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So I just noticed that the set of functions with a fixed point $$f(x_0)=x_0,$$ are closed under composition $$(f\circ g)(x):=g(f(x)),$$ and with $e(x)=x$, the inverible functions even seem to form a (non-commutative) group.

Then if one chooses another point $x_1$ and restricts the set to the functions which also have $x_1$ as a fixed point, then it is again closed and so on.

If I have one parameterized point (i.e. a curve, or even a couple of those), then solving $f_t(x(t))=x(t)$ for the families $f_t$ should give me morphisms between the functions for different values of $t$.

Are there general considerations regarding this?

And is this somehow related to the characterization of points of a manifold via the ideal of functions which evaluate to $0$ that point?

Edit 1: Might be just a general property of homeomorphisms or something, although I don't associating picking out isolated fixed points with these kind of things.

Edit 2: I now see that this might relate a translation/transformation of points in the manifold to a transformation of the function algebra over that manifold. This has some features: If you take two points $y_1$ and $y_2$ and transformations along the curves $Y_1(t),Y_2(t)$ with $Y_1(0)=y_1, Y_1(1)=y_2$ and $Y_2(0): =y_2, Y_2(1)=y_1$ (they move into each other), then the fuction set with both fixed points $Y_1(t),Y_2(t)$ makes a loop as $\{Y_1(0),Y_2(0)\}=\{Y_1(1),Y_2(1)\}=\{y_1,y_2\}$. The particular form of the curves have an impact on how the function set looks in between.

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    Picking a fixed point just corresponds to removing that point from the domain and range of your functions (when looking at the invertible functions that is).2012-08-09
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    Congratulations: you have discovered the category of pointed sets/spaces/manifolds/etc.2012-08-09
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    @Nick Kidman: What do you mean by a morphism between functions?2012-08-09
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    @Tobias: But that's only true for in the category of sets, not in the category of topological space, for instance. Right?2012-08-09
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    @Rasmus: If you have a curve $x(t)$ connecting two points (=family of points) and you can solve the equation I wrote down for $f_t$ (for all the functions which have the fixed point $x(t)$ for any $t$), it seems you also got a family of functions. This shifts a bunch of functions into another bunch of functions in a specific way - that's what I called morphism. E.g. $f(x)=x^5$ has fixed point $x_0=x(t)=1$ and $f_t(x)=(x-t)^5+t$ has $f_0=f$ and fixed point $x(t)=1+t$. I don't know in what sense or in which cases of $x(t)$ a unique set of $f_t$'s follow.2012-08-09
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    As for the relation to functions on manifolds (or arbitrary metric spaces) that evaluate to 0: for $f: M \to M$ define $\tilde f: M \to \mathbb R$ via $d(x,f(x))$, where $d$ is the metric on $M$.2012-08-09
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    You are looking at the stabilizer subgroup for a group action on a set.2012-09-13
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    Without any structure, you're just looking at the set of functions that fix a subset of whatever it is you're looking at crossed with $\mathbb{R}$, to accommodate your desired one-parameter family of fixed points. Perhaps you'd like to specify the domain and codomain/structure of your functions?2013-11-20

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