复与实非阿基米德动力系统及其应用

Dynamical system is a mathematical branch which studies the temporel evolution of a physical system,it has a broad scientific background and a long history.The concerned problems require us to deal with longtime,global and non-linear limit hehaviors of the system.The present project will study the non-Archimedean dynamical systems, it will involve the following mathematical theories like complex analysis,non-archimedean fields,topological dynamical systems,differentiable dynamical systems, ergodic theory, commutative algebra and probability theory etc. Its aim consists of investigating fundamental problems of non-Archimidean dynamical systems and even of general Cantor dynamical systems, and developping a general theory, by using the classical and recent results in the above mentioned mathematical theories and especially the theory of complex analysis and complex dynamical systems. Especially, we will emphasize the comparison of complex and real non-archimedean dynamics. The following topics are proposed to study: dynamical systems defined by entire functions on Berkovich space, minimality and minimal decomposition of compatible systems， Cantor systems from albegraic point of view, rational and picewise rational systems, Collatz problem, thermodynamical formalism of chaotic systems, stochastic behavior and multifractal property of the systems etc. The application domain of this theoretic research program includes computer science,cryptography and mathematical physics etc.

动力系统是研究系统随时间的变化而演变的数学科学，具有广泛的科学背景和悠远的研究历史，所研究的问题需要处理非线性大范围的长时间极限行为。 本项目的研究对象是非阿基米德数域上的动力系统。它涉及面广，交叉性强。它涉及到复分析，非阿基米德域，拓扑动力系统，微分动力系统，遍历理论，交换代数，概率论等数学分支，本项目的研究宗旨在于利用各数学分支的基本理论和最新成果，特别是运用复动力系统和复分析的理论，探讨复与实非阿基米德动力系统以及一般Cantor动力系统的基本问题，发展基本理论。对比研究复与实非阿基米德数域上动力系统的相同与差异。具体要研究的课题包括:Berkovich空间上超越整函数动力系统，相容系统的极小性和极小分解，Cantor系统的代数化研究，有理系统和分段有理系统，Collatz问题，混沌系统的热力学理论，系统的统计行为和重分形性质等。该理论研究的应用背景包括计算科学，密码学，数学物理等。

我们研究探讨了非阿基米德域上若干动力系统问题和分析问题，也考虑了欧式空间的相应问题。特别地是，我们证明了Fuglded谱猜想在一维p-进空间上是成立的(一维欧氏空间的问题依然是开问题); 研究了p-进数域上及其有限扩上不同动力系统的遍历性、极小分解和混沌性(包括收敛级数的动力系统，有理函数动力系统等); 证明一阶波动序列正交于平均Liapounov稳定的动力系统，特别是正交于所有的圆周自同胚 (与Sarnak猜想密切相关)； 证明了一个拓扑Wiener-Wintner加权遍历定理，作为应用之一，遍历幂零系统的轨道可以是无穷阶波动序列；用鞅方法成功地研究了一般的函数项级数的几乎处处收敛性，特别是遍历级数的几乎处处收敛性；证明了整数群上平稳行列式点过程几乎必然是满足加权遍历定理的好序列。

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