Rational periodic points for quadratic maps

Jung Kyu Canci[1]

  • [1] Université Lille 1 Laboratoire Paul Painlevé, Mathématiques 59655 Villeneuve d’Ascq Cedex (France)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 3, page 953-985
  • ISSN: 0373-0956

Abstract

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Let K be a number field. Let S be a finite set of places of K containing all the archimedean ones. Let R S be the ring of S -integers of K . In the present paper we consider endomorphisms of ℙ 1 of degree 2 , defined over K , with good reduction outside S . We prove that there exist only finitely many such endomorphisms, up to conjugation by PGL 2 ( R S ) , admitting a periodic point in ℙ 1 ( K ) of order > 3 . Also, all but finitely many classes with a periodic point in ℙ 1 ( K ) of order 3 are parametrized by an irreducible curve.

How to cite

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Canci, Jung Kyu. "Rational periodic points for quadratic maps." Annales de l’institut Fourier 60.3 (2010): 953-985. <http://eudml.org/doc/116297>.

@article{Canci2010,
abstract = {Let $K$ be a number field. Let $S$ be a finite set of places of $K$ containing all the archimedean ones. Let $R_S$ be the ring of $S$-integers of $K$. In the present paper we consider endomorphisms of $\mathbb\{P\}_1$ of degree $2$, defined over $K$, with good reduction outside $S$. We prove that there exist only finitely many such endomorphisms, up to conjugation by $\{\rm PGL\}_2(R_S)$, admitting a periodic point in $\mathbb\{P\}_1(K)$ of order $&gt;3$. Also, all but finitely many classes with a periodic point in $\mathbb\{P\}_1(K)$ of order $3$ are parametrized by an irreducible curve.},
affiliation = {Université Lille 1 Laboratoire Paul Painlevé, Mathématiques 59655 Villeneuve d’Ascq Cedex (France)},
author = {Canci, Jung Kyu},
journal = {Annales de l’institut Fourier},
keywords = {Rational maps; moduli spaces; $S$-unit equations; reduction modulo $\mathfrak\{p\}$; rational maps; -unit equations; reduction modulo },
language = {eng},
number = {3},
pages = {953-985},
publisher = {Association des Annales de l’institut Fourier},
title = {Rational periodic points for quadratic maps},
url = {http://eudml.org/doc/116297},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Canci, Jung Kyu
TI - Rational periodic points for quadratic maps
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 3
SP - 953
EP - 985
AB - Let $K$ be a number field. Let $S$ be a finite set of places of $K$ containing all the archimedean ones. Let $R_S$ be the ring of $S$-integers of $K$. In the present paper we consider endomorphisms of $\mathbb{P}_1$ of degree $2$, defined over $K$, with good reduction outside $S$. We prove that there exist only finitely many such endomorphisms, up to conjugation by ${\rm PGL}_2(R_S)$, admitting a periodic point in $\mathbb{P}_1(K)$ of order $&gt;3$. Also, all but finitely many classes with a periodic point in $\mathbb{P}_1(K)$ of order $3$ are parametrized by an irreducible curve.
LA - eng
KW - Rational maps; moduli spaces; $S$-unit equations; reduction modulo $\mathfrak{p}$; rational maps; -unit equations; reduction modulo
UR - http://eudml.org/doc/116297
ER -

References

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  1. Robert L. Benedetto, Reduction, dynamics, and Julia sets of rational functions, J. Number Theory 86 (2001), 175-195 Zbl0978.37039MR1813109
  2. B. J. Birch, J. R. Merriman, Finiteness theorems for binary forms with given discriminant, Proc. London Math. Soc. (3) 24 (1972), 385-394 Zbl0248.12002MR306119
  3. Enrico Bombieri, Walter Gubler, Heights in Diophantine Geometry, (2006), Cambridge University Press, Cambridge Zbl1115.11034MR2216774
  4. Jung Kyu Canci, Cycles for rational maps with good reduction outside a prescribed set, Monatsh. Math. 149 (2007), 265-287 Zbl1171.11041MR2284648
  5. Pietro Corvaja, Umberto Zannier, A lower bound for the height of a rational function at S -unit points, Monatsh. Math. 144 (2005), 203-224 Zbl1086.11035MR2130274
  6. Laura DeMarco, Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity, Math. Ann. 326 (2003), 43-73 Zbl1032.37029MR1981611
  7. J.-H. Evertse, K. Győry, Effective finiteness results for binary forms with given discriminant, Compositio Math. 79 (1991), 169-204 Zbl0746.11020MR1117339
  8. Jan-Hendrik Evertse, On sums of S -units and linear recurrences, Compositio Math. 53 (1984), 225-244 Zbl0547.10008MR766298
  9. Marc Hindry, Joseph H. Silverman, Diophantine Geometry, 201 (2000), Springer-Verlag, New York Zbl0948.11023MR1745599
  10. Serge Lang, Algebra, 211 (2002), Springer-Verlag, New York Zbl0984.00001MR1878556
  11. Daniel A. Marcus, Number fields, (1977), Springer-Verlag, New York Zbl0383.12001MR457396
  12. John Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math. 2 (1993), 37-83 Zbl0922.58062MR1246482
  13. Patrick Morton, Joseph H. Silverman, Rational periodic points of rational functions, Internat. Math. Res. Notices (1994), 97-110 Zbl0819.11045MR1264933
  14. Patrick Morton, Joseph H. Silverman, Periodic points, multiplicities, and dynamical units, J. Reine Angew. Math. 461 (1995), 81-122 Zbl0813.11059MR1324210
  15. A. J. van der Poorten, H. P. Schlickewei, The growth condition for recurrence sequences, (1982) 
  16. Wolfgang Schmidt, Diophantine Approximation, 785 (1980), Springer, Berlin Zbl0421.10019MR568710
  17. Wolfgang M. Schmidt, Diophantine approximations and Diophantine equations, 1467 (1991), Springer-Verlag, Berlin Zbl0754.11020MR1176315
  18. Jean-Pierre Serre, Lectures on the Mordell-Weil Theorem, (1997), Friedr. Vieweg & Sohn, Braunschweig Zbl0676.14005MR1757192
  19. Joseph H. Silverman, The space of rational maps on ℙ 1 , Duke Math. J. 94 (1998), 41-77 Zbl0966.14031MR1635900
  20. Joseph H. Silverman, The arithmetic of dynamical systems, 241 (2007), Springer, New York Zbl1130.37001MR2316407

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