Dynamical Properties of Morphisms on Higher Dimensional Projective Space over Local Field

Zheng Jun ZHAO, Xiang CHEN

Acta Mathematica Sinica, Chinese Series ›› 2022, Vol. 65 ›› Issue (6) : 967-978.

PDF(566 KB)
PDF(566 KB)
Acta Mathematica Sinica, Chinese Series ›› 2022, Vol. 65 ›› Issue (6) : 967-978. DOI: 10.12386/A20210073

Dynamical Properties of Morphisms on Higher Dimensional Projective Space over Local Field

  • Zheng Jun ZHAO, Xiang CHEN
Author information +
History +

Abstract

Let Cv be an algebraically closed non-archimedean field, complete with respect to a valuation v. Let φ:PNPN be a morphism of degree greater than one defined over Cv, Φ a lift of φ. Let GΦ be the Green function of Φ and ρ the chordal metric on PN(Cv). In this paper, we first study the properties of reduction of points in high dimensional projective space and reduction of automorphisms of PN with degree one. With the help of Green function GΦ of Φ, we introduce the arithmetic distance of morphisms and investigate its property. The necessary and sufficient condition which Φ has good reduction is obtained in this paper. We also describe explicitly the Filled Julia set of Φ by its Green function.

Key words

local field / morphism / projective space / Green function

Cite this article

Download Citations
Zheng Jun ZHAO, Xiang CHEN. Dynamical Properties of Morphisms on Higher Dimensional Projective Space over Local Field. Acta Mathematica Sinica, Chinese Series, 2022, 65(6): 967-978 https://doi.org/10.12386/A20210073

References

[1] Baker M., A lower bound for average values of dynamical Green’s function, Math. Res. Lett., 2006, 13: 245–257.
[2] Benedetto R., Fatou components in p-adic dynamics, Ph.D. thesis, Brown University, Providence, RI, 1998.
[3] Benedetto R., p-adic dynamics and Sullivan’s no wandering domains theorem, Compositio Math., 2000, 122: 281–298.
[4] Benedetto R., DeMarco L., Jones R., et al., Current trends and open problems in arithmetic dynamics, Bull. Amer. Math. Soc., 2019, 56: 611–685.
[5] Benedetto R., Reduction, dynamics, and Julia sets of rational functions, J. Number Theory, 2001, 86: 175–195.
[6] Benedetto R., Preperiodic points of polynomials over global fields, J. Reine Angew. Math., 2007, 608: 123–153.
[7] Benedetto R., Heights and preperiodic points of polynomials over function fields, Internat. Math. Res. Notices, 2005, 62: 3855–3866.
[8] Call G., Silverman J. H., Canonical heights on varieties with morphisms, Compositio Math., 1993, 89: 163–205.
[9] Herman M., Yoccoz J. C., Generalizations of some theorems of small divisors to non-Archimedean fields, In: Geometric Dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., Vol. 1007, 408–447, Springer, Berlin, 1983.
[10] Hsia L. C., Closure of periodic points over a non-archimedean field, J. London Math. Soc. (2), 2000, 62: 685–670.
[11] Kawaguchi S., Local and global canonical height functions for affine space regular automorphisms, Algebra Number Theory, 2013, 7: 1225–1252.
[12] Kawaguchi S., Silverman J. H., Nonarchimedean Green functions and dynamics on projective space, Math. Z., 2009, 262: 173–197.
[13] Kawaguchi S., Silverman J. H., Canonical heights and the arithmetic complexity of morphisms on projective space, Pure Appl. Math. Q., 2009, 5: 1201–1217.
[14] Kawaguchi S., Canonical height functions for affine plane automorphisms, Math. Ann., 2006, 335: 285–310.
[15] Kawaguchi S., Silverman J. H., Dynamics of projective morphisms having identical canonical heights, Proc. London Math. Soc., 2007, 95: 519–544.
[16] Lee C., An upper bound for the height for regular affine automorphisms of An, Math. Ann., 2013, 355: 1–16.
[17] Lee J., An alternative proof of non-Archimedean Montel theorem for rational dynamics, Proc. Japan Acad. Ser. A, 2016, 92: 56–58.
[18] Morton P., Silverman J. H., Periodic points, multiplicities, and dynamical units, J. Reine Angew. Math., 1995, 461: 81–122.
[19] Morton P., Silverman J. H., Rational periodic points of rational functions, Internat. Math. Res. Notices, 1994, 2: 97–110.
[20] Rivera-Letelier J., Dynamique des fonctions rationnelles sur des corps locaux, PhD thesis, Universite de Paris XI, Orsay, 2000.
[21] Rivera-Letelier J., Dynamique des fonctions rationnelles sur des corps locaux, Astérisque, 2003, 287: 147– 230.
[22] Serre J. P., Lectures on the Mordell–Weil Theorem, Third Edition, Aspect of Mathematics, Vol. E15, Springer Fachmedien Wiesbaden, Wiesbaden, 1997.
[23] Silverman J. H., The Arithmetic of Dynamical Systems, Grad. Texts in Math., Vol. 241, Springer, New York, 2007.
[24] Silverman J. H., Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space, Ergodic Theory Dynam. Systems, 2014, 34: 647–678.
[25] Silverman J. H., Height bounds and preperiodic points for families of jointly regular affine maps, Pure Appl. Math. Q., 2006, 2: 135–145.
PDF(566 KB)

704

Accesses

0

Citation

Detail

Sections
Recommended

/