This is the first talk of the reading group on Ruelle resonances for contact Anosov flows.

I will introduce the notion of Ruelle resonances for expanding maps of the circle, as well as some generic tools to study them. They are defined as eigenvalues of transfer operators ; by varying the function spaces these transfer operators act on, we can reveal these resonances. In addition, the notion of quasi-compactness of operators will be central, and I will introduce some tools to prove this property (Doblin-Fortet inequalities, Hennion's theorem).