Morse-Bott inequalities for endomorphisms

Let $K$ be a finite simplicial complex, let $g\text{\hspace{0.17em}⁣}:K\to K$ be a simplicial map and let $f$ be a discrete Morse-Bott function on $K$ satisfying $f(g(\sigma ))\le f(\sigma )$ for all simplices $\sigma$ in $K$. We establish a set of inequalities (generalizing the Morse-Bott inequalities which we recover as a particular case when $g$ is the identity) relating the dynamics of $g$ and $f$.