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

Abstract

We consider the reducibility problem of cocycles by isometries of Gromov hyperbolic metric spaces in the Lǐsic setting. We show that provided that the boundary cocycle (that acts on a compact space) is reducible in a suitable Hölder class, then the original cocycle by isometries (that acts on an unbounded space) is also reducible.