A conjecture strengthening the Zariski dense orbit problem for birational maps of dynamical degree one

TL;DR

This paper proposes a strengthened version of the Zariski dense orbit conjecture for birational maps with dynamical degree one, proving it for automorphisms of semiabelian varieties and exploring related dynamics.

Contribution

It formulates a new conjecture for dynamical systems of degree one and proves it for automorphisms of semiabelian varieties, also relating dynamical degree to invariant subvarieties.

Findings

Conjecture holds for automorphisms of semiabelian varieties.

Dynamical degree exceeds one iff invariant subvarieties are Zariski dense.

Applications to twisted homogeneous coordinate rings.

Abstract

We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety $X$ defined over an algebraically closed field $K$ of characteristic $0$, endowed with a birational self-map $\phi$ of dynamical degree $1$, we expect that either there exists a non-constant rational function $f:X\dashrightarrow \mathbb{P}^1$ such that $f\circ \phi=f$, or there exists a proper subvariety $Y\subset X$ with the property that for any invariant proper subvariety $Z\subset X$, we have that $Z\subseteq Y$. We prove our conjecture for automorphisms $\phi$ of dynamical degree $1$ of semiabelian varieties $X$. Also, we prove a related result for regular dominant self-maps $\phi$ of semiabelian varieties $X$: assuming $\phi$ does not preserve a non-constant rational function, we have that the dynamical degree of $\phi$ is larger than $1$ if and only if the union of all $\phi$-invariant proper subvarieties of $X$ is Zariski dense. We give applications of our results to representation theoretic questions about twisted homogeneous coordinate rings associated to abelian varieties.