  We demonstrate via numerical simulation that in the \textit{strongly}
nonlinear limit, the $\beta$-FPU system in thermal equilibrium behaves
surprisingly like weakly nonlinear waves in properly renormalized normal
variables. This arises because the collective effect of strongly nonlinear
interactions effectively renormalizes linear dispersion frequency and leads to
effectively weak interaction among these renormalized waves. Furthermore, we
show that the dynamical scenario for thermalized $\beta$-FPU chains is
spatially highly localized discrete breathers riding chaotically on spatially
extended, renormalized waves.
