Summary of Included Data

The database has 593 distinct maps (one per conjugacy class). By degree, type and field of definition:

How much is known about each problem is on the Status of Problems page.

Data sources

This file lists the data and data sources.

Published Sources

• Quadratic polynomials z^2 + c
  • Over QQ
    • Bjorn Poonen. The classificiation of rational preperiodic points of quadratic polynomials over Q: a refined conjecture. Math. Z., 228(1):11--29, 1998
      • The examples contained in the main table
    • Ralph Walde and Paula Russo. Rational periodic points of the quadratic function Qc(x) = x^2 + c. Amer. Math. Monthly, 101(4):318--331, 1994.
      • Table 1 (12 maps with rational fixed points and a rational 2-cycle), Table 2 (8 maps with a rational 3-cycle) and Example 2 (x^2 - 2). Their graphs are all already in the database; the maps already included get this citation (x^2 - 21/16, x^2 - 29/16, x^2 - 301/144, x^2 - 2), and the other 16 are not included
    • Patrick Morton. Arithmetic properties of periodic points of quadratic maps. Acta Arith., 62(4):343--372, 1992.
      • The examples of the factorization and Galois group of the period 3 dynatomic polynomial: x^2 + a for a = 0, 1, -7/2, -1/8 over QQ, a = -1 - 11 sqrt(3)/24 over Q(sqrt(3)) and a = -(67 +- sqrt(17))/32 over Q(sqrt(17)). Their graphs are all already in the database; x^2 and x^2 + 1 are included and get this citation, the others are not included
  • Quadratic Number Fields
    • 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. (arXiv:0909.5050)
      • x^2 - 29/16 (a rational 3-cycle) and x^2 - 71/48 (a 6-cycle over Q(sqrt(33))), both already included, so they get this citation. The example c = (omega + 1)/4 over Q(sqrt(-3)) is not included: its graph is already in the database
    • Michael Stoll. Rational 6-cycles under iteration of quadratic polynomials. LMS J. Comput. Math., 11:367--380, 2008. (arXiv:0803.2836)
      • The 6-cycle of x^2 - 71/48 over Q(sqrt(33)), already included, so it gets this citation
    • John R. Doyle, Xander Faber, and David Krumm. Preperiodic points for quadratic polynomials over quadratic fields. New York J. Math., 20:507--605, 2014
      • The examples contained in the appendix
    • John R. Doyle. Preperiodic points for quadratic polynomials with small cycles over quadratic fields. Math. Z., 289(1-2):729--786, 2018. (arXiv:1509.07098)
      • Appendix A.2 (representative data): the same 67 (field, c) examples as Doyle, Faber, and Krumm's appendix, all included, each over its own quadratic field (the 30 maps already included get this citation)
    • John R. Doyle. Preperiodic points for quadratic polynomials over cyclotomic quadratic fields. Acta Arith., 196(3):219--268, 2020. (arXiv:1801.09003)
      • Appendix B, Table 1: a representative c for each of the 17 graphs realized over Q(i) or Q(omega). The maps already included get this citation (x^2 + c over QQ for c = 1/4, 0, -1, -2, -10/9, -13/9, -301/144, -21/16, -29/16; c = i, -1/4 + 3i/8 over Q(i); c = 2, -5/12 over Q(omega)). New graph structures: x^2 + 1/4 over Q(omega) (4(1)) and over Q(i) (6(2,1)). The representatives c = -6 and c = -3 are not included: their graphs are already in the database
• Cubic Polynomials
  • Over QQ
    • Robert Benedetto, Ben Dickman, Sasha Joseph, Ben Krause, Dan Rubin, and Xinwen Zhou. Computing points of small height for cubic polynomials. Involve, 2:37--64, 2009.
      • 103 distinct preperiodic graphs from https://mgolech.github.io/cubics/index.html
• Polynomials of degree 2 through 15
  • Over QQ
    • John R. Doyle and Trevor Hyde. Polynomials with many rational preperiodic points. Trans. Amer. Math. Soc., 378(8):5981--6011, 2025. (arXiv:2201.11707)
