Continuity of the phonon gap

Abstract

The phonon gap G for an invariant set Λ of a symplectic twist map of R d × R d with action functional W is the infimum of ‖D 2 W x( z) ξ‖ 2 over zϵΛ and variations ξ with ‖ ξ‖ 2=1. It is proved here that if Λ k , kϵ N , is a sequence of compact invariant sets converging in Hausdorff topology to a compact invariant set Λ, then G( Λ k ) converges to G( Λ). The result implies that the phonon gap is an excellent quantifier of uniform hyperbolicity. Several generalisations are sketched.