一维扩张映射驱动的Viana映射

The phrase "Viana map" appearing in the title, which is the main object to study in this research program, means any skew-product of quadratic polynomial driven by some (non-uniform) expanding interval/circle map. The first theme of this program is looking for weaker assumptions on the Viana maps under which the corresponding dynamical systems remain non-uniformly expanding. That means, such a the system has two positive Lyapunov exponents at Lebesgue almost every point and admits an SRB measure. The key tool we plan to use in tackling this problem is to combine the theory of one-dimensional dynamics with certain complex analytic techniques. Then we shall give a further discussion on the statistical properties of such Viana maps and their corresponding SRB measures, where we mainly focus on topics such as decay of correlations and statistical stability.

项目名称中的 “Viana 映射” 一词在这里泛指由（非一致）扩张的区间或圆周映射驱动的二次斜积映射，是本项目研究的主要对象。我们首先关注的课题是如何适当放宽已知结果中对Viana映射的限制条件，使其仍具有非一致扩张性：即几乎处处具有两个正的Lyapunov 指数，并由此可得到SRB 测度的存在性。在该问题的研究过程中，我们将以一维动力系统的理论与适当的复分析技术相结合作为主要研究工具。在此基础上，我们将对所得到的Viana 映射关于其SRB 测度的统计性质进一步讨论做进一步研究，其中主要关注相关性衰减的速度与随机稳定性等方面的内容。

本项目主要研究内容为一族Viana映射的动力学性质，该族映射的形式为两个二次区间映射的斜积。本项目致力于研究在测度意义下，非一致双曲性在该族映射中的丰富性。本项目的主要研究成果为：对任意奇数次多项式形式的耦合函数，当两个二次区间映射的参数对取自某个具有正Lebesgue测度的集合时，相应的Viana映射具有两个正Lyapunov指数和有限多个遍历的绝对连续不变测度。