Abstract
Given a compactly supported area-preserving diffeomorphism of the disk, we prove an integral formula relating the asymptotic action to the asymptotic winding number. As a corollary, we obtain a new proof of Fathi’s integral formula for the Calabi homomorphism on the disk.
Highlights
Let φ : D → D be a diffeomorphism of the disk D ⊂ R2 compactly supported on the interior and preserving the standard area form ω0 = d x ∧ dy
The real valued function φ → C(φ) is a homomorphism on Diffc(D, ω0), the group of area-preserving diffeomorphisms of D compactly supported in the interior and is named the Calabi homomorphism after [4]
Introduced by Calabi in [4], the Calabi homomorphism is an important tool in the study of the structure of the group of Hamiltonian diffeomorphisms of a symplectic manifold
Summary
Let φ : D → D be a diffeomorphism of the disk D ⊂ R2 compactly supported on the interior and preserving the standard area form ω0 = d x ∧ dy. If λ is any primitive of ω0, we can define the action aφ,λ(z) of a point z ∈ D with respect to λ as the value at z of the unique primitive of the exact form φ∗λ − λ that vanishes near the boundary of D. If one sees φ as the time-one map of the isotopy φt that is obtained by integrating the vector field coming from a compactly supported time-dependent Hamiltonian (z, t) → Ht (z), the action has the expression aφ,λ(z) =. The value aφ,λ(z) depends on the choice of the primitive λ, but it is independent of λ at the fixed points of φ
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