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Preperiodicity in Arithmetic Dynamics /

Olechnowicz, Mateusz Grzegorz,

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Preperiodicity in Arithmetic Dynamics /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Preperiodicity in Arithmetic Dynamics // Mateusz Grzegorz Olechnowicz.
Author:	Olechnowicz, Mateusz Grzegorz,
Description:	1 electronic resource (280 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 85-05, Section: B.
Contained By:	Dissertations Abstracts International85-05B.
Subject:	Mathematics. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=30687208
ISBN:	9798380835237

Preperiodicity in Arithmetic Dynamics /
Olechnowicz, Mateusz Grzegorz,

Preperiodicity in Arithmetic Dynamics /Mateusz Grzegorz Olechnowicz. - 1 electronic resource (280 pages)

Source: Dissertations Abstracts International, Volume: 85-05, Section: B.

Motivated by the Uniform Boundedness Conjecture of Morton and Silverman, we prove various results related to preperiodic points of algebraic dynamical systems.Our main contribution concerns the distribution of preperiodic points in one-parameter families of rational functions ft defined over a number field K. We show that for all t outside of a set of natural density zero, every K-rational preperiodic point of ft is the specialization of some K(T)-rational preperiodic point of f. Assuming a weak form of the Uniform Boundedness Conjecture, we also calculate the average number of K-rational preperiodic points of f, giving some examples where this holds unconditionally. To illustrate the theory, we give new estimates on the average number of preperiodic points for the quadratic family ft(z) = z2 + t over the field of rational numbers. This builds on work of Le Boudec and Mavraki.Our second contribution is a new technique for constructing endomorphisms with many preperiodic points. Working over a field of characteristic 0, we use Lagrange interpolation to produce degree-$d$ polynomial maps of An with a finite orbit of size nearly n times the naive bound. This extends a result of Hutz.For our final contribution, we introduce a new generalization of the notion of preperiodic hypersurface and explore some of its basic ramifications. We also prove that among nonlinear endomorphisms of projective space, those with a periodic critical point are Zariski dense. This answers a question of Ingram.

English

ISBN: 9798380835237Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Arithmetic dynamics

Preperiodicity in Arithmetic Dynamics /
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100 1 $a Olechnowicz, Mateusz Grzegorz, $e author. $0 (orcid)0000-0002-0788-7463 $3 3818020
245 1 0 $a Preperiodicity in Arithmetic Dynamics / $c Mateusz Grzegorz Olechnowicz.
264 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2023
300 $a 1 electronic resource (280 pages)
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500 $a Source: Dissertations Abstracts International, Volume: 85-05, Section: B.
500 $a Advisors: Ingram, Patrick Committee members: Tsimerman, Jacob; Tiozzo, Giulio.
502 $b Ph.D. $c University of Toronto (Canada) $d 2023.
520 $a Motivated by the Uniform Boundedness Conjecture of Morton and Silverman, we prove various results related to preperiodic points of algebraic dynamical systems.Our main contribution concerns the distribution of preperiodic points in one-parameter families of rational functions ft defined over a number field K. We show that for all t outside of a set of natural density zero, every K-rational preperiodic point of ft is the specialization of some K(T)-rational preperiodic point of f. Assuming a weak form of the Uniform Boundedness Conjecture, we also calculate the average number of K-rational preperiodic points of f, giving some examples where this holds unconditionally. To illustrate the theory, we give new estimates on the average number of preperiodic points for the quadratic family ft(z) = z2 + t over the field of rational numbers. This builds on work of Le Boudec and Mavraki.Our second contribution is a new technique for constructing endomorphisms with many preperiodic points. Working over a field of characteristic 0, we use Lagrange interpolation to produce degree-$d$ polynomial maps of An with a finite orbit of size nearly n times the naive bound. This extends a result of Hutz.For our final contribution, we introduce a new generalization of the notion of preperiodic hypersurface and explore some of its basic ramifications. We also prove that among nonlinear endomorphisms of projective space, those with a periodic critical point are Zariski dense. This answers a question of Ingram.
546 $a English
590 $a School code: 0779
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
653 $a Arithmetic dynamics
653 $a Critical points
653 $a Dynamical systems
653 $a Lagrange interpolation
653 $a Number theory
653 $a Rational functions
690 $a 0405
690 $a 0642
710 2 $a University of Toronto (Canada). $b Mathematics. $e degree granting institution. $3 3818021
720 1 $a Ingram, Patrick $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 85-05B.
790 $a 0779
791 $a Ph.D.
792 $a 2023
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=30687208