Arithmetic degree and its application to Zariski dense orbit conjecture

We prove that for a dominant rational self‐map on a quasi‐projective variety defined over , there is a point whose ‐orbit is well‐defined and its arithmetic degree is arbitrarily close to the first dynamical degree of . As an application, we prove that Zariski dense orbit conjecture holds for a birational map defined over 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.