A regularity problem in the coboundary equation (Q2722494)

A function \(f\) defined over a dynamical system \((x,T,\mu)\) is a coboundary if there exists a measurable function \(h\) such that \(f=h-h\circ T\). This paper answers in two particular examples the following question: if \(f\) is a regular function, what can be said about the regularity of \(h\)? It is first proved that one can assume \(h\) of integrable square. Then, the case of an ergodic automorphism \(T\) of the torus \(\mathbb{T}^d\) is discussed. A new proof based on Fourier analysis of the following is given: if \(f\) is a measurable coboundary which is \(d\)-times differentiable with \(d\)th derivative Hölder, then it is a coboundary in the class of Hölder functions. This result can also be derived from \textit{A. N. Livshits} [Math. Notes 10(1971), 758-763 (1972); translation from Mat. Zametki 10, 555-564 (1971; Zbl 0227.58006)] or \textit{W. A. Veech} [Ergodic Theory Dyn. Syst. 6, 449-473 (1986; Zbl 0616.28009)]. NEWLINENEWLINENEWLINEThe second example dealt with is the time \(-1\) map \(T\) associated with a geodesic flow on a compact surface of constant negative curvature \(-1\). It is proved by using Birkhoff's Theorem that if \(f\) is a Hölder function of order \(\beta\leq 1\), and if \(f\) is a measurable coboundary, then there exists a Hölder function \(\psi\) of order \(\beta/2\) such that \(f= \psi- \psi_0T\).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00034].