Measured solenoidal Riemann surface and holomorphic dynamics.
Item
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Title
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Measured solenoidal Riemann surface and holomorphic dynamics.
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Identifier
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AAI9707156
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identifier
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9707156
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Creator
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Su, Meiyu.
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Contributor
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Adviser: Dennis Sullivan
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Date
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1996
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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 {dollar}f:\bar\doubc\to\bar\doubc{dollar} be an arbitrary rational map of degree {dollar}d\ge2{dollar} on the Riemann sphere {dollar}\bar\doubc.{dollar} We, from a measure theoretic point of view, study the solenoidal structure on the inverse limit space {dollar}\lim\limits\sb\gets(\bar\doubc,f){dollar} consisting of backward strings of f. It is shown that there is an ergodic holomorphic foliated dynamical object, namely a self mapping of a measured solenoidal Riemann surface {dollar}{lcub}\cal L{rcub},{dollar} which continuously injects into the inverse limit space whose leaves are conformally isomorphic to the complex plane {dollar}\doubc,{dollar} and whose image intersects every fiber in a full measure for the naturally defined multiplicity fiber measure class. The induced dynamics F (by the rational map f) saturates fibers of the solenoid {dollar}{lcub}\cal L{rcub}{dollar} into F-invariant classes, the so-called transverse equivalence classes, which gives rise to a transverse equivalence relation in {dollar}{lcub}\cal L{rcub}.{dollar} There is a one-to-one correspondence between the set of transverse equivalence classes in {dollar}{lcub}\cal L{rcub}{dollar} and the space of grand orbits of f in the sphere. Some useful consequences follow from this correspondence.
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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.
