Rigidity in Higher Rank Abelian Group Actions: Volume 1, Introduction and Cocycle Problem
This self-contained monograph presents rigidity theory for a large class of dynamical systems, differentiable higher rank hyperbolic and partially hyperbolic actions. This first volume describes the subject in detail and develops the principal methods presently used in various aspects of the rigidity theory. Part I serves as an exposition and preparation, including a large collection of examples that are difficult to find in the existing literature. Part II focuses on cocycle rigidity, which serves as a model for rigidity phenomena as well as a useful tool for studying them. The book is an ideal reference for applied mathematicians and scientists working in dynamical systems and a useful introduction for graduate students interested in entering the field. Its wealth of examples also makes it excellent supplementary reading for any introductory course in dynamical systems.
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abelian actions abelian group Anosov action Anosov diffeomorphism Anosov flows Assume automorphisms bundle Cartan action Cartan subgroup closing conditions coboundary cocycle cocycle equation coefficients cohomology commuting compact manifold conjugacy conjugate connected Lie group constant cocycle continuous converges cycle defined Definition denote differentiable eigenvalues element ergodic example exists extension finite foliation follows formula Fourier function given group actions H¨older hence higher rank abelian holonomy homomorphism implies integer irreducible isomorphic leaves Lemma Lie algebra linear Livshitz Lyapunov exponents matrix measure metric Moreover nilmanifolds nilpotent non-trivial partially hyperbolic actions polynomial principal bundles properties Proposition rank-one regularity result rigidity satisfies Section semisimple Lie group smooth space stable and unstable subgroup subspace tangent tangent space topology torus transfer map trivial unstable foliations vector fields Weyl chamber flow Ws(x Zk-action