# Semicontractions, nonpositive curvature, and multiplicative ergodic theory [electronic resource] / Bengt Anders Karlsson

Karlsson, Bengt AndersBib ID | vtls000568516 |

稽核項 | 70 p. |

電子版 | |

附註項 | 數位化論文典藏聯盟 |

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$a Semicontractions, nonpositive curvature, and multiplicative ergodic theory $h [electronic resource] / $c Bengt Anders Karlsson

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$a Source: Dissertation Abstracts International, Volume: 61-05, Section: B, page: 2563.

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$a Director: Gregory A. Margulis.

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$a Thesis (Ph.D.)--Yale University, 2000.

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$a In this thesis we prove a multiplicative ergodic theorem for semicontractions (e.g. isometries) of nonpositively curved spaces. It states that for almost every trajectory of an integrable cocycle (random product) of semicontractions, there exists a unique geodesic ray such that the distance from the trajectory at time <italic>n</italic> to the ray is sublinear in <italic>n</italic>. Oseledec's multiplicative ergodic theorem is a special case of this. Further applications concerning Poisson boundaries, Brownian motion, and Hilbert-Schmidt operators are described. This part of the thesis is a joint work of Margulis and the author.

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$a We also show that any semicontraction of a metric space which is proper (that is, closed balls are compact), must respect some Busemann function: the orbit must be contained either in a ball or a horoball. Often these (horo)balls are invariant and in some cases it can be proved that the orbit must converge to a point in the closure of the space. These results significantly generalize some fairly recent works of Beardon, which in turn extend the classical Wolff-Denjoy theorem.

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$a A few further statements of multiplicative ergodic type are obtained, some of which, in view of a short discussion on symmetric spaces of certain operator algebras, may be interesting.

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$a Furthermore, one chapter of the thesis constitutes a self-contained exposition of subadditive ergodic theory giving alternative proofs of Kingman's fundamental theorem and the key ergodic lemma for the multiplicative ergodic theory mentioned above. The chapter on individual semicontractions also gives proofs of several more or less well-known fixed point theorems. We also discuss various axioms for metric spaces which will be used and which are motivated by examples (hyperbolic spaces, Cartan-Hadamard manifolds, Banach spaces, Kobayashi metrics, Hilbert metrics, etc.).

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$a 數位化論文典藏聯盟 $b PQDT $c 中山大學(2001~2002)

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$a Yale University.

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摘要 | In this thesis we prove a multiplicative ergodic theorem for semicontractions (e.g. isometries) of nonpositively curved spaces. It states that for almost every trajectory of an integrable cocycle (random product) of semicontractions, there exists a unique geodesic ray such that the distance from the trajectory at time <italic>n</italic> to the ray is sublinear in <italic>n</italic>. Oseledec's multiplicative ergodic theorem is a special case of this. Further applications concerning Poisson boundaries, Brownian motion, and Hilbert-Schmidt operators are described. This part of the thesis is a joint work of Margulis and the author. We also show that any semicontraction of a metric space which is proper (that is, closed balls are compact), must respect some Busemann function: the orbit must be contained either in a ball or a horoball. Often these (horo)balls are invariant and in some cases it can be proved that the orbit must converge to a point in the closure of the space. These results significantly generalize some fairly recent works of Beardon, which in turn extend the classical Wolff-Denjoy theorem. A few further statements of multiplicative ergodic type are obtained, some of which, in view of a short discussion on symmetric spaces of certain operator algebras, may be interesting. Furthermore, one chapter of the thesis constitutes a self-contained exposition of subadditive ergodic theory giving alternative proofs of Kingman's fundamental theorem and the key ergodic lemma for the multiplicative ergodic theory mentioned above. The chapter on individual semicontractions also gives proofs of several more or less well-known fixed point theorems. We also discuss various axioms for metric spaces which will be used and which are motivated by examples (hyperbolic spaces, Cartan-Hadamard manifolds, Banach spaces, Kobayashi metrics, Hilbert metrics, etc.). |

附註 | Source: Dissertation Abstracts International, Volume: 61-05, Section: B, page: 2563. Director: Gregory A. Margulis. Thesis (Ph.D.)--Yale University, 2000. 數位化論文典藏聯盟 |

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ISBN/ISSN | 0599791152 |