RATE OF APPROACH TO MINIMA AND SINKS.
Item
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Title
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RATE OF APPROACH TO MINIMA AND SINKS.
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Identifier
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AAI8023682
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identifier
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8023682
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Creator
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WISNIEWSKI, HELENA STASIA.
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Contributor
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Michael Shub
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Date
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1980
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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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For certain classes of Axiom A systems defined on C(INFIN) compact manifolds this dissertation determines the asymptotic rate of approach of orbits to the sinks of the systems. The rate is determined by comparing the Riemannian measure of the entire manifold, M, to the measure of the set of points whose orbits remain outside a neighborhood of the attractors for a specified number of iterations (diffeomorphisms) or a specified time (flows). For C('2) Axiom A systems and for all Morse-Smale diffeomorphisms the measure is bounded by an expression of the form K exp(-D N), for K and D positive constants and N the iteration count or the time.;As an example of the results obtained consider a gradient system dx(t)/dt = -gradF(x) where F:M (--->) R is a Morse function. Let (phi)(x) be the induced flow, and let f be the time one map. As a corollary to a more general result we have:;Corrollary. For S = set of initial points whose orbits remain outside a neighborhood of the sinks of f after N iterations, (mu)(S) (LESSTHEQ) K exp(-D N) where K > O and D is any number smaller than.;C = min {lcub}Jac D(,p) f (VBAR) W('u) (p){rcub}.;(' p).;where the p are the non-sink fixed points of f.;The two major theorems for diffeomorphisms follow; analogous theorems for flows are also obtained.;Theorem. Let f be a Morse-Smale diffeomorphism of the C('(INFIN)) compact manifold M. Let V be a neighborhood of the sinks such that f (V) (L-HOOK) V. With (mu) a measure induced by the Riemannian metric on M and U = M-clV, then given (delta) > O and U(,n) = {lcub}x(epsilon) U:f('k) (x) (epsilon) V for k n{rcub}, we have (mu) (U(,n) (LESSTHEQ) K(1 + (delta))('n) exp(-C n) where K > O and.;C = log min {lcub}l/m Jac D(,p) f('m) (VBAR) W('u) (p(,i)){rcub}.;(' p).;the minimum being taken over the sources and periodic points of f.;Theorem. Let f be a C('2) Axiom a diffemorphism on a C('(INFIN)) compact manifold M. If f has no cycles, then (1) There is a filtration , = {lcub}M(,i){rcub} (,i=1) adapted to f such that M(,1) contains all sinks and attractors of f and (2) With S (i,j,n) = M(,i) - f('-n) int M(,i-1) we have u(S(i,j,n) (LESSTHEQ) K exp (-C(,ij) n) where K > 0 and the C(,ij) are positive constants.;The constants C(,ij) are related to the topological pressure of f, but in general they are smaller than the minimum of the pressure of f over the basic sets.
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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
