Morse-Bott inequalities for endomorphisms

TL;DR

The paper addresses how to relate the dynamics of a simplicial endomorphism $g$ to a discrete Morse-Bott function $f$ on a finite ordered simplicial complex by proving generalized Morse-Bott inequalities that extend Pitcher's ideas to the discrete setting. It introduces the local $k$-trace and a filtration induced by $f$, establishing strong and weak Morse-Bott inequalities for endomorphisms across iterates $g^l$ and localizing Lefschetz data to critical objects. The results generalize Forman-Bott discrete Morse-Bott theory to endomorphisms, include full proofs and a worked example, and specialize to the classical case when $g$ is the identity. Practically, these inequalities provide topological constraints and persistence measures for discrete dynamical systems modeled by $g$, linking combinatorial dynamics with Morse-Bott topology in a Lefschetz-type framework.

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 $f(g(σ))\leq f(σ)$ for all simplices $σ$ 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 $f$.

Figures (1)

• Figure 1.1: Simplicial complex $K$ and combinatorial Morse-Smale combinatorial vector field.

Theorems & Definitions (11)

• Lemma 1.1
• proof
• Definition 1.2
• Remark 1.3
• Definition 1.4
• Theorem 1.5: Strong and Weak inequalities
• proof
• Remark 1.6
• Remark 1.7
• Remark 1.8