关于Bertelson-Gromov动力学Morse熵
On Bertelson-Gromov Dynamical Morse Entropy
摘要：
本文主要是对M. Bertelson和M. Gromov关于动态Morse熵这是对该作品中提出的思想的介绍。设$M$是有限维的紧定向连通$C^\infty$流形.假设$f_0 :M \to [0,1]$是满射Morse函数。对于给定的自然数$n$，考虑集合$M^n$，对于$x=(x_0,x_1,...,x_{n-1}) \in M^n$，记为$ f_n (x) = \frac{1}{n} \, \sum_{j=0}^{n-1} f_0 (x_j).$。动态莫尔斯熵描述了在固定区间$I\subset [0,1]$上，当$f_n$。与Betti数熵相关的部分不需要可微结构。人们可以用这种机制来描述统计力学的$XY$模型中定义的势的一般性质。
英文摘要：
In this mainly expository paper we present a detailed proof of several results contained in a paper by M. Bertelson and M. Gromov on Dynamical Morse Entropy. This is an introduction to the ideas presented in that work. Suppose $M$ is compact oriented connected $C^\infty$ manifold of finite dimension. Assume that $f_0 :M \to [0,1]$ is a surjective Morse function. For a given natural number $n$, consider the set $M^n$ and for $x=(x_0,x_1,...,x_{n-1}) \in M^n$, denote $ f_n (x) = \frac{1}{n} \, \sum_{j=0}^{n-1} f_0 (x_j).$ The Dynamical Morse Entropy describes for a fixed interval $I\subset [0,1]$ the asymptotic growth of the number of critical points of $f_n$ in $I$, when $n \to \infty$. The part related to the Betti number entropy does not requires the differentiable structure. One can describe generic properties of potentials defined in the $XY$ model of Statistical Mechanics with this machinery.
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