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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)