Subleading asymptotics of link spectral invariants and homeomorphism groups of surfaces

TL;DR

This paper investigates the detailed asymptotic behavior of link spectral invariants on surfaces, revealing new properties of the Calabi homomorphism and its extensions to homeomorphism groups, with implications for surface dynamics.

Contribution

It provides the first analysis of subleading asymptotics of link spectral invariants on genus zero surfaces and explores their relation to the Calabi homomorphism and homeomorphism groups.

Findings

Subleading asymptotics are bounded for smooth Hamiltonians.

The Ruelle invariant is recovered for certain disc maps.

The Calabi homomorphism has infinitely many extensions to homeomorphism groups.

Abstract

This paper continues the study of link spectral invariants on compact surfaces, introduced in our previous work and shown to satisfy a Weyl law in which they asymptotically recover the Calabi invariant. Here we study their subleading asymptotics on surfaces of genus zero. We show the subleading asymptotics are bounded for smooth time-dependent Hamiltonians, and recover the Ruelle invariant for autonomous disc maps with finitely many critical values. We deduce that the Calabi homomorphism admits infinitely many extensions to the group of compactly supported area-preserving homeomorphisms, and that the kernel of the Calabi homomorphism on the group of hameomorphisms is not simple.