 Today, we are going to discuss two possible problems for Arithmetical Dynamics in this post.

1.1. Existence of (pre)Periodic Points. These are the topics expanding on I.N. Baker’s theorem. Related reading:

(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.

Some possible questions:

(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?