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dc.contributor.authorHossain, Ehsaan 20:20:18 (GMT) 20:20:18 (GMT)
dc.description.abstractThe Dynamical Mordell--Lang Conjecture states that if a polynomial orbit has infinite intersection with a closed set in an algebraic variety, then the intersection must occur periodically. Although this problem is unsolved in general, a "weak" version of Bell--Ghioca--Tucker obtains the periodicity in the case when the orbit intersection has positive density. The thesis regards a circle of problems exhibiting a Bell--Ghioca--Tucker-type phenomenon in group theory and number theory. In Chapter 1, we explain how the BGT Theorem is essentially a noetherian version of the classic Poincare Recurrence Theorem from ergodic theory; in addition to a generalization of the BGT Theorem to amenable semigroups, we prove a combinatorial analog involving idempotent ultrafilters for any semigroup. In Chapter 2, we prove a version of the BGT Theorem for an automorphism of a polycyclic-by-finite groups, which embellishes the point that noetherian objects obey a Dynamical Mordell--Lang principle. In Chapter 3, in joint work with Bell and Chen, we study the sequence of complex numbers obtained by evaluating a rational function along an orbit; examining the intersection of this sequence with a finitely generated group of units, we again obtain a BGT-type result for such dynamical sequences, and recovering classic theorems of Methfessel, Polya, and Bezivin.en
dc.publisherUniversity of Waterlooen
dc.subjectdynamical systemsen
dc.subjectgroup theoryen
dc.titleRecurrence in Algebraic Dynamicsen
dc.typeDoctoral Thesisen
dc.pendingfalse Mathematicsen Mathematicsen of Waterlooen
uws-etd.degreeDoctor of Philosophyen
uws.contributor.advisorBell, Jason
uws.contributor.affiliation1Faculty of Mathematicsen

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