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I am reading about the Poincaré-Bendixson theorem in the plane, I really liked the theorem. I have seen common applications in Sotomayor and Perko's book. But I would like to know what other applications outside of the common literature (mentioned above) there are, I mean interesting applications that you create valuable. If possible, mention a reference to your example. Thank you very much

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Let me chime in with a striking application in topology, originally due to Smale.

First, an exercise: let $I = [0, 1]$ be a closed interval and consider $\mathrm{Diff}_\partial(I)$, the group of self-diffeomorphisms of $I$ fixing the boundary $\partial I = \{0, 1\}$ pointwise. There is an obvious way to make this a topological group, where we demand two diffeomorphisms are nearby if they are nearby upto derivatives of all orders. Then $\mathrm{Diff}_{\partial}(I)$ is contractible.

Here is a harder follow-up: let $D^n$ denote the closed $n$-disk and this time let $\mathrm{Homeo}_\partial(D^n)$ be the topological group of self-homeomorphisms of $D^n$ fixing $\partial D^n$ pointwise, the topology being $C^0$-topology. Then $\mathrm{Homeo}_\partial(D^n)$ is also contractible. (Here is an idea that you can try to make precise: given a homeomorphism $f : D^n \to D^n$ fixing the boundary, we need to construct a canonical path to the identity homeomorphism, in the space $\mathrm{Homeo}_\partial(D^n)$; so find a homotopy of $f$ to identity where every level is a homeomorphism - also called an isotopy. One way to do it is to first make sure $f(0) = 0$ by an isotopy, and then consider $D^n$ as foliated by the concentric spheres around the origin. Applying $f$ will wildly move around the leaves of this foliation, but on the outermost leaf $\partial D^n$, it is identity, so in particular $f$ preserves this one leaf. Using this as a pivot, "comb" the messed-up foliation inward by a linear interpolation trick and make the "mess" disappear into the point $0$, so that at the end you're left with the same foliation as before.)

If you have successfully figured out the follow-up, you'll realize that the above, known as the Alexander trick, could never have worked if we started out with $\mathrm{Diff}_\partial(D^n)$, because any semblance of differentiability will be lost at the origin where all the wild stuff shrinks off. Nevertheless, we have:

Theorem (Smale): $\mathrm{Diff}_\partial(D^2)$ is contractible.

Here is a rough sketch of Smale's original proof. First, one can take $I^2$ instead of $D^2$ and focus only on diffeomorphisms which are identity on an unspecified collar neighborhood of the boundary; these are easily seen to be weak homotopy equivalent. So pick such an $f : I^2 \to I^2$, and just for convenience extend $f$ by identity to $f : \Bbb R^2 \to \Bbb R^2$. We must come up with a canonical isotopy between $f$ and identity, through diffeomorphisms supported on a slightly smaller square in $I^2$.

The main idea is to consider the vector field $X = f_*(\partial/\partial x)$, which is a nonzero vector field that agrees with $\partial/\partial x$ outside a slightly smaller square in $I^2$. Then $f$ can be recovered from $X$ as $f(x, y) = \Phi^x_X(0, y)$. It so happens, now, that any arbitrary nonzero vector field $X$ which agrees with $\partial/\partial x$ outside a slightly smaller square in $I^2$ will be of this form. To see this, the first step is to prove every flowline of $X$ emanating from a point of $0 \times I$ exits through a unique point on $1 \times I$ in finite time. If not, choose an offending flowline $\gamma$. Then by Poincare-Bendixson theorem the forward-time limit set of $\gamma$ is (1) a fixed point, (2) a bunch of disjoint homoclinic/heteroclinic flowlines going between fixed points, or (3) a periodic flowline. (1) and (2) are impossible as $X$ is nowhere vanishing; if (3) happens then the periodic flowline traps a Jordan domain in $\Bbb R^2$, and the flow of $X$ is a self-diffeomorphism of this topological disk, which must have a fixed point, again a contradiction. This proves the claim. Uniqueness of the point it hits $1 \times I$ at is obvious since $X = \partial/\partial x$ in a neighborhood of $1 \times I$.

Next, denote for any $(0, y) \in 0 \times I$ the hitting time $\tau_X(y)$ to be the time it takes for the flowline of $X$ starting at $(0, y)$ to reach $1 \times I$. Then $f(x, y) = \Phi^{\tau_X(y)}_X(0, y)$ is a diffeomorphism of $\Bbb R^2$, except it won't be identity outside a smaller square in $I^2$ as the flow of $X$ might scale in the $x$- and $y$-directions. This is easily fixable; essentially the reason is contractibility of $\mathrm{Diff}_\partial(I)$, where one takes $I$ to be one of the four edges of $I^2$.

This shows $\mathrm{Diff}_\partial(D^2)$ is homotopy equivalent to the space $\mathcal{S}$ of nowhere-zero vector fields on $I^2$ which agrees with $\partial/\partial x$ on a collar neighborhood of $\partial I^2$, which is the same as the subspace of the space of maps $\phi : I^2 \to \Bbb R^2 \setminus \{0\}$ such that $\phi \equiv (1, 0)$ on a collar neighborhood of $\partial I^2$. Map-lifting to the universal cover of $\Bbb R^2 \setminus \{0\}$ and doing an appropriately smoothened straightline-homotopy, we get that $\phi$ is canonically homotopic to the constant map to $(1, 0)$. This proves a contraction of $\mathcal{S}$ to the vector field $\partial/\partial x$, hence $\mathrm{Diff}_\partial(D^2) \simeq \mathcal{S}$ is contractible. $\blacksquare$

Smale conjectured, and Hatcher famously proved that $\mathrm{Diff}_\partial(D^3)$ is also contractible. In higher dimensions such things are in general false, because of existence of exotic spheres. Hatcher's "A 50-Year View of Diffeomorphism Groups" might be a good introduction to what is known. I learnt the above proof from Sanders Kuper's excellent book "Lectures on Diffeomorphism Groups Of Manifolds", where the remaining details of the sketch may be found.

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  • $\begingroup$ Wow what a beautiful application, thank you very much for spending your time on such a detailed answer. I will leave the question open so that others are encouraged to write their most interesting application, meanwhile I will read the book you mentioned, thanks again $\endgroup$
    – Zaragosa
    Commented Feb 22, 2022 at 14:33
  • $\begingroup$ Hello Balarka Sen, what an interesting application. This subject is new to me, I would like to read a book that has all the necessary tools to demonstrate the initial exercise that you posted about $\mathrm{Diff}_\partial(I)$, because to understand Smale's proof well you already gave an excellent reference. Thanks in advance for the help. $\endgroup$
    – Leeland
    Commented Jan 21, 2023 at 17:57

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