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

  • Jorge MelloEmail author
Article
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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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