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Let K be a number field, let ϕ ∈ K (t) be a rational map of degree at least 2, and let α, β ∈ K. We show that if α is not in the forward orbit of β, then there is a positive proportion of primes p of K such that α mod p is not in the forward orbit of β mod p. Moreover, we show that a similar result holds for several maps and several points. We also present… (More)
iv Acknowledgements I would first like to thank my advisor Joseph Silverman, whose knowledge and keen insight were invaluable to the completion of this work. Our weekly discussions never failed to provide more ideas to ponder and a clearer understanding of the current problems. His careful reading also kept many errors out of this work that I would have… (More)
We prove the effectivity of the dynatomic cycles for morphisms of projective varieties. We then analyze the degrees of the dynatomic cycles and multiplicities of formal periodic points and apply these results to the existence of periodic points with arbitrarily large primitive periods.
This article examines dynamical systems on a class of K3 surfaces in P 2 ×P 2 with an infinite automorphism group. In particular, this article develops an algorithm to find Q-rational periodic points using information over F p for various primes p. This algorithm is then optimized to examine the growth of the average number of cycles versus p and to… (More)
I am interested in arithmetic properties of periodic and pre-periodic points arising from iterating morphisms of projective varieties and associated computational problems. The major guiding problem of my research is the uniform boundedness conjecture of Morton and Silverman. Note, for example, that this conjecture implies Merel's theorem on the uniform… (More)
The behavior under iteration of the critical points of a polynomial map plays an essential role in understanding its dynamics. We study the special case where the forward orbits of the critical points are finite. Thurston's theorem tells us that fixing a particular critical point portrait and degree leads to only finitely many possible polynomials (up to… (More)
Research Interests • Galois theory and irreducibility of polynomials (particularly involving iterated morphisms) • Discrete dynamics (particularly iteration of rational functions over global and finite fields) • Automorphism groups of rooted trees and iterated monodromy groups • Elliptic curves and torsion fields • Recurrence sequences Perfect powers in… (More)