      • Table 2 (degrees 2-9, given explicitly) and Table 4 (degrees 10-15, given by the values f(1), ..., f(d+1); recovered by Lagrange interpolation)
      • The degree 2 and 3 examples are conjugate to maps already included (Poonen's x^2 - 29/16 and Benedetto et al.'s II18), so they add no new functions; they attach this citation to those maps
• Quadratic Rational Functions
  • Over QQ
    • 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.
      • First 10 maps from each Period X, Length Y file at https://rlbenedetto.people.amherst.edu/quadratdata/ (122 maps; the maps only, not the orbits)
    • Michelle Manes. Q-rational cycles for degree-2 rational maps having an automorphism. Proc. Lond. Math. Soc. (3), 96:669--696, 2008.
      • Section 4 and Figure 1: one map kz + b/z for each of the 12 possible graphs of rational preperiodic points of a quadratic map with automorphism group C_2 (cycles of length at most 4). New graph structures: (k, b) = (-1/3, 1/3), (-2, 1), (-5/4, 1), (1/3, -1/3), (2/3, -2/3), (-25/24, 1/6), (4/3, -2/15), (7/24, -7/6). z + 1/z and (1/2)(z + 1/z) (conjugate to z^2) are already included and get this citation; z - 1/z and 3z/4 + 1/z are not included (their graphs are already in the database). Figure 1 labels example (j) as (4/3, 2/15); it is (4/3, -2/15), as in the text
    • Jérémy Blanc, Jung Kyu Canci, and Noam D. Elkies. Moduli spaces of quadratic rational maps with a marked periodic point of small order. Int. Math. Res. Not. IMRN, 2015(23):12459--12489, 2015. (arXiv:1305.1054)
      • Maps with a rational 6-cycle. The family of Lemma 4.7 (a 6-cycle and a rational fixed point) has a new graph structure, a 6-cycle and a fixed point each with one rational preimage (14 points); its member of smallest height, p = 1/2, is included. Example 4.10 and the maps of the two rational points of M_2(6) printed in Section 4.3 (a 6-cycle with one preimage of each point, 12 points) have a graph already in the database and are not included. The paper prints those two points in the order [X:Y:Z:W]
    • Jung Kyu Canci and Solomon Vishkautsan. Quadratic maps with a periodic critical point of period 2. Int. J. Number Theory, 13(6):1393--1417, 2017. (arXiv:1510.04726)
      • The graph tables (Section 6, arXiv version): one map for each of the 13 graphs of rational preperiodic points of a quadratic map with a rational periodic critical point of period 2 (6 PCF maps from Lukas-Manes-Yap and 7 others; the classification assumes no rational periodic points of period at least 7), plus the paper's maps for critical period 3 (6 maps) and 4 (5 maps). New graph structures: 2/z^2 (a twist of 1/z^2), the 7 non-PCF period 2 maps (3 to 9 points) and the 5 period 4 maps (7 to 13 points). The other maps are already included (the Lukas-Manes-Yap maps, z^2 - 1, and the period 3 maps of Vishkautsan below), so they add this citation
    • Solomon Vishkautsan. Quadratic rational functions with a rational periodic critical point of period 3. With an appendix by Michael Stoll. J. Théor. Nombres Bordeaux, 31(1):49--79, 2019. (arXiv:1711.06345)
      • Table 1: one map for each of the 6 graphs of rational preperiodic points of a quadratic map with a rational periodic critical point of period 3 (the classification is conditional). R3P1 through R3P5 (5 to 11 points) are new graph structures; R3P0, 1/(z-1)^2, is the Lukas-Manes-Yap PCF map already included, so it adds this citation
    • Hasan Bilgili and Mohammad Sadek. Rational preperiodic points of quadratic rational maps over Q with nonabelian automorphism groups. arXiv:2603.06203, 2026.
