Journal ArticleDOI
A Multiplicative Ergodic Theorem and Nonpositively Curved Spaces
Anders Karlsson,Gregory Margulis +1 more
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In this article, the integrable cocycles u.n; x/ over an ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively curved space Y, e.g. a Cartan-Hadamard space or a uniformly convex Banach space were studied.Abstract:
We study integrable cocycles u.n; x/ over an ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively curved space Y , e.g. a Cartan-Hadamard space or a uniformly convex Banach space. It is proved that for any y 2 Y and almost all x; there exist A 0 and a unique geodesic ray .t; x/in Y starting at y such that lim n!1 1 n d. .An; x/; u.n; x/y/ D 0: In the case where Y is the symmetric space GLN.R/=ON.R/ and the cocycles take values in GLN.R/; this is equivalent to the multiplicative ergodic theorem of Oseledec. Two applications are also described. The first concerns the determination of Poisson boundaries and the second concerns Hilbert-Schmidt operators.read more
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Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents
Luis Barreira,Yakov Pesin +1 more
TL;DR: In this paper, the authors present a self-contained and comprehensive account of modern smooth ergodic theory, which provides a rigorous mathematical foundation for the phenomenon known as deterministic chaos -the appearance of 'chaotic' motions in pure deterministic dynamical systems.
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Boundaries of hyperbolic groups
Ilya Kapovich,Nadia Benakli +1 more
TL;DR: The authors survey the known results about boundaries of word-hyperbolic groups, and present a survey of the boundary of word hyperbolic group boundaries in the context of word embeddings.
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The Poisson formula for groups with hyperbolic properties
TL;DR: In this article, the authors developed a new method of identifying the Poisson boundary based on entropy estimates for conditional random walks, which leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature.
Journal ArticleDOI
Almost isometric actions, property (T), and local rigidity
David Fisher,Gregory Margulis +1 more
TL;DR: In this paper, it was shown that any Riemannian isometric action of a discrete group with property T of Kazhdan is locally rigid on a compact manifold X and a foliated version of this result was used in their proof of local rigidity for standard actions of higher rank semisimple Lie groups and their lattices.
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Non-expanding maps and Busemann functions
TL;DR: In this article, stronger versions and alternative simple proofs of some results of Beardon, [Be1] and [Be2] were given for contractions of locally compact metric spaces.
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Uniformly convex spaces
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Fundamentals of differential geometry
TL;DR: In this paper, the authors provide an introduction to basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas: for instance, the existence, uniqueness, and smoothness theorem for differential equations and the flow of a vector field; the basic theory of vector bundles including the existence of tubular neighborhoods for a submanifold; the calculus of differential forms; basic notions of symplectic manifolds, including the canonical 2-form; sprays and covariant derivatives for Riemannian and pseudo