Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools

The isolated spectrum of transfer operators is known to play a critical role in determining mixing properties of piecewise smooth dynamical systems. The so-called Dellnitz-Froyland ansatz places isolated eigenvalues in correspondence with structures in phase space that decay at rates slower than local expansion can account for. Numerical approximations of transfer operator spectrum are often insufficient to distinguish isolated spectral points, so it is an open problem to decide to which eigenvectors the ansatz applies. We propose a new numerical technique to identify the isolated spectrum and large-scale structures alluded to in the ansatz. This harmonic analytic approach relies on new stability properties of the Ulam scheme for both transfer and Koopman operators, which are also established here. We demonstrate the efficacy of this scheme in metastable one- and two-dimensional dynamical systems, including those with both expanding and contracting dynamics, and explain how the leading eigenfunctions govern the dynamics for both real and complex isolated eigenvalues.