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, padic 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.)  


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


Publications  
[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, LiangChung 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, LiangChung 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 fieldofdefinition/fieldofmoduli 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 RubinsteinSalzedo, 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 halfplane
with Michael Ching J. Comb. 3 (2012), no. 1, 1–48. (journal  arXiv) 


Thesis  
[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]. 



Coauthors 
Talia Blum (Stanford)  Andrew Bridy (Yale) 
Michael Ching (Amherst)  Xander Faber (IDA/CCS) 
Paul Fili (Oklahoma State)  Dragos Ghioca (British Columbia) 
LiangChung 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 RubinsteinSalzedo (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 