Mathematical Research Letters

Volume 29 (2022)

Number 6

On the projective derivative cocycle for circle diffeomorphisms

Pages: 1859 – 1879



Andrés Navas (Dpto. de Matemática, Universidad de Santiago de Chile, Santiago, Chile; and Unidad Cuernavaca Instituto de Matemáticas, Universidad Nacional Autónoma de México, Cuernavaca, México)

Mario Ponce (Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Macul, Chile)


We study the projective derivative as a cocycle of Möbius transformations over groups of circle diffeomorphisms. By computing precise expressions for this cocycle, we obtain several results about reducibility and almost reducibility to a cocycle of rotations. We also introduce an extension of this cocycle to the diagonal action on the $3$-torus for which we generalize the previous results.

Andrés Navas was funded by the projects FONDECYT 1200114 (in Chile) as well as FORDECYT 265667 and the PREI of the DGAPA at UNAM (in México).

Mario Ponce was funded by the projects FONDECYT 1180922 and ANILLO ACT172001 CONICYT.

Received 15 December 2020

Accepted 10 June 2021

Published 4 May 2023