Vol. 12, No. 7, 2018

Download this article
Download this article For screen
For printing
Recent Issues

Volume 12
Issue 7, 1559–1821
Issue 6, 1311–1557
Issue 5, 1001–1309
Issue 4, 751–999
Issue 3, 493–750
Issue 2, 227–492
Issue 1, 1–225

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors' Addresses
Editors' Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
Author Index
To Appear
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Algebraic dynamics of the lifts of Frobenius

Junyi Xie

Vol. 12 (2018), No. 7, 1715–1748

We study the algebraic dynamics of endomorphisms of projective spaces with coefficients in a p-adic field whose reduction in positive characteristic is the Frobenius. In particular, we prove a version of the dynamical Manin–Mumford conjecture and the dynamical Mordell–Lang conjecture for the coherent backward orbits of such endomorphisms. We also give a new proof of a dynamical version of the Tate–Voloch conjecture in this case. Our method is based on the theory of perfectoid spaces introduced by P. Scholze. In the appendix, we prove that under some technical condition on the field of definition, a dynamical system for a polarized lift of Frobenius on a projective variety can be embedded into a dynamical system for some endomorphism of a projective space.

algebraic dynamics, perfectoid space
Mathematical Subject Classification 2010
Primary: 37P55
Secondary: 37P20, 37P35
Received: 2 October 2017
Revised: 15 June 2018
Accepted: 17 July 2018
Published: 27 October 2018
Junyi Xie
Institut de Recherche Mathématique de Rennes
CNRS - Université de Rennes 1
Bâtiment 22-23 du campus de Beaulieu