Spectral rigidity and nonrigidity of dynamical systems

This thesis studies spectral rigidity and nonrigidity phenomena in dynamical systems. The central question is whether a dynamical system can be determined, up to a natural conjugacy, from its spectrum. We consider three related spectra: the length spectrum, the action spectrum, and the Lyapunov spectrum. The first part of the thesis concerns Liouville metrics on the two-dimensional torus. It is a long-standing folklore conjecture that Liouville metrics are the only integrable metrics on the torus. We prove a length-spectral rigidity result for linear conformal deformations of Liouville metrics by exploiting the dynamical properties of the rational tori -- analogues of the resonant convex caustics in billiards. We also establish a complementary classification result showing that marked-length-isospectral Liouville metrics are characterized by rearrangements of the one-dimensional functions appearing in their conformal factors, generalizing a theorem of Abbondandolo-Mazzucchelli. In particular, the second result gives nonrigidity examples within the class of Liouville metrics. The second part of the thesis studies the standard map from the viewpoint of action and Lyapunov spectra. We construct nontrivial deformations of the standard map which preserve the symplectic actions (respectively, the Lyapunov exponents) of infinitely many periodic orbits accumulating on an invariant curve. The proof combines a resonant normal form construction with Picard iteration schemes to obtain a sequence of periodic orbits accumulating on an invariant curve with a Liouville rotation number. Within the resonant normal forms we capture the dependence of these periodic orbits on the resonant Fourier coefficients of the dynamics on the invariant curve and, using the contraction mapping principle, obtain a suitable deformation achieving the prescribed spectral data associated with this sequence of orbits. The result can be viewed as a symplectic twist-map analogue of a length-spectral nonrigidity phenomenon for Riemannian manifolds and convex billiards, and it motivates the existence problem for similar 'partially length-isospectral' deformations of strictly convex billiard tables.