Minimal \(C^{1}\)-diffeomorphisms of the circle which admit measurable fundamental domains (Q2838962)

Given a probability space \((X, \mu)\) and a transformation \(T:X\rightarrow X\), \(\mu\) is said to be quasi-invariant if the push forward \(T_*\mu\) is equivalent to \(\mu\). In such a case, \(T\) is called ergodic with respect to \(\mu\), if every \(T\)-invariant Borel subset of \(X\) is either null or co-null. If a transformation \(T\) leaves \(\mu\) quasi-invariant, then a Borel subset \(C\) of \(X\) is called a measurable fundamental domain if \((T^nC)_{n\in\mathbb Z}\) is a pairwise disjoint family with union a co-null set.NEWLINENEWLINENEWLINETheorem: For any irrational number \(\alpha\), there exists a minimal \(C^1\) diffeomorphism of the circle with rotation number \(\alpha\) which admits a measurable fundamental domain with respect to the Lebesgue measure.NEWLINENEWLINENEWLINEThe remaining of the paper deals with the details of the proof of this theorem. The authors also ask the following questions: Does there exist a minimal non-ergodic \(C^{1+r}\) diffeomorphism for \(0<r<1\)? More generally, for any \(d\geq2\) and \(r>d^{-1}\), any free \(\mathbb Z^d\)-action by a \(C^{1+r}\) diffeomorphism is known to be minimal. Do there exist non-ergodic actions?