Recurrence in Algebraic Dynamics
| dc.contributor.author | Hossain, Ehsaan | |
| dc.date.accessioned | 2020-07-28T20:20:18Z | |
| dc.date.available | 2020-07-28T20:20:18Z | |
| dc.date.issued | 2020-07-28 | |
| dc.date.submitted | 2020-07-21 | |
| dc.description.abstract | The 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.identifier.uri | http://hdl.handle.net/10012/16086 | |
| dc.language.iso | en | en |
| dc.pending | false | |
| dc.publisher | University of Waterloo | en |
| dc.subject | algebra | en |
| dc.subject | dynamics | en |
| dc.subject | dynamical systems | en |
| dc.subject | group theory | en |
| dc.title | Recurrence in Algebraic Dynamics | en |
| dc.type | Doctoral Thesis | en |
| uws-etd.degree | Doctor of Philosophy | en |
| uws-etd.degree.department | Pure Mathematics | en |
| uws-etd.degree.discipline | Pure Mathematics | en |
| uws-etd.degree.grantor | University of Waterloo | en |
| uws.contributor.advisor | Bell, Jason | |
| uws.contributor.affiliation1 | Faculty of Mathematics | en |
| uws.peerReviewStatus | Unreviewed | en |
| uws.published.city | Waterloo | en |
| uws.published.country | Canada | en |
| uws.published.province | Ontario | en |
| uws.scholarLevel | Graduate | en |
| uws.typeOfResource | Text | en |