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Research – Arithmetic Dynamics

Research Track leader

Arnab Dey Sarkar Pursuing Ph.D. at St. Louis on Arithmetic Dynamics Publication:
A Dichotomy for the Gelfand–Kirillov Dimensions of Simple Modules over Simple Differential Rings Other Papers

Understand the problem

Arithmetic dynamics is a field that amalgamates two areas of mathematics, discrete dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, and/or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.
Given an endomorphism \( f \) on a set \( X \) ; \( f:X\to X \)
a point x in X is called preperiodic point if it has “finite forward orbit under iteration of $f$ with mathematical notation if there exist distinct \( n \) and \( m \) such that
\( f^{n}(x)=f^{m}(x) \) (i.e it is eventually periodic”).
We are trying to classify for which maps are there not as many preperiodic points as their should be in dimension 1. It is a more precise version of I.N. Baker’s theorem which states “Let P be a polynomial of degree at least two and suppose that P has \( \textbf{\underline{no}} \) periodic points of period n. Then n=2 and P is conjugate to \( z^2-z \).”.

Useful information

How it works

  • Training bursts (3 to 4 weekly training sessions once every two months)
  • Weekly presentation/problem solving (moderated)
  • Bi-weekly advisor meeting
  • Students are required to present at least once per month
  • Students are required to submit an expository article once every 6 months

Some possible questions

Existence of (pre)Periodic Points. These are the topics expanding on I.N. Baker’s (1) Is there a lower bound on the number of minimal periodic points of period n in terms
of the degree of the map?
(2) What about Morton’s criteria for using generalized dynatomic polynomials (i.e.,
preperiodic points)? i.e., find a polynomial which vanishes at the c values where
there is collapse of (m, n) periodic points.
(3) What existence of periodic points for fields with characteristic p > 0? (such as finite
fields, or p-adic fields)
(4) In higher dimensions, are there always periodic points of every period? See Ivashkovich.
What about morphisms versus rational maps?

Papers

  (1) Silverman Arithmetic of Dynamical Systems: p 165. Bifurcation polynomials. Exercise 4.12 outlines some known properties and open questions (** = open question).

(2) Patrick Morton: Arithmetic properties of periodic points of quadratic maps, II –
article describes where there is collapse for the family of quadratic polynomials
(3) Vivaldi and Morton: Bifurcations and discriminants for polynomial maps
(4) Hagihara – Quadratic rational maps lacking period 2 orbits
(5) John Doyle https://arxiv.org/abs/1501.06821 – a related topic but not quite the
same
(6) https://arxiv.org/abs/1703.04172 – perhaps related to the question in characteristic p > 0.
(7) On fixed points of rational self-maps of complex projective plane – Ivashkovich http:
//arxiv.org/abs/0911.5084 Its the only goal is to provide examples of rational selfmaps of complex projective plane of any given degree without (holomorphic) fixed
points. This makes a contrast with the situation in one dimension.

Books

  • Abstract Algebra (Dummit Foote)
  • Commutative Algebra (Atiyah)
  • An Invitation to Algebraic Geometry (Smith Kahanpaa)
  • The Arithmetic of Dynamical Systems (Silverman)