Here is a list of articles that relate to the dynamics of quadratic rational maps. Some are not arithmetic in nature, but are important for arithmetic considerations as well. Let me know if there are of articles on this topic you think should be in the list.
| [1] | Bjorn Poonen. The classification of rational preperiodic points of quadratic polynomials over Q: a refined conjecture. Math. Z., 228(1):11-29, 1998. [ bib | DOI | http ] |
| [2] | Michael Stoll. Rational 6-cycles under iteration of quadratic polynomials. LMS J. Comput. Math., 11:367-380, 2008. [ bib | DOI | http ] |
| [3] | E. V. Flynn, Bjorn Poonen, and Edward F. Schaefer. Cycles of quadratic polynomials and rational points on a genus-2 curve. Duke Math. J., 90(3):435-463, 1997. [ bib | DOI | http ] |
| [4] | Benjamin Hutz and Patrick Ingram. On Poonen’s conjecture concerning rational preperiodic points of quadratic maps. Rocky Mountain J. Math., 43(1):193-204, 2013. [ bib | DOI | http ] |
| [5] | Xander Faber, Benjamin Hutz, Patrick Ingram, Rafe Jones, Michelle Manes, Thomas J. Tucker, and Michael E. Zieve. Uniform bounds on pre-images under quadratic dynamical systems. Math. Res. Lett., 16(1):87-101, 2009. [ bib | DOI | http ] |
| [6] | Patrick Morton. On certain algebraic curves related to polynomial maps. Compositio Math., 103(3):319-350, 1996. [ bib | http ] |
| [7] | Ralph Walde and Paula Russo. Rational periodic points of the quadratic function Qc(x)=x2+c. Amer. Math. Monthly, 101(4):318-331, 1994. [ bib | DOI | http ] |
| [8] | Xander Faber. Benedetto’s trick and existence of rational preperiodic structures for quadratic polynomials. Proc. Amer. Math. Soc., 143(2):685-694, 2015. [ bib | DOI | http ] |
| [9] | Robert L. Benedetto, Ruqian Chen, Trevor Hyde, Yordanka Kovacheva, and Colin White. Small dynamical heights for quadratic polynomials and rational functions. Exp. Math., 23(4):433-447, 2014. [ bib | DOI | http ] |
| [10] | Laura DeMarco. The moduli space of quadratic rational maps. J. Amer. Math. Soc., 20(2):321-355, 2007. [ bib | DOI | http ] |
| [11] | Jung Kyu Canci. Rational periodic points for quadratic maps. Ann. Inst. Fourier (Grenoble), 60(3):953-985, 2010. [ bib | http ] |
| [12] | Timo Erkama. Arithmetic progressions in cycles of quadratic polynomials. Rev. Roumaine Math. Pures Appl., 54(5-6):441-450, 2009. [ bib ] |
| [13] | Dustin Gage and Daniel Jackson. Computer generated images for quadratic rational maps with a periodic critical point. Acta Math. Acad. Paedagog. Nyházi. (N.S.), 27(1):77-88, 2011. [ bib ] |
| [14] | Michelle Manes. Moduli spaces for families of rational maps on P1. J. Number Theory, 129(7):1623-1663, 2009. [ bib | DOI | http ] |
| [15] | Michelle Manes. Q-rational cycles for degree-2 rational maps having an automorphism. Proc. Lond. Math. Soc. (3), 96(3):669-696, 2008. [ bib | DOI | http ] |
| [16] | Patrick Morton. Arithmetic properties of periodic points of quadratic maps. II. Acta Arith., 87(2):89-102, 1998. [ bib ] |
| [17] | Patrick Morton. Arithmetic properties of periodic points of quadratic maps. Acta Arith., 62(4):343-372, 1992. [ bib ] |
| [18] | Clayton Petsche and Brian Stout. On quadratic rational maps with prescribed good reduction. Proc. Amer. Math. Soc., 143(3):1145-1158, 2015. [ bib | DOI | http ] |
| [19] | John R. Doyle, Xander Faber, and David Krumm. Preperiodic points for quadratic polynomials over quadratic fields. New York J. Math., 20:507-605, 2014. [ bib | .html ] |
| [20] | Jan Kiwi. Puiseux series dynamics of quadratic rational maps. Israel J. Math., 201(2):631-700, 2014. [ bib | DOI | http ] |
| [21] | Robert L. Devaney, Núria Fagella, Antonio Garijo, and Xavier Jarque. Sierpiński curve Julia sets for quadratic rational maps. Ann. Acad. Sci. Fenn. Math., 39(1):3-22, 2014. [ bib | DOI | http ] |
| [22] | Selim Berker, Adam L. Epstein, and Kevin M. Pilgrim. Remarks on the period three cycles of quadratic rational maps. Nonlinearity, 16(1):93-100, 2003. [ bib | DOI | http ] |
| [23] | John Milnor. Geometry and dynamics of quadratic rational maps. Experiment. Math., 2(1):37-83, 1993. With an appendix by the author and Lei Tan. [ bib | http ] |
| [24] | David Lukas, Michelle Manes, and Diane Yap. A census of quadratic post-critically finite rational functions defined over Q. LMS Journal of Computation and Mathematics, 17:314-329, 2014. [ bib | DOI | http ] |
| [25] | J. Blanc, J.K. Canci, and N. D. Elkies. Moduli spaces of quadratic rational maps with a marked periodic point of small order. 2014. [ bib | http ] |
| [26] | Zhiming Wang and Robin Zhang. On quadratic periodic points of quadratic polynomials. 2015. [ bib | http ] |
| [27] | Thierry Bousch. Sur quelques problèmes de dynamique holomorphe. PhD thesis, Universite de Paris-Sud, 1992. [ bib | .html ] |
| [28] | Chatchawan Panraksa. Arithmetic dynamics of quadratic polynomials and dynamical units. 2011. [ bib | .pdf ] |
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Solomon Vishkautsan 