John R. Doyle              

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Research Interests

My main research focus is in the area of arithmetic dynamics, though my work also involves arithmetic geometry and algebraic number theory.

Arithmetic dynamics is the study of the dynamics of rational functions defined over fields of arithmetic interest, like the rational numbers, number fields, p-adic fields, and function fields. Much of my work has involved dynamical moduli spaces, which parametrize classes of rational maps together with "level structure," which in the dynamical setting typically refers to marked preperiodic points. My interest in these spaces comes from their applications to one of the major open problems in arithmetic dynamics: the dynamical uniform boundedness conjecture of Morton and Silverman.

All articles are available on the arXiv
(Note that the arXiv version may differ slightly from the published version.)

[21] New families satisfying the dynamical uniform boundedness principle over function fields
with Xander Faber
[20] Polynomials with many rational preperiodic points
with Trevor Hyde
[19] Stochastic equidistribution and generalized adelic measures
with Paul Fili and Bella Tobin
You can find some images of stochastic Julia sets here.

[18] Dynatomic polynomials, necklace operators, and universal relations for dynamical units
with Paul Fili and Trevor Hyde
New York J. Math. 28 (2022), 534–556. (journal | arXiv)
[17] Dynamical moduli spaces and polynomial endomorphisms of configurations
with Talia Blum, Trevor Hyde, Colby Kelln, Henry Talbott, and Max Weinreich
Arnold Math. J., published electronically (2022). (journal | arXiv)
[16] Multivariate polynomial values in difference sets
with Alex Rice
Discrete Anal. 2021:11, 46pp. (journal | arXiv)
[15] A question for iterated Galois groups in arithmetic dynamics
with Andrew Bridy, Dragos Ghioca, Liang-Chung Hsia, and Thomas J. Tucker
Canad. Math. Bull. 64 (2021), no. 2, 401–417. (journal | arXiv)
[14] Finite index theorems for iterated Galois groups of unicritical polynomials
with Andrew Bridy, Dragos Ghioca, Liang-Chung Hsia, and Thomas J. Tucker
Trans. Amer. Math. Soc. 374 (2021), no. 1, 733–752. (journal | arXiv)
[13] Moduli spaces for dynamical systems with portraits
with Joseph H. Silverman
Illinois J. Math. 64 (2020), no. 3, 375–465. (journal | arXiv)
[12] Preperiodic points for quadratic polynomials over cyclotomic quadratic fields
Acta Arith. 196 (2020), no. 3, 219–268. (journal | arXiv)
[11] Gonality of dynatomic curves and strong uniform boundedness of preperiodic points
with Bjorn Poonen
Compos. Math. 156 (2020), 733–743. (journal | arXiv)
[10] A uniform field-of-definition/field-of-moduli bound for dynamical systems on P^N
with Joseph H. Silverman
J. Number Theory 195 (2019), 1–22. (journal | arXiv)
[9] Dynamical modular curves for quadratic polynomial maps
Trans. Amer. Math. Soc. 371 (2019), no. 8, 5655–5685. (journal | arXiv)
[8] Reduction of dynatomic curves
with Holly Krieger, Andrew Obus, Rachel Pries, Simon Rubinstein-Salzedo, and Lloyd West
Ergodic Theory Dynam. Systems 39 (2019), no. 10, 2717–2768. (journal | arXiv)
[7] Preperiodic points for quadratic polynomials with small cycles over quadratic fields
Math. Z. 289 (2018), no. 1–2, 729–786. (journal | arXiv)
[6] Preperiodic portraits for unicritical polynomials over a rational function field
Trans. Amer. Math. Soc. 370 (2018), no. 5, 3265–3288. (journal | arXiv)
[5] Configuration of the crucial set for a quadratic rational map
with Kenneth Jacobs and Robert Rumely
Res. Number Theory 2 (2016), 2:11. (journal | arXiv)
[4] Preperiodic portraits for unicritical polynomials
Proc. Amer. Math. Soc. 144 (2016), no. 7, 2885–2899. (journal | arXiv)
[3] Computing algebraic numbers of bounded height
with David Krumm
Math. Comp. 84 (2015), no. 296, 2867–2891. (journal | arXiv)
[2] Preperiodic points for quadratic polynomials over quadratic fields
with Xander Faber and David Krumm
New York J. Math. 20 (2014), 507–605. (journal | arXiv)
[1] Apollonian circle packings of the half-plane
with Michael Ching
J. Comb. 3 (2012), no. 1, 1–48. (journal | arXiv)

[0] Dynamics of quadratic polynomials over quadratic fields (pdf; 210 pages)
The results of my thesis appear in the three articles [7], [9], and [12], and the interested reader is encouraged to read those instead. The main purpose of posting my thesis here is that the details of certain computations are omitted in [7], but they appear in full in my thesis.

If you decide to peruse my thesis, please be aware that there are some minor errors (though nothing that invalidates the main results) that have been fixed for publication. Also note that some of the notation — specifically, the names of the dynamical modular curves — has since been changed and will appear differently in the published articles. Finally, some of the results have since been strengthened. In particular, the proof of Conjecture 2.34 appears in [9].

Talia Blum (Stanford) Andrew Bridy (Yale)
Michael Ching (Amherst) Xander Faber (IDA/CCS)
Paul Fili (Oklahoma State) Dragos Ghioca (British Columbia)
Liang-Chung Hsia (NTNU) Trevor Hyde (Chicago)
Ken Jacobs Colby Kelln (Cornell)
Holly Krieger (Cambridge) David Krumm (Reed)
Andrew Obus (CUNY Baruch College) Bjorn Poonen (MIT)
Rachel Pries (Colorado State) Alex Rice (Millsaps)
Simon Rubinstein-Salzedo (Euler Circle) Robert Rumely (Georgia)
Joseph H. Silverman (Brown) Henry Talbott (Michigan)
Bella Tobin (Oklahoma State) Thomas J. Tucker (Rochester)
Max Weinreich (Brown) Lloyd West