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The Dynamical and Arithmetical Degrees for Eigensystems of Rational Self-maps

  • Jorge MelloEmail author
Article

Abstract

We define arithmetical and dynamical degrees for dynamical systems with several rational maps on smooth projective varieties, study their properties and relations, and prove the existence of a canonical height function associated with divisorial relations in the Néron-Severi Group over Global fields of characteristic zero, when the rational maps are morphisms. For such, we show that for any Weil height \(h_X^+ = \max \{1, h_X\}\) with respect to an ample divisor on a smooth projective variety X, any dynamical system \({\mathcal {F}}\) of rational self-maps on X with dynamical degree \(\delta _{{\mathcal {F}}}\), \({\mathcal {F}}_n\) its set of \(n-\)iterates, and any \(\epsilon >0\), there is a positive constant \(C=C(X, h_X, {\mathcal {F}}, \epsilon )\) such that
$$\begin{aligned} \mathop \sum \limits _{f \in {\mathcal {F}}_n} h^+_X(f(P)) \le C. k^n.(\delta _{{\mathcal {F}}} + \epsilon )^n . h^+_X(P) \end{aligned}$$
for all points P whose \({\mathcal {F}}\)-orbit is well defined.

Keywords

Canonical heights Dynamical degree Arithmetic degree Néron-Severi group Preperiodic rational points 

Notes

Acknowledgements

The author was supported by CAPES, ARC Discovery Grant DP180100201 and UNSW in this research.

References

  1. Arthur, B.: Canonical vector heights on algebraic K3 surfaces with Picard number two. Canad. Math. Bull. 46, 495–508 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Arthur, B.: Rational points on K3 surfaces in \({\mathbb{P}}^1 \times {\mathbb{P}}^1 \times {\mathbb{P}}^1\). Math. Ann. 305, 541–558 (1996)MathSciNetCrossRefGoogle Scholar
  3. Bombieri, E., Gubler, W.: Heights: in Diophantine Geometry, Number 4 in New Mathematical Monographs. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  4. Call, G., Silverman, J.: Canonical heights on varieties with morphisms. Compositio Math. 89, 163–205 (1993)MathSciNetzbMATHGoogle Scholar
  5. Fulton, W.: Intersection theory, Ergeb. Math. Grenzgeb. (3)2, 2nd edn. Springer-Verlag, Berlin (1998)CrossRefGoogle Scholar
  6. Guedj, V.: Ergodic properties of rational mappings with large topological degree. Ann.Math. (2) 161(3), 1589–1607 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Hartshorne, R.: Algebraic Geometry. Springer-Verlag, New York (1977)CrossRefzbMATHGoogle Scholar
  8. Hindry, M., Silverman, J.: Diophantine Geometry: An Introduction, Graduate Texts in Mathematics, vol. 201. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  9. Kawaguchi, S.: Canonical height functions for affine plane automorphisms. Math. Ann. 335(2), 285–310 (2006a)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Kawaguchi, S.: Canonical heights, invariant currents, and dynamical eigensystems of morphisms for line bundles. J. Reine Angew. Math. 597, 135–173 (2006b)MathSciNetzbMATHGoogle Scholar
  11. Kawaguchi, S.: Projective surface automorphisms of positive topological entropy from an arithmetic viewpoint. Am. J. Math. 130(1), 159–186 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Kawaguchi, S., Silverman, J.: On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties. J. Reine Angew. Math. 713, 21–48 (2016)MathSciNetzbMATHGoogle Scholar
  13. Kawaguchi, S., Silverman, J.: Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties. Trans. Amer. Math. Soc. 368, 5009–5035 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Lang, S.: Fundamentals of Diophantine Geometry. Springer-Verlag, New York(1983) Google Scholar
  15. Matsuzawa, Y.: On upper bounds of arithmetic degrees, to appear in Amer. J. Math. (2016). arXiv:1606.00598
  16. Silverman, J.: Examples of dynamical degree equals arithmetic degree. Mich Math. J. 63(1), 41–63 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Silverman, J.: Dynamical degrees, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space. Ergod. Th. Dyn. Syst. 34(2), 647–678 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Silverman, J.: Heights and the specialization maps for families of abelian varieties. J. Reine Angew. Math. 342, 197–211 (1983)MathSciNetzbMATHGoogle Scholar
  19. Silverman, J.: Rational points on K3 surfaces: a new canonical height. Invent. Math. 105, 347–373 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Silverman, J.: The Arithmetic of Dynamical Systems. Graduate Texts in Mathematics, vol. 241. Springer, New York (2007)CrossRefzbMATHGoogle Scholar
  21. Silverman, J.: The Arithmetic of Elliptic Curves, Grad. Texts in Math., vol. 106, 2nd edn. Springer-Verlag, Dordrecht (2009)CrossRefGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2019

Authors and Affiliations

  1. 1.School of Mathematics and Statistics UNSWUniversity of New South WalesSydneyAustralia

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