Dynamical Morse entropy

We consider actions of a tileable amenable group $\mathrm{\Gamma }$ on a topological space $X$. For a continuous function on $X$, we define the entropy of the number of homologically detectable critical point of the average of that function over $\mathrm{\Gamma }$. This number is bounded below by the sum of the Betti number entropy. This result is thus a generalization of a standard Morse inequality in differential geometry to this setting.