均值型映射关于算术均值的不变及嵌入流问题

The dynamic system composed of mean-type mappings has a long history and has been widely used in many fields, such as economics, physics, chemistry, medicine and so on. Studying Gauss iteration and invariant equation, the iterative root and embedded flows of means, we can kindly know the dynamics of the dynamic system composed of mean-type mappings. The project focuses on the dynamics of the dynamic system composed of some known mean-type mappings, which will be studied from three aspects: the invariant equation of Cauchy mean with respect to arithmetic mean, the invariant equation of generalized Bajraktarevic mean with respect to arithmetic mean, and embedded flows of Matkowski means. To solve the first two problems, we will use the theory of determinant function to break through the difficulty of that there exist four unknown functions in this kind of problems. For the third problem, we will firstly study the square iterative root of Matkowski means using the theory of invariant principle, and then construct its rational iterations, and finally get the concrete form of embedded flow. This project will also provide conditions for training young scholars in mathematics discipline of Southwest University of Science and technology, help the construction of the major dynamic system in basic mathematics of this college, enhance the overall development level and influence of mathematics discipline of Southwest University of Science and technology, and promote the balanced development of mathematics research level in China.

均值型映射构成的动力系统研究历史悠久，应用广泛，现涉及的领域有经济学、物理、化学及医学等。研究均值的Guass迭代与不变方程、迭代根与嵌入流问题能很好地了解均值型映射构成的动力系统的动力学行为。本项目着重研究几类均值构成的动力系统的动力学性质，具体研究内容大致分为三个方面：一研究Cauchy均值关于算术均值的不变方程；二研究推广的Bajraktarevic均值关于算术均值的不变方程；三研究Matkowski均值的嵌入流问题。前两个问题拟引用行列式函数理论来突破该类问题中未知函数为四个的难点，第三个问题拟由Matkowski均值满足的不变恒定式理论入手研究二次迭代根问题，接着构造其有理数次的迭代，最终得到嵌入流的具体形式。本项目还将为西南科技大学数学学科培养青年学者提供条件，助力西南科技大学基础数学动力系统方向的建设，提升该校数学学科的整体发展水平和影响，促使我国数学研究水平的均衡发展。

均值是数学学科中一个基础又重要的概念，比如一些常见范数或度量的定义就是均值，均值有着明确的几何意义，其研究历史悠久并应用广泛，目前涉及均值的领域还有经济学、静电学、热传导、化学及医学等。均值的Gauss迭代的收敛问题与均值的不变方程关系紧密，所以研究均值的不变方程可以更好地了解均值型映射构成的动力系统的动力学性质。Cauchy均值和Bajraktarevic均值分别是Lagrangian均值和拟算术均值的推广，对应的不变方程问题涉及四个未知函数，目前仍是未完全解决的公开问题。推广的Bajraktarevic均值含有两个衍生函数和一个概率测度，是包含了Cauchy均值和Bajraktarevic均值的更广泛的一类均值。本项目采用Wronski行列式相关知识构造辅助函数，根据衍生测度的中心距对应的取值分类讨论了推广的Bajraktarevic均值关于算术均值的不变方程，得到了所有可能解满足的条件。作为部分结论的应用，我们得到了当衍生分母函数满足谐振子方程时，Cauchy均值和Bajraktarevic均值分别关于算术均值的不变方程解的性质。重要研究成果见论文[On the invariance of generalized quasiarithmetic means]。同时，对于Z^2保测动力系统，为了刻画零熵系统的复杂性，我们定义了方向上的测度熵维数，并比较了方向熵维数和方向拓扑熵维数。该部分成果还在进一步整理和完善中。本课题的研究有助于均值函数方程和拓扑动力系统基础理论的拓展和创新，结合自身独特的分析方法和Wronski行列式函数的引入，有效地解决了商均值类的不变方程问题。