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. 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 and the exponential mixing. Moreover, as an immediate consequence, we obtain a full list of statistical properties for the equilibrium states: 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).

我们建立了由通用全纯自同态在\(\mathbb{P}^{k} = \mathbb{P}^{k} (\mathbb{C})\)上诱导的转移算子的谱间隙的存在，并且合适的连续重量及其对各种功能空间的扰动，即使在一维中也是新的。由于谱间隙，我们建立了指数收敛速度，用于点向后轨道的均匀分布，以实现共形测量和指数混合。此外，作为直接结果，我们获得了平衡状态统计属性的完整列表：CLT、Berry-Esseen 定理、局部 CLT、ASIP、LIL、LDP、几乎肯定的 CLT。这些属性中的许多属性即使在一维中也是新的，有些甚至在零权重函数的情况下（即，用于测量最大熵）也是新的。