Morse-Bott inequalities for endomorphisms

TL;DR

This paper generalizes Morse-Bott inequalities to relate the dynamics of a simplicial map and a discrete Morse-Bott function on a finite complex, extending classical results to a broader setting.

Contribution

It introduces inequalities connecting the dynamics of a simplicial map with a discrete Morse-Bott function, generalizing classical Morse-Bott inequalities to non-identity maps.

Findings

Established inequalities relating $g$ and $f$ dynamics.

Recovered classical Morse-Bott inequalities as a special case.

Extended Morse-Bott theory to simplicial complexes with endomorphisms.

Abstract

Let $K$ be a finite simplicial complex, let $g\colon K\to K$ be a simplicial map and let $f$ be a discrete Morse-Bott function on $K$ satisfying $f(g(\sigma))\leq f(\sigma)$ 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 $f$.