MORSE - Theoretical methods in Hamiltonian dynamics

• Applicants: Professor Dr. Alberto Abbondandolo; Professor Dr. Luca Asselle
• Subject Area: Mathematics
• Term: from 2017 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 380257369

• Final Report Year: 2023

Final Report Abstract

Many of the questions that were listed in the research proposal were successfully addressed, and some unforseen results were also obtained. The main results concern the following topics: (i) Morse homology for the Hamiltonian action functional on the cotangent bundle of a closed manifold. (ii) New proofs of the Weinstein conjecture on cotangent bundles and the Hamiltonian Arnold conjecture on the complex projective space. (iii) Homotopy classification of proper Fredholm maps into a Hilbert space. (iv) Existence of closed magnetic geodesics on orbifolds. (v) Generalization of Mather’s graph theorem to subcritical energies. (vi) Rigidity and flexibility results for magnetic systems all of whose orbits are closed. (vii) Invariant stable foliations and CW-structure induced by a gradient-like Morse- Smale flow. (viii) S-balanced configurations in the n-body problem. (ix) Linear stability of relative equilibria for the n-body problem.

Publications

• On geodesic flows with symmetries and closed magnetic geodesics on orbifolds. Ergodic Theory and Dynamical Systems, 40(6), 1480-1509.
  ASSELLE, LUCA & SCHMÄSCHKE, FELIX
  
• Minimal Boundaries in Tonelli Lagrangian Systems. International Mathematics Research Notices, 2021(20), 15746-15787.
  Asselle, Luca; Benedetti, Gabriele & Mazzucchelli, Marco
  
• Integrable Magnetic Flows on the Two-Torus: Zoll Examples and Systolic Inequalities. The Journal of Geometric Analysis, 31(3), 2924-2940.
  Asselle, Luca & Benedetti, Gabriele
  
• On the rigidity of Zoll magnetic systems on surfaces. Nonlinearity, 33(7), 3173-3194.
  Asselle, Luca & Lange, Christian
  
• The Palais–Smale condition for the Hamiltonian action on a mixed regularity space of loops in cotangent bundles and applications. Calculus of Variations and Partial Differential Equations, 59(4).
  Asselle, Luca & Starostka, Maciej
  
• Morse Theory for S-balanced Configurations in the Newtonian n-body Problem. Journal of Dynamics and Differential Equations, 35(1), 907-946.
  Asselle, Luca & Portaluri, Alessandro
  
• Normal forms for strong magnetic systems on surfaces: trapping regions and rigidity of Zoll systems. Ergodic Theory and Dynamical Systems, 42(6), 1871-1897.
  ASSELLE, LUCA & BENEDETTI, GABRIELE
  
• Stable foliations and CW-structure induced by a Morse–Smale gradient-like flow. Journal of Topology and Analysis, 15(04), 1037-1093.
  Abbondandolo, Alberto & Majer, Pietro
  
• Bifurcations of balanced configurations for the Newtonian n-body problem in R^4. Journal of Fixed Point Theory and Applications, 24(2).
  Asselle, Luca; Fenucci, Marco & Portaluri, Alessandro
  
• On a comparison principle and the uniqueness of spectral flow. Mathematische Nachrichten, 295(4), 785-805.
  Starostka, Maciej & Waterstraat, Nils
  
• Spectral stability, spectral flow and circular relative equilibria for the Newtonian n-body problem. Journal of Differential Equations, 337, 323-362.
  Asselle, Luca; Portaluri, Alessandro & Wu, Li
  
• The homotopy classification of proper Fredholm maps of index one. Topological Methods in Nonlinear Analysis, 1-37.
  Alberto, Abbondandolo & Thomas, Rot
  
• The Arnold conjecture in CP^n and the Conley index. Discrete and Continuous Dynamical Systems - B, 28(4), 2603-2620.
  Asselle, Luca; Izydorek, Marek & Starostka, Maciej
  
• Morse homology for the Hamiltonian action in cotangent bundles. Calculus of Variations and Partial Differential Equations, 64(4).
  Asselle, Luca & Starostka, Maciej