The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane

Abstract

In this paper we prove the following theorem. Let $f:{\mathbb{A}}^2\to {\mathbb{A}}^2$ be a dominate polynomial endomorphisms defined over an algebraically closed field $k$ of characteristic $0$. If there are no nonconstant rational function $g:{\mathbb{A}}^2-rightarrow{\mathbb{P}}^1$ satisfying $g\circ f=g$, then there exists a point $p\in {\mathbb{A}}^2(k)$ whose orbit under $f$ is Zariski dense in ${\mathbb{A}}^2$. This result gives us a positive answer to a conjecture of Amerik, Bogomolov and Rovinsky ( and Zhang) for polynomial endomorphisms on the affine plane.

arXiv admin note: substantial text overlap with arXiv:1503.00773

References in corpus (4)

• Questions on self maps of algebraic varieties
• p-adic interpolation of iterates
• The algebraic dynamics of generic endomorphisms of P^n
• Intersection of valuation rings in $k[x,y]$

Cited by in corpus (8)

• On the Medvedev-Scanlon Conjecture for Minimal Threefolds of Non-Negative Kodaira Dimension
• The existence of Zariski dense orbits for endomorphisms of projective surfaces (with an appendix in collaboration with Thomas Tucker)
• Density of orbits of endomorphisms of commutative linear algebraic groups
• Finiteness properties of pseudo-hyperbolic varieties
• Surjective endomorphisms of projective surfaces: the existence of infinitely many dense orbits
• Space spanned by characteristic exponents
• Advances in the equivariant minimal model program and their applications in complex and arithmetic dynamics
• Algebraic dynamics of skew-linear self-maps