On Bertelson-Gromov Dynamical Morse Entropy

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.