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

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
A dynamical variant of the Pink–Zilber conjecture

### Dragos Ghioca and Khoa Dang Nguyen

Vol. 12 (2018), No. 7, 1749–1771
##### Abstract

Let ${f}_{1},\dots ,{f}_{n}\in \overline{ℚ}\left[x\right]$ be polynomials of degree $d>1$ such that no ${f}_{i}$ is conjugate to ${x}^{d}$ or to $±{C}_{d}\left(x\right)$, where ${C}_{d}\left(x\right)$ is the Chebyshev polynomial of degree $d$. We let $\phi$ be their coordinatewise action on ${\mathbb{A}}^{n}$, i.e., $\phi :{\mathbb{A}}^{n}\to {\mathbb{A}}^{n}$ is given by $\left({x}_{1},\dots ,{x}_{n}\right)↦\left({f}_{1}\left({x}_{1}\right),\dots ,{f}_{n}\left({x}_{n}\right)\right)$. We prove a dynamical version of the Pink–Zilber conjecture for subvarieties $V$ of ${\mathbb{A}}^{n}$ with respect to the dynamical system $\left({\mathbb{A}}^{n},\phi \right)$, if $min\left\{dim\left(V\right),codim\left(V\right)-1\right\}\le 1$.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/ant

We have not been able to recognize your IP address 18.207.133.27 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

##### Keywords
dynamical Pink–Zilber conjecture, heights
##### Mathematical Subject Classification 2010
Primary: 11G50
Secondary: 11G35, 14G25
##### Milestones
Received: 1 November 2017
Revised: 6 April 2018
Accepted: 6 June 2018
Published: 27 October 2018
##### Authors
 Dragos Ghioca Department of Mathematics University of British Columbia Vancouver BC Canada Khoa Dang Nguyen Department of Mathematics and Statistics The University of Calgary Calgary AB Canada