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[Экзаменационное задание .pdf]
A cohomological equation over a dynamics codes the possibility of transforming a general map (dynamics) into a simpler one. For instance, the most basic example is that over a circle diffeomorphism: solving this equation allows conjugating it to a rotation.
In general, it is impossible to solve the equation, but in many cases finding approximate solutions is possible. We will see that this is related to almost reducing the dynamics into a simpler one.
More importantly, we will show a nonlinear geometric framework
in which several classical results do persist (roughly, isometries
of nonpositively curved spaces). Among concrete results/applications,
we will show that: