Cocycles' stability for partially hyperbolic systems
The cohomological equation for an unknown function \(\varphi\) on a space \(X\) reads \(\varphi(x) - h(x) + h(f(x)) = 0\), where \(h\) is the given transfer function on \(X\) and \(f\) is a given transformation \(X \to X\). A solution \(\varphi\) is also called a coboundary and two functions are cohomologous if their difference is a coboundary. Results on this type of cohomology of particular importance are the Livshitz theorems on hyperbolic diffeomorphisms of manifolds. There are related results for some non-hyperbolic systems too: the present paper establishes Livshitz-type theorems for partially hyperbolic systems. For a large class of discrete systems and flows the space of Hölder coboundaries is closed and admits a natural geometrical description. The authors also derive transitivity results for the pair of stable and unstable foliations.
scientific article; zbMATH DE number 913759