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# How much "Morse theory" can be accomplished given only a continuous transformation of a space?

If $M$ is a Riemannian manifold and $f:M\to \mathbb{R}$ a Morse-Smale function (which is just a rigorous way to say "generic smooth function"), then Morse theory essentially recovers the manifold itself from relatively basic information about the gradient flow diffeomorphisms of $f$. To sketch briefly: for each pair of critical points $p$ and $q$ of $f$ (i. e., fixed points of the diffeomorphisms), we can consider the subset $S_{p,q}$ of $M$ that is attracted to $p$ and repelled from $q$ under the diffeomorphisms. ("Repelled from $q$" just means attracted to $q$ under the inverse of the diffeomorphisms.) These $S_{p,q}$ essentially constitute a decomposition of $M$ as a cell complex. If you just want the homology groups, then you can get away with just considering pairs of critical points whose indices differ by one, and if you just want the Euler characteristic, then you only need local information around each critical point (to define its index). The index of a critical point $p$ is the number of negative eigenvalues of the Hessian (which does not actually depend on the coordinates chosen or even the metric), and it is also the dimension of the submanifold of points in any small neighborhood of $p$ that are attracted to $p$ under the gradient flow diffeomorphisms.

I want to know how much of that can be done if we don't have $f:M\to \mathbb{R}$, but just some transformation $F:M\to M$ homotopic to the identity, and if $M$ isn't necessarily even a manifold (but probably compact and metrizable). Given information about the fixed points of $F$ (or other dynamical information?), how much of the topology of $M$ can be recovered? (Can we still try to define the "index" of a fixed point of $F$ by looking at the set of points that are attracted to it as $F$ is iterated?)

Some thoughts:

• For some $M$, there might well be maps $F$ that have no fixed points at all. If the Euler characteristic can be recovered from the fixed points of $F$, then such $M$ would have to have an Euler characteristic of zero. (Is that the case??) So the fixed points of $F$ are not very useful in such cases, but are there more general dynamical features of $F$ that relate to the topology of $M$?

• Some $M$ might admit perfectly continuous, even smooth, $F$ with chaotic dynamics.

• If $F$ has a unique fixed point $x_0$ and for every $x\in M$, $F^n(x)\to x_0$, then $M$ is contractible (recalling our assumption that $F$ is homotopic to the identity).

• Can we get better results by considering an even more restrictive class of transformations? Of course, I don't want to go as far as to say that $F$ belongs to some group of gradient flow diffeomorphisms on a manifold, but maybe we can try to relax that by supposing there exists $f:M\to \mathbb{R}$ such that $f\circ F \geq f$. (That condition makes sense even if $M$ is not a manifold.)