# zbMATH — the first resource for mathematics

Integer points in backward orbits. (English) Zbl 1246.37102
Summary: A theorem of J. Silverman states that a forward orbit of a rational map $$\varphi (z)$$ on $$\mathbb P^1(K)$$ contains finitely many $$S$$-integers in the number field $$K$$ when ($$\varphi \circ\varphi )(z)$$ is not a polynomial. We state an analogous conjecture for the backward orbits using a general $$S$$-integrality notion based on the Galois conjugates of points. This conjecture is proven for the map $$\varphi (z)=z^d$$, and consequently Chebyshev polynomials, by uniformly bounding the number of Galois orbits for $$z^n - \beta$$ when $$\beta \neq 0$$ is a non-root of unity. In general, our conjecture is true provided that the number of Galois orbits for $$\varphi ^n(z) - \beta$$ is bounded independently of $$n$$.

##### MSC:
 37P35 Arithmetic properties of periodic points 11S82 Non-Archimedean dynamical systems
Full Text:
##### References:
 [1] Baker, A., Transcendental number theory, (1975), Cambridge University Press Cambridge · Zbl 0297.10013 [2] Baker, M., A lower bound for average values of dynamical greenʼs functions, Math. res. lett., 13, 2-3, 245-257, (2006) · Zbl 1173.11041 [3] Baker, M.; Ih, S.; Rumley, R., A finiteness property of torsion points, Algebra number theory, 2, 2, 217-248, (2008) · Zbl 1182.11030 [4] Baker, M.; Rumely, R., Equidistribution of small points, rational dynamics, and potential theory, Ann. inst. Fourier (Grenoble), 56, 3, 625-688, (2006) · Zbl 1234.11082 [5] Bombieri, E.; Guber, W., Heights in Diophantine geometry, (2006), Cambridge University Press Cambridge [6] Chambert-Loir, A., Mesures et équidistribution sur LES espaces de berkovich, J. reine angew. math., 595, 215-235, (2006) · Zbl 1112.14022 [7] Dobrowolski, E., On a question of Lehmer and the number of irreducible factors of a polynomial, Acta arith., 34, 4, 391-401, (1979) · Zbl 0416.12001 [8] Favre, C.; Rivera-Letelier, J., Equidistribution quantitative des points de petite hauteur sur la droite projective, Math. ann., 335, 2, 311-361, (2006) · Zbl 1175.11029 [9] Ghioca, D.; Tucker, T.J., Equidistribution and integral points for Drinfeld modules, Trans. amer. math. soc., 360, 9, 4863-4887, (2008) · Zbl 1178.11046 [10] Jones, R., The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. lond. math. soc. (2), 78, 2, 523-544, (2008) · Zbl 1193.37144 [11] Lang, S., Algebra, Grad. texts in math., vol. 211, (2002), Springer-Verlag New York · Zbl 0984.00001 [12] Lehmer, D.H., Factorization of certain cyclotomic functions, Ann. of math. (2), 34, 3, 461-479, (1933) · Zbl 0007.19904 [13] Lyubich, M., Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic theory dynam. systems, 3, 351-385, (1983) · Zbl 0537.58035 [14] Masser, D.W., Counting points of small height on elliptic curves, Bull. soc. math. France, 117, 2, 247-265, (1989) · Zbl 0723.14026 [15] Petsche, C., S-integral preperiodic points for dynamical systems over number fields, Bull. lond. math. soc., 40, 5, 749-758, (2008) · Zbl 1243.11073 [16] Silverman, J.H., Integer points, Diophantine approximation, and iteration of rational maps, Duke math. J., 71, 3, 793-829, (1993) · Zbl 0811.11052 [17] Silverman, J.H., The arithmetic of dynamical systems, (2007), Springer New York · Zbl 1130.37001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.