      • The examples of each of the 5 portraits (Table 1) of rational preperiodic points of a quadratic map over QQ with automorphism group S_3; all such maps are twists of 1/z^2. Three are new graph structures: 6z/(z^2 - 3) (three fixed points, 6 points), 2z/(z^2 - 1) (4 points) and the map with (d, k) = (-1031116321/672121026723, 1) (a 3-cycle, 6 points). The example with a fixed point and a 2-cycle is conjugate over QQ to 1/z^2, already included, so it adds this citation. The example with 2 points is not included: its graph is already in the database and the map is a twist of 1/z^2
• Polynomials of degree 2 through 13 and rational functions of degree 2 through 5
  • Over QQ
    • Benjamin Hutz. A genetic algorithm for generating extreme examples in arithmetic dynamics. arXiv:2601.11482, 2026.
      • The maps in the appendix (Extended Data Set) tables Many Preperiodic Points, Long Periodic Cycles, and Long Preperiodic Tail
      • Each map is recovered by interpolation from the orbit the paper lists for it (164 distinct maps: 125 polynomials, 39 rational functions)
      • Small Height Ratio tables: the 3 smallest ratios for each degree (polynomials of degree 2 through 12, rational functions of degree 2 through 5; 2 maps for quadratic rational functions, and the degree 13 table is empty), 44 maps. The stored smallest height is the ratio of the point 0, the start of the paper's orbit. The records for degree 2 and 3 polynomials and degree 2 rational maps match the best previously known (the paper's summary table) and are also credited to those searches: Benedetto, Chen, Hyde, Kovacheva and White (2014) for degree 2, Benedetto et al. (2009) for cubic polynomials
      • Plus the degree 6 polynomial record of the paper's Section 5 summary table, orbit [0, 1, -1, -4, -7, -3, -6, -9] (ratio 1.8372 * 10^-9), which the appendix doesn't list; every other record in that table is the appendix's best
      • Two misprints in the Small Height Ratio tables are corrected: for degree 2 polynomials the orbit [0, 1, -1, 2] (ratio 0.01102) should be [0, 1, -2, 6], the orbit of the map printed with it; for degree 11 the ratio 7.8701 * 10^17 should be 7.8701 * 10^-17
• Unicritical polynomials x^d + c over number fields
  • John R. Doyle and Wade Hindes. Preperiodic integers for x^d+c in large degree. arXiv:2510.14067, 2025.
    • Table 1 (Theorem 1.1): the 12 nonempty skeletons of the rational preperiodic points of x^d + c for algebraic integers c and large d, one map for each, found from the conditions in the proofs (arXiv:2408.14657, Section 4) in the smallest cyclotomic field and then the smallest d: (1)a x^3 - 6 over QQ, (2)b x^2 - 1 over QQ (already included, gets this citation), (1,1) x^2 + 1 over Q(zeta_3) (already included over QQ), (2)a x^4 + 1 and (1)c x^6 + zeta_3^2 over Q(zeta_3), (2,2) x^7 + zeta_12^3 - zeta_12 and (2,1,1) x^11 - zeta_12^3 over Q(zeta_12), (1)d x^6 + zeta_3 and (3) x^7 + zeta_9^5 + zeta_9^2 over Q(zeta_9), (1)b x^4 - zeta_16^4 - 1 over Q(zeta_16), (1)e x^6 + zeta_3 and (2)c x^6 - 1 over Q(zeta_36)
    • The rational preperiodic points of the last seven (fields of degree 4 to 12) are not yet computed: the standard computation does not finish (over 30 minutes each), so they need a different method
• Postcritically finite (PCF) maps over QQ - complete classifications, up to conjugacy over QQbar
  • Quadratic polynomials
    • Patrick Ingram. A finiteness result for post-critically finite polynomials. Int. Math. Res. Not. IMRN, 2012(3):524--543, 2012.
      • Section 4: z^2 + c is PCF for c in {0, -1, -2} only (these are already in the Poonen data; this adds the citation). Also the seven monic centered PCF cubics z^3 + Az + B (6 classes)
  • Cubic polynomials
    • Jacqueline Anderson, Michelle Manes, and Bella Tobin. Cubic post-critically finite polynomials defined over Q. ANTS XIV, Open Book Ser. 4:23--38, 2020.
      • All 15 classes (Theorem / Table 1)
