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

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.

This is a preview of subscription content, access via your institution.

References

  1. Arthur, B.: Canonical vector heights on algebraic K3 surfaces with Picard number two. Canad. Math. Bull. 46, 495–508 (2003)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  3. Bombieri, E., Gubler, W.: Heights: in Diophantine Geometry, Number 4 in New Mathematical Monographs. Cambridge University Press, Cambridge (2006)

    MATH  Google Scholar 

  4. Call, G., Silverman, J.: Canonical heights on varieties with morphisms. Compositio Math. 89, 163–205 (1993)

    MathSciNet  MATH  Google Scholar 

  5. Fulton, W.: Intersection theory, Ergeb. Math. Grenzgeb. (3)2, 2nd edn. Springer-Verlag, Berlin (1998)

    Book  Google Scholar 

  6. Guedj, V.: Ergodic properties of rational mappings with large topological degree. Ann.Math. (2) 161(3), 1589–1607 (2005)

    MathSciNet  Article  Google Scholar 

  7. Hartshorne, R.: Algebraic Geometry. Springer-Verlag, New York (1977)

    Book  Google Scholar 

  8. Hindry, M., Silverman, J.: Diophantine Geometry: An Introduction, Graduate Texts in Mathematics, vol. 201. Springer, New York (2000)

    Book  Google Scholar 

  9. Kawaguchi, S.: Canonical height functions for affine plane automorphisms. Math. Ann. 335(2), 285–310 (2006a)

    MathSciNet  Article  Google Scholar 

  10. Kawaguchi, S.: Canonical heights, invariant currents, and dynamical eigensystems of morphisms for line bundles. J. Reine Angew. Math. 597, 135–173 (2006b)

    MathSciNet  MATH  Google Scholar 

  11. Kawaguchi, S.: Projective surface automorphisms of positive topological entropy from an arithmetic viewpoint. Am. J. Math. 130(1), 159–186 (2008)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google Scholar 

  14. Lang, S.: Fundamentals of Diophantine Geometry. Springer-Verlag, New York(1983)

  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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  18. Silverman, J.: Heights and the specialization maps for families of abelian varieties. J. Reine Angew. Math. 342, 197–211 (1983)

    MathSciNet  MATH  Google Scholar 

  19. Silverman, J.: Rational points on K3 surfaces: a new canonical height. Invent. Math. 105, 347–373 (1991)

    MathSciNet  Article  Google Scholar 

  20. Silverman, J.: The Arithmetic of Dynamical Systems. Graduate Texts in Mathematics, vol. 241. Springer, New York (2007)

    Book  Google Scholar 

  21. Silverman, J.: The Arithmetic of Elliptic Curves, Grad. Texts in Math., vol. 106, 2nd edn. Springer-Verlag, Dordrecht (2009)

    Book  Google Scholar 

Download references

Acknowledgements

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

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jorge Mello.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author would like to say thanks to J. Silverman and S. Kawaguchi.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mello, J. The Dynamical and Arithmetical Degrees for Eigensystems of Rational Self-maps. Bull Braz Math Soc, New Series 51, 569–596 (2020). https://doi.org/10.1007/s00574-019-00165-w

Download citation

Keywords

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