有限型亚纯映射族的分岔轨迹
Bifurcation loci of families of finite type meromorphic maps
AI总结：
本文证明有限型亚纯映射族中J-稳定性开且稠密，推广了有理映射与整函数的相关结果，关键在于研究周期轨道离开定义域这一新型分岔及其与奇异值分岔的关系。
摘要：
我们证明，在具有有限多个奇异值的单复变量亚纯映射的自然族中，J-稳定性是开且稠密的，并且更一般地，对于有限型映射也成立。这推广了Mañé-Sad-Sullivan关于黎曼球面上有理映射的结果以及Eremenko和Lyubich关于复平面上有限型整函数的结果，并在本质上解决了具有有限多个奇异值的单复变量全纯动力系统中结构稳定性密度的问题。该结果是作为对一种新型分岔的详细研究的推论而得到的，这种分岔在同时存在极点和本性奇点时出现（即周期轨道沿参数曲线离开映射的定义域），特别是它与奇异值动力学中的分岔之间的关系。这些新的分岔参数的存在需要与先前用于有理映射或整函数映射的工作本质上不同的方法。
英文摘要：
We show that $J-$ stability is open and dense in natural families of meromorphic maps of one complex variable with a finite number of singular values, and even more generally, to finite type maps. This extends the results of Mañé-Sad-Sullivan for rational maps of the Riemann sphere and those of Eremenko and Lyubich for entire maps of finite type of the complex plane, and essentially closes the problem of density of structural stability for holomorphic dynamical systems in one complex variable with finitely many singular values. This result is obtained as a consequence of a detailed study of a new type of bifurcation that arises with the presence of both poles and essential singularities (namely periodic orbits exiting the domain of definition of the map along a parameter curve), and in particular its relation with the bifurcations in the dynamics of singular values. The presence of these new bifurcation parameters require essentially different methods to those used in previous work for rational or entire maps.
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