abstract: In a seminal paper Ruelle showed that the long time asymptotic behaviour of analytic hyperbolic systems can be understood in terms of the eigenvalues of a certain operator acting on a suitable Banach space of holomorphic functions. Ruelle also showed that these eigenvalues, also known as Pollicott-Ruelle resonances, are bounded from above by a decaying exponential.
In this talk I will focus on lower bounds for the Ruelle eigenvalues. More precisely, I will explain how to prove that there exists a dense set of analytic expanding circle maps for which the Ruelle eigenvalues enjoy exponential lower bounds.
This is based on work with W. Just, J. Slipantschuk, and F. Naud.