On upper bounds of arithmetic degrees

Let $X$ be a smooth projective variety over $\overline{\mathbb Q}$, and $f:X -rightarrow X$ be a dominant rational map. Let $\delta_{f}$ be the first dynamical degree of $f$ and $h_{X}:X( \overline{\mathbb Q})\to [1,\infty)$ be a Weil height function on $X$ associated with an ample divisor on $X$. We prove several inequalities which give upper bounds of the sequence $(h_X (f^n(P)))_{n\geq0}$ where $P$ is a point of $X( \overline{\mathbb Q})$ whose forward orbit by $f$ is well-defined. As a corollary, we prove that the upper arithmetic degree is less than or equal to the first dynamical degree; $\overline{\alpha}_{f}(P) \leq \delta_{f}$. Furthermore, if the Picard number of $X$ is one, $f$ is algebraically stable and $\delta_{f}>1$, we prove that the limit defining canonical height $\lim_{n\to \infty} h_{X} (f^{n}(P)) \big/ \delta_f^n$ converges.