TOPOLOGICAL EQUIVALENCE OF FLOWS ON HOMOGENEOUS SPACES, DIVERGENCE OF ONE-PARAMETER SUBGROUPS, AND ASYMPTOTIC HOMOTOPY CLASSES (LIE GROUPS, DYNAMICAL SYSTEMS).
Item
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Title
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TOPOLOGICAL EQUIVALENCE OF FLOWS ON HOMOGENEOUS SPACES, DIVERGENCE OF ONE-PARAMETER SUBGROUPS, AND ASYMPTOTIC HOMOTOPY CLASSES (LIE GROUPS, DYNAMICAL SYSTEMS).
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Identifier
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AAI8601623
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identifier
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8601623
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Creator
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BENARDETE, DIEGO.
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Contributor
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Michael Shub
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Date
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1985
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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Let (GAMMA) and (GAMMA)' be lattices, and (phi) and (phi)' one-parameter subgroups, of the connected Lie groups G and G'. If Conditions (a), (b), or (c) hold, Theorem A states that if the induced flows on the homogeneous spaces G/(GAMMA) and G'/(GAMMA)' are topologically equivalent, then they are topologically equivalent by an affine map. (a) G and G' are one-connected and nilpotent. (b) G and G' are one-connected and solvable, and for all X in L(G) and X' in L(G'), ad(X) and ad(X') have only real eigenvalues. (c) G and G' are centerless and semisimple with no compact direct factor and no direct factor G(,i) isomorphic to PSL(2,R) such that (GAMMA)G(,i) is closed in G. There is no compact, connected subgroup in the centralizer of (phi).;Theorem A depends on Theorem B, which concerns divergence properties of one-parameter subgroups. We say (phi) is isolated if and only if for any (phi)' which recurrently approaches (phi) for positive and negative time, (phi) equals (phi)' up to sense-preserving reparameterization. Theorem B(a) states that if G is one-connected and nilpotent, or one-connected and solvable with exp: L(G) (--->) G a diffeomorphism, then every (phi) of G is isolated. Let G be connected and semisimple and (phi)(t) = exp(tX). Then Theorem B(b) states that (phi) is isolated, if X,Y = 0 and ad(Y) being semisimple imply that ad(Y) had some eigenvalue not pure imaginary and not 0. If G has finite center, (phi) is isolated if there is no compact connected subgroup in the centralizer of (phi).;We also develop the rudiments of a theory of asymptotic homotopy classes and use it to give an alternate proof of Theorem A in the nilpotent case.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
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Program
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Mathematics
