speed resp. at certain sequences of speeds,
and discuss how these results complement the rigidity results explained by Benedetti in his talk. Then we
will show how Zoll's and Guillemin's ideas could be used to show that the space of magnetic systems which
are periodic at a given speed is infinite dimensional. This is joint work with Gabriele Benedetti (Heidelberg). 

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