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On the distribution of orbits in affine varieties

Published online by Cambridge University Press:  03 July 2014

CLAYTON PETSCHE
CLAYTON PETSCHE*
Affiliation:
Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA email petschec@math.oregonstate.edu
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Abstract

Given an affine variety $X$, a morphism ${\it\phi}:X\rightarrow X$, a point ${\it\alpha}\in X$, and a Zariski-closed subset $V$ of $X$, we show that the forward ${\it\phi}$-orbit of ${\it\alpha}$ meets $V$ in at most finitely many infinite arithmetic progressions, and the remaining points lie in a set of Banach density zero. This may be viewed as a weak asymptotic version of the dynamical Mordell–Lang conjecture for affine varieties. The results hold in arbitrary characteristic, and the proof uses methods of ergodic theory applied to compact Berkovich spaces.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 7 , October 2015 , pp. 2231 - 2241
Copyright
© Cambridge University Press, 2014 

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