Morse–Bott inequalities for endomorphisms

Let K be a finite simplicial complex, let g:K→ K g : K → K be a simplicial map and let f be a discrete Morse–Bott function on K satisfying f(g(σ ))≤ f(σ ) f ( g ( σ ) ) ≤ f ( σ ) for all simplices σ 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 .