On Bertelson-Gromov Dynamical Morse Entropy

TL;DR

This paper provides an expository overview of the proof of results related to Dynamical Morse Entropy, which measures the growth of critical points of averaged Morse functions on manifolds, with applications to statistical mechanics models.

Contribution

It offers a detailed proof and explanation of the concepts behind Dynamical Morse Entropy, expanding on prior work by Bertelson and Gromov, and clarifies its relation to Betti number entropy and statistical mechanics.

Findings

Describes asymptotic growth of critical points in Morse functions

Connects Morse entropy to Betti number entropy without differentiability

Applies to potentials in the XY model of statistical mechanics

Abstract

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.