#### Vol. 12, No. 7, 2018

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Algebraic dynamics of the lifts of Frobenius

### Junyi Xie

Vol. 12 (2018), No. 7, 1715–1748
##### Abstract

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.

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