论文标题
$ \ mathbb {p}^k（\ mathbb {c}）$和类似于多项式的地图的内态植物家庭中的强概率稳定性
Strong probabilistic stability in holomorphic families of endomorphisms of $\mathbb{P}^k(\mathbb{C})$ and polynomial-like maps
论文作者
论文摘要
我们证明，在$ \ mathbb {p}^k（\ mathbb {c}）$的稳定家族中，所有不变的度量都严格大于$（k-1）\ log log d $在给定参数的$（k-1）\ log d $，都可以在所有参数下与所有参数遵循所有参数。结果，朱莉娅集合中几乎所有点（相对于任何参数的任何措施）都可以在没有相交的情况下进行霍明态遵循。这概括了Berteloot，Dupont和最大熵度量的第一作者的先前结果，并在此设置中提供了与Berger-Dujardin-Lyubich的HénonMaps概率稳定性相似的。我们的证明依赖于稳定性/分叉理论在任何维度上的稳定性/分叉理论，也取决于lyapunov指数的显式下限，这是由于其度量理论熵而导致的，这是由于deThélin和dupont而导致的。我们的结果的本地版本也适用于Julia设置支持的所有措施，并以严格的阳性Lyapunov指数，而不会收取临界后集合。大拓扑度的多项式图的家族中的类似结果。在这种情况下，作为我们证明的一部分，我们还为lyapunov指数的积极性提供了足够的条件，该指数在任何方面都在任何维度上对多项式的映射进行了类似于多项式的映射，从而将其推广到此设置，从而使DeThélin和Dupont的类似结果对$ \ \ Mathbb bbbbb {p}^k（p}^k（c）
We prove that, in stable families of endomorphisms of $\mathbb{P}^k(\mathbb{C})$, all invariant measures whose measure-theoretic entropy is strictly larger than $(k-1)\log d$ at a given parameter can be followed holomorphically with the parameter in all the parameter space. As a consequence, almost all points (with respect to any such measure at any parameter) in the Julia set can be followed holomorphically without intersections. This generalizes previous results by Berteloot, Dupont, and the first author for the measure of maximal entropy, and provides a parallel in this setting to the probabilistic stability of Hénon maps by Berger-Dujardin-Lyubich. Our proof relies both on techniques from the theory of stability/bifurcation in any dimension and on an explicit lower bound for the Lyapunov exponents for an ergodic measure in terms of its measure-theoretic entropy, due to de Thélin and Dupont. A local version of our result holds also for all measures supported on the Julia set with just strictly positive Lyapunov exponents and not charging the post-critical set. Analogous results hold in families of polynomial-like maps of large topological degree. In this case, as part of our proof, we also give a sufficient condition for the positivity of the Lyapunov exponents of an ergodic measure for a polynomial-like map in any dimension in term of its measure-theoretic entropy, generalizing to this setting the analogous result by de Thélin and Dupont valid on $\mathbb{P}^k(\mathbb{C})$.
