Length spectra and strata of flat metrics

In this paper we consider strata of flat metrics coming from quadratic differentials (semi-translation structures) on surfaces of finite type. We provide a necessary and sufficient condition for a set of simple closed curves to be spectrally rigid over a stratum with enough complexity, extending a result of Duchin–Leininger–Rafi. Specifically, for any stratum with more zeroes than the genus, the (Sigma ) -length-spectrum of a set of simple closed curves (Sigma ) determines the flat metric in the stratum if and only if (Sigma ) is dense in the projective measured foliation space. We also prove that flat metrics in any stratum are locally determined by the (Sigma ) -length-spectrum of a finite set of closed curves (Sigma ) .