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In this paper we study the Birkhoff coordinates (Cartesian action anglecoordinates) of the Toda lattice with periodic boundary condition in the limitwhere the number $N$ of the particles tends to infinity. We prove that thetransformation introducing such coordinates maps analytically a complex ball ofradius $R/N^\alpha$ (in discrete Sobolev-analytic norms) into a ball of radius$R'/N^\alpha$ (with $R,R'>0$ independent of $N$) if and only if $\alpha\geq2$.Then we consider the problem of equipartition of energy in the spirit ofFermi-Pasta-Ulam. We deduce that corresponding to initial data of size $R/N^2$,$0<R\ll 1$, and with only the first Fourier mode excited, the energy remainsforever in a packet of Fourier modes exponentially decreasing with the wavenumber. Finally we consider the original FPU model and prove that energyremains localized in a similar packet of Fourier modes for times one order ofmagnitude longer than those covered by previous results which is the time offormation of the packet. The proof of the theorem on Birkhoff coordinates isbased on a new quantitative version of a Vey type theorem by Kuksin andPerelman which could be interesting in itself.

Birkhoff coordinates for the Toda Lattice in the limit of infinitely many particles with an application to FPU