Abstract:

Let $X$ be a smooth projective variety defined over $\\overline\{\\Bbb\{Q\}\}$, and $f\\colon X\\dashrightarrow X$ be a dominant rational map. Let $\\delta_f$ be the first dynamical degree of $f$ and $h_X\\colon X(\\overline\{\\Bbb\{Q\}\})\\rightarrow [1,\\infty)$ 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 $X(\\overline\{\\Bbb\{Q\}\})$ 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, we prove the canonical height functions of rational self-maps exist under certain conditions. For example, when the Picard number of $X$ is one, $f$ is algebraically stable (in the sense of Fornaess-Sibony) and $\\delta_f>1$, the limit defining canonical height $\\lim_\{n\\to\\infty\}h_X(f^n(P))\\big/\\delta_f^n$ converges.

摘要：

假设$ X $是在$ \\ overline \ {\\ Bbb \ {Q \} \} $上定义的平滑投影变种，而$ f \\冒号X \\ dashrightarrow X $是占主导地位的有理图。设$ \\ delta_f $为$ f $和$ h_X \\冒号X（\\ overline \ {\\ Bbb \ {Q \} \}）\\ rightarrow [1，\\ infty）$的第一个动态度。是$ X $上的Weil高度函数，并具有$ X $上的足够除数。我们证明了几个不等式，它们给出序列$（h_X（f ^ n（P）））_ \ {n \\ geq0 \} $的上限，其中$ P $是$ X（\\ overline \ {\ \ Bbb \ {Q \} \}）$的定义很好，其$ f $的正向轨道。作为推论，我们证明了较高的算术度小于或等于第一个动力学度；$ \\ overline \ {\\ alpha \} _ f（P）\\ leq \\ delta_f $。此外，我们证明了在一定条件下存在有理自映射的规范高度函数。例如，当$ X $的皮卡德数为1时，