A Livšic type theorem for germs of analytic diffeomorphisms

Abstract

We deal with the problem of the validity of Livšic's theorem for cocycles of diffeomorphisms satisfying the orbit periodic obstruction over an hyperbolic dynamics. We give a result in the positive direction for cocycles of germs of analytic diffeomorphisms at the origin.

Livšic theorem for low-dimensional diffeomorphism cocycles

“…Moreover, the control of distortion techniques used in [NT95,dlLW10] yield a loss of regularity in the solution of the cohomological equation. A recent result for infinite dimensional groups which does not fit in the previous description is due to Navas and Ponce [NP13]. They prove a Livšic theorem for cocycles taking values in the group of analytic germs at the origin.…”

Livšic theorem for diffeomorphism cocycles

“…Some extensions of these results were obtained by de la Llave and Windsor in [dlLW10], but always under certain localization hypotheses. Navas and Ponce dealt in [NP13] with the case of cocycles taking values in the group of analytic germs at the origin.…”

Relating boundary and interior solutions of the cohomological equation for cocycles by isometries of negatively curved spaces. The Lǐsic case*

“…For instance, in the minimal case, a lot of work had been assembled under the theory of Gottschalk and Hedlund (see [11,17,20]), and the KAM theory (see [2]). In the hyperbolic case, the Liv ˇsic theory has produced a great amount of results (see [6,7,10,13,14,17,19,21,23,24]). Recently the solution to the cohomological equation in the Abelian case has been addressed by some authors as a central problem in the partially hyperbolic dynamical system case (see [16,25]).…”