  • Quartic polynomials
    • Benjamin Fraser. On PCF polynomials. PhD thesis, York University, 2024.
      • All 16 classes (Theorem 4.9, Table 4.1; bicritical classes in the Table 4.2 form)
  • Quadratic rational functions
    • David Lukas, Michelle Manes, and Diane Yap. A census of quadratic post-critically finite rational functions defined over Q. LMS J. Comput. Math., 17(A):314--329, 2014.
      • All 12 classes (Theorem 1), including the 3 polynomials
  • No complete classification of degree 5 (or higher) PCF polynomials over QQ was found
• Maps with a given exact automorphism group, over QQ - one for each finite subgroup of PGL_2
  • Joao Alberto de Faria and Benjamin Hutz. Automorphism groups and invariant theory on PN. J. Algebra Appl., 17:1850162, 2018. (arXiv:1509.06670)
    • Figure 1 (Exact Automorphism Groups): the octahedral (S_4, degree 5) and icosahedral (A_5, degree 11) maps
    • The cyclic maps C_n: (x^(n+1) + xy^n : y^(n+1)) and dihedral maps D_2n: (y^(n-1) : x^(n-1)) are infinite families whose degree depends on n, so the first few members are included: C_n for n = 2 through 8 (degrees 3 through 9) and D_2n for n = 3 through 8 (degrees 2 through 7). The formula gives degree 1 for D_4, so D_4 is not included
  • Brandon Gontmacher, Benjamin Hutz, Grayson Jorgenson, Srinjoy Srimani, and Simon Xu. Automorphism loci for degree 3 and degree 4 endomorphisms of the projective line. New York J. Math., 27:1613--1702, 2021.
    • The tetrahedral map (z^3 - 3)/(-3z^2) (A_4, degree 3), defined over QQ. It replaces the map of de Faria-Hutz Figure 1, which is defined over QQ(sqrt(-3)): their Theorem 4.8 says no map over QQ has tetrahedral automorphism group, and this map shows otherwise
    • One map for every automorphism locus in degrees 3 and 4: the single conjugacy classes 1/z^3 (D_4), (z^3 - 3)/(-3z^2) (A_4) and 1/z^4 (D_5), and for each family the member of smallest height whose automorphism group is exactly the family's group - degree 3: (z^3 + 1)/z^2 (C_3), (2z^2 + 1)/(z^3 + 2z) and (2z^2 - 1)/(z^3 - 2z) (C_2 x C_2), z^3/(z^2 + 1) and 1/(z^3 + z) (C_2); degree 4: (z^4 + 1)/z^3 (C_4), z^4 (S_3), z^4/(z^3 + 1) (C_3), (z^4 + 1)/z (C_2)
    • Rational preperiodic graph structures over QQ of these families (Sections 6 and 7: classifications for the one-parameter families, census tables for the others): the 106 structures not already in the database, one map each (46 for degree 3 rational maps, 59 for degree 4 rational maps, 1 for a degree 4 polynomial)
  • Masayo Fujimura and Kiyoko Nishizawa. Moduli spaces and symmetry loci of polynomial maps. In Proceedings of ISSAC 1997, pages 342--348. ACM, 1997.
    • Polynomials: a polynomial of degree d conjugate to z^d has the dihedral group of order 2(d-1) as automorphism group, and any other has a cyclic group C_m with m dividing d-1. One example of every possible group for each degree d up to 10, all coefficients 1: z^d (dihedral), z^d + z (C_(d-1)), z^d + z^(m+1) (C_m for the other divisors m > 1 of d-1)
  • Nikita Miasnikov, Brian Stout, and Phillip Williams. Automorphism loci for the moduli space of rational maps. Acta Arith., 180(3):267--296, 2017. (arXiv:1408.5655)
    • Example 1: (z^5 - 5z)/(1 - 5z^4), the single conjugacy class of degree 5 with automorphism group S_4 (its A_4 locus is the same point). It is the de Faria-Hutz octahedral map, already included, so it adds this citation
  • John Milnor. Geometry and dynamics of quadratic rational maps. Experiment. Math., 2(1):37--83, 1993.
    • The automorphism groups of quadratic rational maps (the symmetry locus: C_2, and S_3 at one point). The degree 2 rational map with automorphism group exactly C_2: z + 1/z, the smallest height example (S_3 is the de Faria-Hutz map 1/z^2)

