# Transfer operator for ultradifferentiable expanding maps of the circle

## TL;DR

This paper constructs a specialized Hilbert space for smooth functions where the transfer operator of a smooth expanding circle map acts compactly, enabling detailed spectral analysis and bounds on dynamical determinants.

## Contribution

It introduces a new Hilbert space framework using Denjoy-Carleman classes for analyzing transfer operators of smooth expanding maps.

## Key findings

- Transfer operator acts as a compact operator on the constructed Hilbert space.
- Quantitative bounds on singular values of the transfer operator.
- Bound on the growth of the dynamical determinant.

## 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 $\mathcal{L}$.