Kaloshin Group

Dynamical Systems, Celestial Mechanics, and Spectral Rigidity

The Kaloshin group has achieved considerable progress in studying spectral rigidity and integrability for billiards. Motivated by a famous question: Can you hear the shape of a drum?” we helped prove the Lyapunov rigidity for expanding circle maps satisfying the sparsity condition. In loose terms, it means that if the Lyapunov spectrum, a collection of Lyapunov exponents of all periodic points as proposed by McMullen, is sparse, which is a quantitative form of absence of multiple eigenvalues, then a small perturbation of such a map having the same Lyapunov spectrum is smoothly conjugate to the original map. We also helped construct drums for which there are silent elements of the Laplace spectrum. On the other hand, we studied the existence of locally integral billiards near the period two orbit and the analyticity of the conjugacy map to its Birkhoff normal form. Remarkably, we proved that, generically, such a conjugacy map is not analytic, but only Gerver. Furthermore, we established a diamond KAM theorem and investigated the existence of mushroom KAM tori in the absence of Kolmogorov nondegeneracy. We are also making progress in the construction of isospectral deformations preserving a countable collection of spectral data.
Besides we extended previous results of the group to symplectic billiards. We are also performing a deep analysis of geodesic flows on tori having degenerate behavior with the potential to build a counter-example to a well-known Ivrii conjecture.