Personal Communications

• Quadratic Rational Functions
  • Over QQ
    • Solomon Vishkautsan, personal communication Sept 2026
      • f(z) = (z^2 - 21z + 20)/(z^2 + 7z): 16 rational preperiodic points, two 3-cycles

Citations

• [Morton1992] Patrick Morton. Arithmetic properties of periodic points of quadratic maps. Acta Arith., 62(4):343--372, 1992. MR1199627
• [Milnor1993] John Milnor. Geometry and dynamics of quadratic rational maps. With an appendix by the author and Lei Tan. Experiment. Math., 2(1):37--83, 1993. MR1246482
• [WR1994] Ralph Walde and Paula Russo. Rational periodic points of the quadratic function Qc(x) = x^2 + c. Amer. Math. Monthly, 101(4):318--331, 1994. MR1270956
• [FN1997] Masayo Fujimura and Kiyoko Nishizawa. Moduli spaces and symmetry loci of polynomial maps. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), pages 342--348. ACM, New York, 1997. MR1810003
• [Poonen1998] Bjorn Poonen. The classificiation of rational preperiodic points of quadratic polynomials over Q: a refined conjecture. Math. Z., 228(1):11--29, 1998. MR1617987
• [Manes2008] Michelle Manes. Q-rational cycles for degree-2 rational maps having an automorphism. Proc. Lond. Math. Soc. (3), 96:669--696, 2008. MR2407816
• [Stoll2008] Michael Stoll. Rational 6-cycles under iteration of quadratic polynomials. LMS J. Comput. Math., 11:367--380, 2008. MR2465796
• [Benedetto2009] Robert Benedetto, Ben Dickman, Sasha Joseph, Ben Krause, Dan Rubin, and Xinwen Zhou. Computing points of small height for cubic polynomials. Involve, 2:37--64, 2009. MR2501344
• [Ingram2012] Patrick Ingram. A finiteness result for post-critically finite polynomials. Int. Math. Res. Not. IMRN, 2012(3):524--543, 2012. MR2885981
• [Hutz2013] 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. MR3065461
• [BCHKW2014] 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. MR3277939
• [Doyle2014] John R. Doyle, Xander Faber, and David Krumm. Preperiodic points for quadratic polynomials over quadratic fields. New York J. Math., 20:507--605, 2014. MR3218788
• [Lukas2014] David Lukas, Michelle Manes, and Diane Yap. A census of quadratic post-critically finite rational functions defined over Q. LMS J. Comput. Math., 17(A):314--329, 2014. MR3240812
• [BCE2015] Jérémy Blanc, Jung Kyu Canci, and Noam D. Elkies. Moduli spaces of quadratic rational maps with a marked periodic point of small order. Int. Math. Res. Not. IMRN, 2015(23):12459--12489, 2015. MR3431627
• [CV2017] Jung Kyu Canci and Solomon Vishkautsan. Quadratic maps with a periodic critical point of period 2. Int. J. Number Theory, 13(6):1393--1417, 2017. MR3656200
• [Miasnikov2017] Nikita Miasnikov, Brian Stout, and Phillip Williams. Automorphism loci for the moduli space of rational maps. Acta Arith., 180:267--296, 2017. MR3709645
• [Doyle2018] John R. Doyle. Preperiodic points for quadratic polynomials with small cycles over quadratic fields. Math. Z., 289(1-2):729--786, 2018. MR3803810
• [dFH2018] Joao Alberto de Faria and Benjamin Hutz. Automorphism groups and invariant theory on PN. J. Algebra Appl., 17:1850162, 38 pp., 2018. MR3846410
• [Vishkautsan2019] Solomon Vishkautsan. Quadratic rational functions with a rational periodic critical point of period 3. With an appendix by Michael Stoll. J. Théor. Nombres Bordeaux, 31(1):49--79, 2019. MR3994719
• [AMT2020] Jacqueline Anderson, Michelle Manes, and Bella Tobin. Cubic post-critically finite polynomials defined over Q. In ANTS XIV - Proceedings of the Fourteenth Algorithmic Number Theory Symposium, volume 4 of Open Book Ser., pages 23--38. Math. Sci. Publ., Berkeley, CA, 2020. MR4235104
• [Doyle2020] John R. Doyle. Preperiodic points for quadratic polynomials over cyclotomic quadratic fields. Acta Arith., 196(3):219--268, 2020. MR4161298
• [GHJSX2021] Brandon Gontmacher, Benjamin Hutz, Grayson Jorgenson, Srinjoy Srimani, and Simon Xu. Automorphism loci for degree 3 and degree 4 endomorphisms of the projective line. New York J. Math., 27:1613--1702, 2021. MR4359208
• [Fraser2024] Benjamin Fraser. On PCF polynomials. PhD thesis, York University, Toronto, 2024.
• [DoHi2025] John R. Doyle and Wade Hindes. Preperiodic integers for x^d+c in large degree. arXiv:2510.14067, 2025.
• [DoHy2025] John R. Doyle and Trevor Hyde. Polynomials with many rational preperiodic points. Trans. Amer. Math. Soc., 378(8):5981--6011, 2025. MR4929863
• [BS2026] Hasan Bilgili and Mohammad Sadek. Rational preperiodic points of quadratic rational maps over Q with nonabelian automorphism groups. arXiv:2603.06203, 2026.
• [Hutz2026] Benjamin Hutz. A genetic algorithm for generating extreme examples in arithmetic dynamics. arXiv:2601.11482, 2026.
• [Vishkautsan2026] Solomon Vishkautsan, personal communication Sept 2026