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Learn MoreToday, 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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