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The Fermi-Pasta-Ulam-Tsingou (FPUT) problem addresses fundamental questionsin statistical physics, and attempts to understand the origin of recurrences inthe system have led to many great advances in nonlinear dynamics andmathematical physics. In this work we revisit the problem and studyquasiperiodic recurrences in the weakly nonlinear $\alpha-$FPUT system in moredetail. We aim to reconstruct the quasiperiodic behaviour observed in theoriginal paper from the canonical transformation used to remove the three waveinteractions, which is necessary before applying the wave turbulence formalism.We expect the construction to match the observed quasiperiodicity if we are inthe weakly nonlinear regime. Surprisingly, in our work we show that this is notalways the case and in particular, the recurrences observed in the originalpaper cannot be constructed by our method. We attribute this disagreement tothe presence of small denominators in the canonical transformation used toremove the three wave interactions before arriving at the starting point ofwave turbulence. We also show that these small denominators are present even inthe weakly nonlinear regime, and they become more significant as the systemsize is increased. We also discuss our results in the context of the problem ofequilibration in the $\alpha-$FPUT system, and point out some mathematicalchallenges when the wave turbulence formalism is applied to explainthermalization in the $\alpha-$FPUT problem. We argue that certain aspects ofthe $\alpha-$FPUT system such as presence of the stochasticity threshold,thermalization in the thermodynamic limit and the cause of quasiperiodicity arenot clear, and that they require further mathematical and numerical studies.

Quasiperiodicity in the $\alpha-$Fermi-Pasta-Ulam-Tsingou problem revisited: an approach using ideas from wave turbulence