论文标题
$ \ mathbb {p}^k $ ii的内态性的平衡状态：光谱稳定性和限制定理
Equilibrium states of endomorphisms of $\mathbb{P}^k$ II: spectral stability and limit theorems
论文作者
论文摘要
我们通过通用的全态内态性和适当的连续重量及其对各种功能空间的扰动，在$ \ mathbb p^k = \ mathbb p^k（\ mathbb c）$上建立了光谱差距的存在，这在维度甚至是新的。要克服的主要问题是复杂对象的刚度，因为转移操作员是对操作员$ f _*$的非形态扰动。此外，该系统是非均匀双曲线的，一个系统可能在朱莉娅集合上具有关键点。我们的规范的构建需要引入和研究几个中间新规范，以及从多功能和插值理论中仔细的思想结合。据我们所知，这是第一次应用多能量方法来解决混合的实时复杂问题。由于光谱差距，我们建立了融合的指数速度，以使朝向共形度量的向后轨道等分。此外，我们获得了平衡状态的统计特性的完整列表：指数混合，CLT，Berry-Esseen定理，本地CLT，ASIP，LIL，LIL，LDP，几乎可以肯定的CLT。这些属性中的许多都是新的，即使在尺寸为1的情况下，甚至在零重量函数的情况下（即，对于最大熵的度量）。
We establish the existence of a spectral gap for the transfer operator induced on $\mathbb P^k = \mathbb P^k (\mathbb C)$ by a generic holomorphic endomorphism and a suitable continuous weight and its perturbations on various functional spaces, which is new even in dimension one. The main issue to overcome is the rigidity of the complex objects, since the transfer operator is a non-holomorphic perturbation of the operator $f_*$. The system is moreover non-uniformly hyperbolic and one may have critical points on the Julia set. The construction of our norm requires the introduction and study of several intermediate new norms, and a careful combination of ideas from pluripotential and interpolation theory. As far as we know, this is the first time that pluripotential methods have been applied to solve a mixed real-complex problem. Thanks to the spectral gap, we establish an exponential speed of convergence for the equidistribution of the backward orbits of points towards the conformal measure. Moreover, we obtain a full list of statistical properties for the equilibrium states: exponential mixing, CLT, Berry-Esseen theorem, local CLT, ASIP, LIL, LDP, almost sure CLT. Many of these properties are new even in dimension one, some even in the case of zero weight function (i.e., for the measure of maximal entropy).
