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Here's $f(x) = x^2,$ it has the property that $f(-x) = f(x).$

I have a couple questions related to this topic:

1.) Is there a broader name for symmetries of the form $f(h(x)) = f(x)$ for invertible $h$?

2.) Is there a broader name of symmetries that keep a manifold invariant, for the cases that a manifold is defined by a system of equations like $x^2 + y = z$, $z^2 - 1 = y^2$ and so on?

3.) How do we keep the "symmetry" invariant of the specific coordinates? Every parabola has a line of symmetry about its vertex, but in the example I provided, the line of symmetry is stuck at the line $x = 0,$ how do we classify and find symmetries that are irregardless of the specific coordinates, to say that a parabola has a symmetry about some line centered at its vertex?

4.) How do we know we've found all symmetries of a given manifold, at least in 2D or 3D cases?

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  1. In this generality they're just called "symmetries." You'll have to get more specific to get a specific answer to this.
  2. It depends. Continuous symmetries are called homeomorphisms. Smooth symmetries are called diffeomorphisms. Generally, symmetries in some category are called automorphisms.
  3. It depends. Generally we try to find a description of the object we're interested in that doesn't depend on a choice of coordinates. In general it's a hard question to find all symmetries of some object.
  4. Again, in general it's a hard question to find all symmetries of some object. You know this the same way you know anything else in mathematics: by proving it. If you ask about a more specific example then more specific things can be said.
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  • $\begingroup$ Okay, let's take one question one at a time then. I'm interested in symmetries of the form $f(h(x)) = f(x),$ the generalization of $f(-x) = f(x).$ What branch of math most closely studies this? $f(-x) = f(x)$ is an example of a discrete symmetry, but could it be a holomorphic symmetry if you interpolate over complex numbers? $\endgroup$ Oct 10, 2022 at 18:03
  • $\begingroup$ @askquestions4: yes, one could ask for continuous families of symmetries, or smooth or even holomorphic. This is still a very general question so many branches of math are relevant, as I mentioned in my answer to your other question: group theory, Lie theory, representation theory, harmonic analysis... it would help if you got even more specific than this, e.g. naming a specific function $f$ and a specific class of symmetries $h$ you want to characterize (algebraic, continuous, smooth, holomorphic, etc.). $\endgroup$ Oct 10, 2022 at 18:17
  • $\begingroup$ But what I'm interested in is as general as I presented it. I guess we could be more specific to start with $f$ and $h$ being analytic. Given $f$, what is the name or process by which we find symmetries such that $f(h(x)) = f(x)$? $\endgroup$ Oct 10, 2022 at 18:19
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    $\begingroup$ @askquestions4: it's just called "finding the symmetries," I guess. Again, you are asking an extremely general question. I am going to specifically limit attention to the function $f(x) = x^2$ on $\mathbb{R}$ for simplicity. A basic observation is that any function $h$ satisfying $f(h(x)) = f(x)$ must permute the level sets $f^{-1}(r) = \{ \pm \sqrt{r} \}$. Equivalently, for any particular $x$ we must have $h(x) = x$ or $h(x) = -x$. It follows that if $h$ is continuous and invertible then the only possibilities are $h(x) = x$ for all $x$ or $h(x) = -x$ for all $x$. $\endgroup$ Oct 10, 2022 at 18:30
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    $\begingroup$ This example was pretty easy to deal with because the level sets were so small. If $f$ had been a function $\mathbb{R}^n \to \mathbb{R}$ then its level sets would be manifolds (or more complicated objects) and things would get more complicated. $\endgroup$ Oct 10, 2022 at 18:31

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