Morse–Bott inequalities for endomorphisms

Abstract Let K be a finite simplicial complex, let $$g:K\rightarrow K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>:</mml:mo> <mml:mi>K</mml:mi> <mml:mo>→</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> be a simplicial map and let f be a discrete Morse–Bott function on K satisfying $$f(g(\sigma ))\le f(\sigma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>(</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> <mml:mo>≤</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for all simplices $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> 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 .