Resonances of dynamical systems and Fredholm-Riesz operators on rigged Hilbert spaces

Resonances of dynamical systems are defined as the singularities of the analytically continued resolvent of the restriction of the Frobenius-Perron operator to suitable test-function spaces. A sufficient condition for resonances to arise from a meromorphic continuation to the entire plane is that the Frobenius-Perron operator is a Fredholm-Riesz operator on a rigged Hilbert space. After a discussion of spectral theory in locally convex topological vector spaces, we illustrate the approach for a simple chaotic system, namely the Rényi map.