标题： 多项式斜积与较小的相对次数 显示英文标题

标题： Polynomial skew products with small relative degree

摘要： 我们研究了一个适当超吸引的全纯芽$f$在$(\mathbb{C}^2,0)$中的局部动力学，该芽拥有一个完全不变的直线$L$，使得$f^*L = d L$与$d\ge 2$相关，并且使得$f|_L$在$0$处有一个阶为$2 \le c < d$的超吸引固定点。 我们证明任何这样的映射在形式上共轭于如下形式的斜积$(z^d, P(z,w))$，其中$P \in \mathbb{C}[[z]][w]$是关于$w$的次数为$c$的多项式，因此它在伯克霍夫仿射直线（关于$\mathbb{C}(\!(z)\!)$）上诱导了一个自然的动力学。 这种非阿基米德斜积最近由 Birkett 和 Nie-Zhao 进行了研究。 在非阿基米德方面，我们关注伯克霍夫单位开球上的动力学限制（这自然包含了所有原点处的不可约解析芽）。 我们展示了一个不变紧集$\mathcal{K}$，在其之外的所有点都趋向于$L$，并且该集合支持一个自然的遍历不变测度。 通过仔细分析局部相交数，我们证明了迭代曲线的重数增长由临界集的递归性质所控制。 特别是，当$f$的任何关键分支都不属于$\mathcal{K}$时，$\mathcal{K}$中的任何点都对应于在$0$处具有有界乘数的曲线。 我们随后回到复数图像，并证明在某个不变的全纯极小正闭$(1,1)$-当前$T$的外部，所有轨道都以超指数速度收敛到$0$$c$。 在与上述临界分支相同的假设下，我们证明$T$可以作为$\mathcal{K}$中曲线上的积分当前的平均值，相对于自然不变测度。 特别是，$T$在原点外是均匀层状的。 显示英文摘要

摘要： We investigate the local dynamics of a proper superattracting holomorphic germ $f$ in $(\mathbb{C}^2,0)$ possessing a totally invariant line $L$ such that $f^*L = d L$ with $d\ge 2$, and such that $f|_L$ has a superattracting fixed point at $0$ of order $2 \le c < d$. We prove that any such map is formally conjugated to a skew product of the form $(z^d, P(z,w))$, where $P \in \mathbb{C}[[z]][w]$ is polynomial in $w$ of degree $c$, hence it induces a natural dynamics on the Berkovich affine line over $\mathbb{C}(\!(z)\!)$. Such non-Archimedean skew products were recently studied by Birkett and Nie-Zhao. On the non-Archimedean side, we focus on the restriction of the dynamics on the Berkovich open unit ball (which naturally contains all irreducible analytic germs at the origin). We exhibit an invariant compact set $\mathcal{K}$ outside of which all points tend to $L$, and which supports a natural ergodic invariant measure. By a careful analysis of local intersection numbers, we prove that the growth of multiplicity of iterated curves is controlled by the recurrence properties of the critical set. In particular, when no critical branch of $f$ belongs to $\mathcal{K}$, any point in $\mathcal{K}$ corresponds to a curve of uniformly bounded multiplicity at $0$. We then return to the complex picture and show the existence of an invariant pluripolar positive closed $(1,1)$-current $T$, outside of which all orbits converge to $0$ at super-exponential speed $c$. Under the same assumption on the critical branches as above, we prove that $T$ admits a geometric representation as an average of currents of integration over the curves in $\mathcal{K}$, with respect to the natural invariant measure. In particular, $T$ is uniformly laminar outside the origin.