Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T04:12:05.501Z Has data issue: false hasContentIssue false
  • English
  • Français

$J$-stability of expanding maps in non-Archimedean dynamics

Published online by Cambridge University Press:  20 June 2017

JUNGHUN LEE
JUNGHUN LEE*
Affiliation:
Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The aim of this paper is to show $J$-stability of expanding rational maps over an algebraically closed, complete and non-Archimedean field of characteristic zero. More precisely, we will show that for any expanding rational map, there exists a neighborhood of it such that the dynamics on the Julia set of any rational map in the neighborhood is the same as the dynamics of the expanding rational map as a non-Archimedean analogue of a corollary of Mañé, Sad and Sullivan’s result [On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér. (4)16 (1983), 193–217] in complex dynamics.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 4 , April 2019 , pp. 1002 - 1019
Copyright
© Cambridge University Press, 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benedetto, R. L.. Hyperbolic maps in p-adic dynamics. Ergod. Th. & Dynam. Sys. 21 (2001), 1–11.Google Scholar
Benedetto, R. L.. Examples of wandering domains in p-adic polynomial dynamics. C. R. Math. Acad. Sci. Paris 335 (2002), 615–620.Google Scholar
Benedetto, R. L.. An Ahlfors islands theorem for non-Archimedean meromorphic functions. Trans. Amer. Math. Soc. 360 (2008), 4099–4124.Google Scholar
Benedetto, R. L.. A criterion for potentially good reduction in nonarchimedean dynamics. Acta Arith. 165 (2014), 251–256.Google Scholar
Bézivin, J.-P.. Sur les points périodiques des applications rationnelles en dynamique ultramétrique. Acta Arith. 100 (2001), 63–74.Google Scholar
Bézivin, J.-P.. Sur la compacité des ensembles de Julia des polynômes p-adiques. Math. Z. 246 (2004), 273–289.Google Scholar
Briend, J.-Y. and Hsia, L.-C.. Weak Néron models for cubic polynomial maps over a non-Archimedean field. Acta Arith. 153 (2012), 415–428.Google Scholar
Baker, M. and Rumely, R.. Potential Theory and Dynamics on the Berkovich Projective Line. American Mathematical Society, Providence, RI, 2010.Google Scholar
Doyle, J. R., Jacob, K. and Rumely, R.. Configuration of the crucial set for a quadratic rational map. Res. Number Theory 2 (2016), 11.Google Scholar
Graczyk, J. and S̀watek, G.. The Real Fatou Conjecture. Princeton University Press, Princeton, NJ, 1998.Google Scholar
Hsia, L.-C.. Closure of periodic points over a non-Archimedean field. J. Lond. Math. Soc. (2) 62 (2000), 685–700.Google Scholar
Kiwi, J.. Puiseux series polynomial dynamics and iteration of complex cubic polynomials. Ann. Inst. Fourier (Grenoble) 56 (2006), 1337–1404.Google Scholar
McMullen, C.. Complex Dynamics and Renormalization. Princeton University Press, Princeton, NJ, 1994.Google Scholar
Mañé, R., Sad, P. and Sullivan, D.. On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér. (4) 16 (1983), 193–217.Google Scholar
Okuyama, Y.. Repelling periodic points and logarithmic equidistribution in non-archimedean dynamics. Acta Arith. 152 (2012), 267–277.Google Scholar
Rivera-Letelier, J.. Dynamique des fonctions rationnelles sur des corps locaux. Astérisque 287 (2003), 147–230.Google Scholar
Robert, A. M.. A Course in p-adic Analysis. Springer, New York, 2000.Google Scholar
Silverman, J. H.. The Arithmetic of Dynamical Systems. Springer, New York, 2007.Google Scholar