We consider the degree sequences of the tame automorphisms preserving an affine quadric threefold. Using some valuative estimates derived from the work of Shestakov and Umirbaev and the action of this group on a $\mathrm{CAT}<!--FUNCTION\; APPLICATION-->(0)$, Gromov-hyperbolic square complex constructed by Bisi, Furter and Lamy, we prove that the dynamical degrees of tame elements avoid any value strictly between 1 and $\frac{4}{3}$. As an application, these methods allow us to characterize when the growth exponent of the degree of a random product of finitely many tame automorphisms is positive.

中文翻译：

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三重仿射二次曲面的驯服自同构的度增长

我们考虑保留仿射二次曲面的驯服自同构的度序列。使用从 Shestakov 和 Umirbaev 的工作以及该小组对$猫<!--FUNCTION\; APPLICATION-->（0）$，Bisi、Furter 和 Lamy 构造的 Gromov-双曲平方复形，我们证明了驯服元素的动力度严格避免 1 到 1 之间的任何值$\frac{4}{3}$。作为一种应用，这些方法使我们能够表征有限多个驯服自同构的随机乘积的度的增长指数何时为正。