A central question in quantum chaos is to understand the behavior of high-frequency Laplace eigenfunctions on hyperbolic manifolds. Quantum cat maps are a model system that behave like hyperbolic manifolds: the known tools apply equally well to both systems. However, some quantum cat maps exhibit a strange periodicity, resulting in concentration of eigenfunctions—behavior not expected for eigenfunctions on hyperbolic manifolds. We discuss new work distinguishing these two systems. The key ingredient is an uncertainty principle that distinguishes the Fourier transform from a modified Fourier transform with nonlinear phase function. 

Joint with Semyon Dyatlov.