标题： 实有理曲面自同构：正性和线性 显示英文标题

标题： Real Rational Surface Automorphisms : Positivity and Linearity

摘要： 我们研究了一类从保持尖点三次曲线的二次有理映射$\pcc$得到的有理曲面自同构的实动力学性质，其临界轨道长度为$(1,m,n)$，其中$1+m+n\ge 10$。 转到实轨域并沿不变的三次曲线切割，我们得到一个基本群为自由群的可定向曲面的微分同胚。 我们的关键工具是一个在基本群上的有限生成不变正半群$S_{m,n}$，在该半群上诱导作用的迭代通过拼接而不发生抵消作用。 这种正性产生了一个非负的原始转移矩阵，因此Perron-Frobenius理论提供了基本群上诱导作用的显式指数增长速率$\lambda>1$。 因此，实映射具有正的拓扑熵。 我们将生成元的组合结构打包成一个“核心-尾部归纳原理”，这使我们只需进行有限的基检查即可同时处理七个轨道数据族。 最后，使用Bestvina-Handel和Dehn-Nielsen-Baer对应关系，我们证明了具有$m+n$个奇数的诱导外自同构由切割后的曲面的伪Anosov同胚实现。 显示英文摘要

摘要： We study the real dynamics of a family of rational surface automorphisms obtained from quadratic birational maps of $\pcc$ that preserve a cuspidal cubic and whose critical orbits have lengths $(1,m,n)$ with $1+m+n\ge 10$. Passing to the real locus and cutting along the invariant cubic, we obtain a diffeomorphism of an orientable surface whose fundamental group is free. Our key device is a finitely generated invariant, positive semigroup $S_{m,n}$ in the fundamental group on which an iterate of induced action acts by concatenation without cancellation. This positivity yields a nonnegative primitive transition matrix, so Perron-Frobenius theory supplies an explicit exponential growth rate $\lambda>1$ for the induced action on the fundamental group. Consequently, the real map has positive topological entropy. We package the combinatorics of the generators in a ``Core-Tail Induction Principle," which allows us to treat simultaneously seven orbit-data families with only finite base checks. Finally, using Bestvina-Handel and the Dehn-Nielsen-Baer correspondence, we show that the induced outer automorphism with $m+n$ odd is realized by a pseudo-Anosov homeomorphism of the cut surface.