区间映射之间拓扑半共轭的构造、光滑性和稳定性

Many topics in dynamical systems involve problems of conjugacy, for instance, structural stability, linearization, normal form, ergodic theory, and iterative functional equation. Problems of conjugacy is directly linked to the conjugacy equation. This project investigates the construction, smoothness and stability of topological (semi-)conjugacy between multimodal mappings and between multimodal mapping and piecewise continuous mappings of the interval. We will use an iterative construction method and methods of symbolic dynamical system respectively to construct all topological (semi-)conjugacy between these two families of mappings, and generalize Koebe's distortion theorem with the cross ratio tool to estimate the degree of smoothness of these topological (semi-)conjugacies, then apply the notion "continuity degree" to prove the stability of these topological (semi-)conjugacies. These investigations can deeply characterize the dynamic nature of one-dimensional nonlinear mappings, more finely classify interval mappings, and help us to link conjugacy equation with two-scale difference equations and singularity functions.

动力系统中的许多主题涉及到共轭问题，如结构稳定性、线性化、正规形、遍历理论和迭代函数方程。共轭问题与相应的共轭方程有直接联系。本项目将研究多峰映射和逐段连续的区间映射之间拓扑（半）共轭的构造、光滑性和稳定性。拟用迭代构造法和符号动力系统的方法分别构造这两类映射族间的所有拓扑（半）共轭；并利用交比工具来改进Koebe扭曲定理，估计某些拓扑（半）共轭的光滑度；然后利用"连续度"的概念，证明拓扑（半）共轭的稳定性。这些研究能够深入刻画一维非线性映射的动力学性质；能够更加精细地对区间映射进行分类；也有助于寻找共轭方程与两尺度差分方程和奇异函数的联系。

动力系统中的许多主题涉及到共轭问题，如结构稳定性、线性化、正规形、遍历理论和迭代函数方程。共轭问题与相应的共轭方程有直接联系。本项目研究了非对称的Bernoulli变换、斜帐篷映射、梯形映射、多峰映射这四类的区间映射的拓扑（半）共轭的构造及其光滑性。用迭代构造法或符号动力系统的方法分别构造这四类映射族间的拓扑（半）共轭；并利用符号动力系统的实数表示和水平集，证明了共轭函数的奇异性、不可微性、Hölder连续性，估计某些拓扑（半）共轭的Hölder指数；利用不动点原理，构造了一致收敛的函数序列，并且给出了这些函数序列逼近拓扑（半）共轭的误差估计。这些研究深入刻画一维非线性映射的动力学性质；更加精细地对区间映射进行分类。证明了非对称的Bernoulli变换之间的拓扑共轭是严格递增的奇异函数，它是de Rham函数自然的推广，并指出de Rham函数与Salem函数虽然构造方式不一样，但是从拓扑共轭的角度看，它们是对应同一函数。

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