Abstract
We study the cohomological equation for discrete horocycle maps on \begin{document}$ {\rm SL}(2, \mathbb{R}) $\end{document} and \begin{document}$ {\rm SL}(2, \mathbb{R})\times {\rm SL}(2, \mathbb{R}) $\end{document} via representation theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions of the cohomological equation of horocycle maps in representations of \begin{document}$ {\rm SL}(2, \mathbb{R}) $\end{document} . Our estimates improve on previous results and are sharp up to a fixed, finite loss of regularity. Moreover, they are tame on a co-dimension one subspace of \begin{document}$ \operatorname{\mathfrak s\mathfrak l}(2, \mathbb{R}) $\end{document} , and we prove tame cocycle rigidity for some two-parameter discrete actions, improving on a previous result. Our estimates on the cohomological equation of horocycle maps overcome difficulties in previous papers by working in a more suitable model for \begin{document}$ {\rm SL}(2, \mathbb{R}) $\end{document} in which all cases of irreducible, unitary representations of \begin{document}$ {\rm SL}(2, \mathbb{R}) $\end{document} can be studied simultaneously. Finally, our results combine with those of a very recent paper by the authors to give cohomology results for discrete parabolic actions in regular representations of some general classes of simple Lie groups, providing a fundamental step toward proving differential local rigidity of parabolic actions in this general setting.
Highlights
Cohomology arises in various problems in dynamical systems, such as those concerning the existence of invariant measures and mixing of suspension flows, and it is of central importance to rigidity and stability questions, see [4], [2], [3] and [11]
We prove non-tame estimates for the solution to the cohomological equation of horocycle maps that are sharp up to a fixed finite loss of regularity, see Theorem 1.2, improving on results in [18] and [9]
Even though the solution is not tame, we prove that estimates are tame in a co-dimension one subspace of the Lie algebra, see Theorem 1.3, which is used in [19] to deduce tame cocycle rigidity for abelian discrete parabolic actions in the higher-rank setting
Summary
Cohomology arises in various problems in dynamical systems, such as those concerning the existence of invariant measures and mixing of suspension flows, and it is of central importance to rigidity and stability questions, see [4], [2], [3] and [11]. There is an abundance of rigidity results for higher-rank (partially) hyperbolic algebraic systems. Local rigidity and cocycle rigidity for higher-rank standard hyperbolic algebraic actions were proved in the 1990’s by Katok and Spatzier [12], [13]. The results were extended to a large class of higher-rank partially hyperbolic actions, see for example [2], [11], [12], [10] and [22]. These results were proven using geometric arguments that rely on the fast separation of orbits
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