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Arithmetical Dynamics: Two possible problems

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.

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