Resonances for Anosov diffeomorphisms on the torus

Eigenvalues of transfer operators, known as Pollicott-Ruelle resonances provide insight into the long-term behaviour of the underlying dynamical system, in particular determining its exponential mixing rates.

I will present a complete description of Pollicott-Ruelle resonances for a class of rational Anosov diffeomorphisms on the two-torus. This allows us to show that every homotopy class of two-dimensional Anosov diffeomorphisms contains (non-linear) maps with the sequence of resonances decaying stretched exponentially, exponentially or having only trivial resonances.