Middle-dimensional squeezing and non-squeezing phenomena in Hamiltonian dynamics on finite dimensional and infinite-dimensional phase spaces

• Applicant: Professor Dr. Alberto Abbondandolo
• Subject Area: Mathematics
• Term: from 2013 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 242354134

• Final Report Year: 2017

Final Report Abstract

Results about middle dimensional non-squeezing in finite dimensions (Subproject A). A main question of this project was whether the middle dimensional symplectic non-squeezing property holds locally. Rigolli has given a positive answer to this question in the analytic category: If ϕ is an analytic symplectic embedding of a domain of (R2n, ω) into (R2n, ω), then the symplectic 2k-dimensional projection of the image by ϕ of any ball in Ω with sufficiently small radius r has ωk -volume at least πk r2k. In the smooth category, the question of the validity of the local middle dimensional nonsqueezing property is still open. In the proposal, we had already observed that this property would follow from a deep conjecture of Viterbo about volume bounds for the symplectic capacities of convex sets. Together with B. Bramham, U. Hryniewicz and P. Salomão, we were able to confirm a local version of this conjecture in dimension 4. As a corollary, we obtained the local middle dimensional non-squeezing property in the smooth category when the space on which we project has dimension 4. This result about the Viterbo conjecture has opened a new unexpected line of research about systolic inequalities in Reeb dynamics and has already produced new results in metric geometry. Results about rigidity of classical systems (Subproject B). Another question in this proposal concerned the rigidity properties of special classes of symplectomorphisms, such as the ones which are given by Tonelli Hamiltonians on cotangent bundles. Together with O. Bernardi and F. Cardin, we obtained results relating the rigidity of Lagrangian submanifolds of optical energy surfaces to recurrence properties of the dynamics on such submanifolds. On the one hand, we related a rigidity phenomenon which had been studied by G. P. Paternain, L. Polterovich and K. F. Siburg to the behaviour of Lipschitz continuous Lyapunov functions. On the other hand, we proved that any Lagrangian submanifold Λ of an optical energy surface cannot be analytically moved outside of the domain bounded by the energy surface when the dynamics on Λ is strongly chain transitive. Results about non-squeezing in infinite dimensions (Subproject C). Another main question in this project concerned the validity of the non-squeezing theorem for symplectomorphisms on infinite dimensional Hilbert spaces. In a joint paper with P. Majer, we obtained a positive answer in the case in which the image of the ball by the symplectomorphism is convex. This paper contains also applications to Hamiltonian PDEs and infinite dimensional counterexamples to symplectic statements which are true in finite dimensions. Rigolli is currently working on an alternative approach to non-squeezing in infinite dimensions based on generating functions.

Publications

• “A non-squeezing theorem for convex symplectic images of the Hilbert ball”, Calc. Var. 54 (2015), 1469–1506
  A. Abbondandolo, P. Majer
  
• “Local middle dimensional symplectic non-squeezing in the analytic setting”
  L. Rigolli
  
• “Sharp systolic inequalities for Reeb flows on the three-sphere”
  A. Abbondandolo, B. Bramham, U. Hryniewicz, P. Salomão
  
• “Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces”, J. Dyn. Diff. Equat. 2016
  A. Abbondandolo, O. Bernardi, F. Cardin
  
• “A systolic inequality for geodesic flows on the two-sphere”, Math. Ann. 367 (2017), 701–753
  A. Abbondandolo, B. Bramham, U. Hryniewicz and P. Salomão