  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.
