Transverse winding transfers along the transport
DisprovedBirkhoffGlobalSection.transported_winding_above_onecelestial-mechanicsdynamical-systemshamiltonian-dynamics
Transverse winding transfers along the transport. If a model solution with transverse winding above one is the strictly monotone reparametrized image of a Levi-Civita solution, then the Levi-Civita solution also has transverse winding above one.
The coordinate change is symplectic up to a positive constant and the time change is orientation-preserving, so neither alters transverse rotation counts. This isolates the winding-invariance analysis from the orbit transport.
Preamble
import Definitions.Def_BirkhoffGlobalSection_DynamicalConvexity import Definitions.Def_BirkhoffGlobalSection_RegularizationModel
Formal statement
namespace BirkhoffGlobalSection
theorem transported_winding_above_one
(μ c : ℝ) (M : RegularizationModel μ c) (S : Set Phase)
(x : ℝ → Phase) (T : ℝ)
(y : ℝ → Phase) (T' : ℝ) (σ : ℝ → ℝ)
(hσ : StrictMono σ) (hσ0 : σ 0 = 0) (hσT : σ T = T')
(hrel : ∀ t : ℝ, y (σ t) = M.toModel (x t))
(hwind : HasTransverseWindingAboveOne M.modelHamiltonian y T') :
HasTransverseWindingAboveOne (leviCivitaHamiltonian μ c) x T := by sorry
end BirkhoffGlobalSection
Source
Closed-orbit transport by the regularization clock and invariance of transverse winding under conformally symplectic coordinate changes with positive time change; see the regularization-coordinate setup of Liu--Salomao, https://arxiv.org/html/2506.17867v2, Section 10.