Surjective endomorphisms of projective surfaces: the existence of infinitely many dense orbits

Abstract

Let $f:X\to X$ be a surjective endomorphism of a normal projective surface. When $\mathrm{deg}f\ge 2$, applying an (iteration of) $f$-equivariant minimal model program (EMMP), we determine the geometric structure of $X$. Using this, we extend the second author's result to singular surfaces to the extent that either $X$ has an $f$-invariant non-constant rational function, or $f$ has infinitely many Zariski-dense forward orbits; this result is also extended to Adelic topology (which is finer than Zariski topology).

Minor revision. Final version

References in corpus (2)

• The existence of Zariski dense orbits for endomorphisms of projective surfaces (with an appendix in collaboration with Thomas Tucker)
• Kawaguchi-Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism

Cited by in corpus (2)

• Advances in the equivariant minimal model program and their applications in complex and arithmetic dynamics
• Potential density of projective varieties having an int-amplified endomorphism