The Hamiltonian (Oh-Schwarz) spectral invariant is constructed from the
Hamiltonian Floer homology.In 2006, Entov and Polterovich proved that it
and its asymptotization are "quasi-quasi-morphisms" and provide some
applications to Hamiltonian dynamics including non-displaceability.
In this talk, we consider the Lagrangian spectral invariant constructed
from the Lagrangian Floer theory. It is a "quasi-quasi-morphism", but we
do not know whether its asymptotization is a "quasi-quasi-morphism" or
not.
We exceed this difficulty considering "quasi-quasi-quasi-morphism" and
provide some applications to Hamiltonian dynamics.