  In previous work, we defined "link spectral invariants" for any compact
surface and used these to study the algebraic structure of the group of
area-preserving homeomorphisms; in particular, we showed that the kernel of
Fathi's mass-flow homomorphism is never simple. A key idea for this was a kind
of Weyl law, showing that asymptotically the link spectral invariants recover
the classical Calabi invariant.
  In the present work, we use the subleading asymptotics in this Weyl law to
learn more about the algebraic structure of these homeomorphism groups in the
genus zero case. In particular, when the surface has boundary, we show that the
kernel of the Calabi homomorphism on the group of hameomorphisms is not simple,
answering an old question of Oh and M\"uller; this contrasts the smooth case,
where the kernel of Calabi is simple. We similarly show that the group of
hameomorphisms of the two-sphere is not simple. Related considerations allow us
to extend the Calabi homomorphism to the full group of compactly supported
area-preserving homeomorphisms, answering a longstanding question of Fathi. In
fact, we produce infinitely many distinct extensions.
  Central to the applications is that we show that the subleading asymptotics
for smooth, possibly time-dependent, Hamiltonians are always $O(1)$, and for
certain autonomous maps recover the Ruelle invariant. The construction of a
hameomorphism with "infinite Ruelle invariant" then shows that a normal
subgroup with $O(1)$ subleading asymptotics is proper.
