Exceptional Points in Arithmetic Dynamics
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2015-05
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[Honolulu] : [University of Hawaii at Manoa], [May 2015]
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Abstract
Let be a morphism of PN defined over a field K. We prove three main results:
When K is a number field, we prove that there is a bound B depending only on such that every twist of has no more than B K-rational preperiodic points. (This result is analagous to a result of Silverman for abelian varieties.) For two specific families of quadratic rational maps over Q, we find the bound B explicitly.
When K is a finite field, we find the limiting proportion of periodic points in towers of finite fields for polynomial maps associated to algebraic groups, namely pure power maps (z) = zd and Chebyshev polynomials.
When K is a number field or Qp for p 6= 3; and L=K is an extension we prove that K fails to be critically reducible at 3. Meanwhile, Q3 is critically reducible at 3.
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Ph.D. University of Hawaii at Manoa 2015.
Includes bibliographical references.
Includes bibliographical references.
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Theses for the degree of Doctor of Philosophy (University of Hawaii at Manoa). Math
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