Cohomological rigidity and the Anosov-Katok construction

Abstract

We provide a general argument for the failure of Anosov-Katok-like constructions (as in \cite{AFKo2015} and \cite{NKInvDist}) to produce Cohomologically Rigid diffeomorphisms in manifolds other than tori. A $C^{\infty }$ smooth diffeomorphism $f$ of a compact manifold $M$ is Cohomologically Rigid iff the equation, known as Linear Cohomological one, \begin{equation*} ψ\circ f - ψ= φ\end{equation*} admits a $C^{\infty }$ smooth solution $\psi$ for every $\phi$ in a codimension $1$ closed subspace of $C^{\infty }(M,\mathbb{C})$. As an application, we show that no Cohomologically Rigid diffeomorphisms exist in the Almost Reducibility regime for quasi-periodic cocycles in homogeneous spaces of compact type, even though the Linear Cohomological equation over a generic such system admits a solution for a dense subset of functions $\phi$. We thus confirm a conjecture by M. Herman and A. Katok in that context and provide some insight in the mechanism obstructing the construction of counterexamples.

43 pages. The proof of corollary B provides a more precise statement and some minor errors have been corrected

References in corpus (2)

• Global aspects of the reducibility of quasiperiodic cocycles in semisimple compact Lie groups
• Local Rigidity of Diophantine translations in higher dimensional tori

Cited by in corpus (1)

• Fibered rotation vector and hypoellipticity for quasiperiodic cocycles in compact Lie groups