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arXiv · 2409.06160

Arithmetic degree and its application to Zariski dense orbit conjecture

Yohsuke Matsuzawa·Junyi Xie
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Abstract

We prove that for a dominant rational self-map $f$ on a quasi-projective variety defined over $\overline{\mathbb{Q}}$, there is a point whose $f$-orbit is well-defined and its arithmetic degree is arbitrarily close to the first dynamical degree of $f$. As an application, we prove that Zariski dense orbit conjecture holds for a birational map defined over $\overline{\mathbb{Q}}$ whose first dynamical degree is strictly larger than its third dynamical degree. In particular, the conjecture holds for birational maps on threefolds whose first dynamical is degree greater than $1$.

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Yohsuke Matsuzawa, Junyi Xie. 2024-09-10. Arithmetic degree and its application to Zariski dense orbit conjecture. https://arxiv.org/abs/2409.06160

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