Transfer operator for ultradifferentiable expanding maps of the circle

Abstract

Given a $\mathcal{C}^\infty$ expanding map $T$ of the circle, we construct a Hilbert space $\mathcal{H}$ of smooth functions on which the transfer operator $\mathcal{L}$ associated to $T$ acts as a compact operator. This result is made quantitative (in terms of singular values of the operator $\mathcal{L}$ acting on $\mathcal{H}$) using the language of Denjoy-Carleman classes. Moreover, the nuclear power decomposition of Baladi and Tsujii can be performed on the space $\mathcal{H}$, providing a bound on the growth of the dynamical determinant associated to $\mathcal{L}$.