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?
1.2. Automorphisms. Readings
(1) deFaria-Hutz : classification/compuation of automorphism groups (https://arxiv. org/abs/1509.06670)
(2) Manes-Silverman arxiv:1607.05772 classification of degree 2 in P 2 with automorphisms and states a number of other open questions in Section 3 that seem tractable.
1 Possible questions: I will have a group of undergrads working on automorphism related questions this summer, so we’ll need to see what they do or do not answer (1) Dimension 1: what about existence/size of automorphim groups in characterstic p > 0. Faber https://arxiv.org/abs/1112.1999. There are a number of questions that could be asked similar to deFaria-Hutz in characteristic p.
(2) Classification of birational automorphism groups (See Manes-Silverman https:// arxiv.org/abs/1607.05772)
(3) Better bound on Field of Moduli degree (see Doyle-Silverman https://arxiv.org/ abs/1804.00700)
You can refer these articles for Arithmetical Dynamics problems.