2025/11/20 by Mélanie Bertelson, Bertelson, Mélanie, Pranav Chakravarthy +3
Mathematics · Physics and Astronomy · #Geometric and Algebraic Topology #Quantum chaos and dynamical systems #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2511.16329
Using generating functions quadratic at infinity for Lagrangian submanifolds of twisted cotangent bundles, we define spectral selectors for compactly supported lcs Hamiltonian diffeomorphisms of the locally conformal symplectizations S1 × ℝ2n+1 and S1 × ℝ2n × S1 of ℝ2n+1 and ℝ2n × S1, and obtain several applications: the construction of a bi-invariant partial order on the group of compactly supported lcs Hamiltonian diffeomorphisms of S1 × ℝ2n+1 and S1 × ℝ2n × S1, of an integer-valued bi-invariant metric on the group of compactly supported lcs Hamiltonian diffeomorphisms of S1 × ℝ2n × S1, and of an integer-valued lcs capacity for domains of S1 × ℝ2n × S1. The lcs capacity is used to prove a lcs non-squeezing theorem in S1 × ℝ2n × S1 analogous to the contact non-squeezing theorem in ℝ2n × S1 discovered in 2006 by Eliashberg, Kim and Polterovich. Along the way we introduce for Liouville lcs manifolds the notions of essential Lee chords between exact Lagrangian submanifolds and of essential translated points of exact lcs diffeomorphisms. We prove that essential translated points always exist for time-1 maps of sufficiently C0-small lcs Hamiltonian isotopies of compact Liouville lcs manifolds and for all compactly supported lcs Hamiltonian diffeomorphisms of S1 × ℝ2n+1 and S1 × ℝ2n × S1. We also obtain an existence result for essential Lee chords between the zero section of a twisted cotangent bundle with compact base and its image by any lcs Hamiltonian isotopy, which can be thought of as a lcs analogue of the Lagrangian and Legendrian Arnold conjectures on usual cotangent and 1-jet bundles.