Stability of injection

Shadow · **Posted:** Thu, 8 Mar 2012 04:39:59 UTC

I'm convinced that smooth injections with compact domains are not stable under small perturbations, but I cannot come up with a counterexample. Does anyone have one available?

And as long as I'm at it, is there a good smooth homeomorphism which is not stable?

outermeasure · **Posted:** Thu, 8 Mar 2012 05:31:50 UTC

Shadow wrote:

I'm convinced that smooth injections with compact domains are not stable under small perturbations, but I cannot come up with a counterexample. Does anyone have one available?

And as long as I'm at it, is there a good smooth homeomorphism which is not stable?

Small perturbation in which what sense?

I think C^1-small perturbations still give you C^1-injections. (The usual inverse function theorem gives the immersion+local injection part. To globalise, use the existence of tubular neighbourhood.)

Shadow · **Posted:** Thu, 8 Mar 2012 05:42:13 UTC

outermeasure wrote:

Shadow wrote:

I'm convinced that smooth injections with compact domains are not stable under small perturbations, but I cannot come up with a counterexample. Does anyone have one available?

And as long as I'm at it, is there a good smooth homeomorphism which is not stable?

Small perturbation in which what sense?

I think C^1-small perturbations still give you C^1-injections.

They are $C^{\infty}$ .

outermeasure · **Posted:** Thu, 8 Mar 2012 05:48:44 UTC

Shadow wrote:

outermeasure wrote:

Shadow wrote:

I'm convinced that smooth injections with compact domains are not stable under small perturbations, but I cannot come up with a counterexample. Does anyone have one available?

And as long as I'm at it, is there a good smooth homeomorphism which is not stable?

Small perturbation in which what sense?

I think C^1-small perturbations still give you C^1-injections.

They are $C^{\infty}$ .

Yes, but you still need to measure "smallness" of your perturbation in some sense. If you measure it in C^0 then I believe horrible things can happen, but if you measure it in C^1 (or stronger) then everything in the garden is lovely.

Shadow · **Posted:** Thu, 8 Mar 2012 05:51:47 UTC

outermeasure wrote:

Shadow wrote:

outermeasure wrote:

Shadow wrote:

I'm convinced that smooth injections with compact domains are not stable under small perturbations, but I cannot come up with a counterexample. Does anyone have one available?

And as long as I'm at it, is there a good smooth homeomorphism which is not stable?

Small perturbation in which what sense?

I think C^1-small perturbations still give you C^1-injections.

They are $C^{\infty}$ .

Yes, but you still need to measure "smallness" of your perturbation in some sense. If you measure it in C^0 then I believe horrible things can happen, but if you measure it in C^1 (or stronger) then everything in the garden is lovely.

Well, not surjectivity at least.

You can use $f_t:[0,1]\to [0,1]$ , f_t(x)=x^3-tx

and there is no small delted nbhd of 0 where f_t

is onto. We're measuring smallness in that way.

This gives me hope that injections are also unstable, and similarly smooth homeomorphisms.

outermeasure · **Posted:** Thu, 8 Mar 2012 06:00:07 UTC

Shadow wrote:

outermeasure wrote:

Shadow wrote:

outermeasure wrote:

Shadow wrote:

I'm convinced that smooth injections with compact domains are not stable under small perturbations, but I cannot come up with a counterexample. Does anyone have one available?

And as long as I'm at it, is there a good smooth homeomorphism which is not stable?

Small perturbation in which what sense?

I think C^1-small perturbations still give you C^1-injections.

They are $C^{\infty}$ .

Yes, but you still need to measure "smallness" of your perturbation in some sense. If you measure it in C^0 then I believe horrible things can happen, but if you measure it in C^1 (or stronger) then everything in the garden is lovely.

Well, not surjectivity at least.

You can use $f_t:[0,1]\to [0,1]$ , f_t(x)=x^3-tx

and there is no small delted nbhd of 0 where f_t

is onto. We're measuring smallness in that way.

This gives me hope that injections are also unstable, and similarly smooth homeomorphisms.

Oops, yes, you want injections not immersions. There are no ways to perturb $f\colon\mathbb{RP}^1\to\mathbb{RP}^1$ , $x\mapsto x^3$ along something which is expansive at 0 and/or infinity.

Shadow · **Posted:** Thu, 8 Mar 2012 06:04:59 UTC

outermeasure wrote:

Shadow wrote:

outermeasure wrote:

Shadow wrote:

outermeasure wrote:

Shadow wrote:

I'm convinced that smooth injections with compact domains are not stable under small perturbations, but I cannot come up with a counterexample. Does anyone have one available?

And as long as I'm at it, is there a good smooth homeomorphism which is not stable?

Small perturbation in which what sense?

I think C^1-small perturbations still give you C^1-injections.

They are $C^{\infty}$ .

Yes, but you still need to measure "smallness" of your perturbation in some sense. If you measure it in C^0 then I believe horrible things can happen, but if you measure it in C^1 (or stronger) then everything in the garden is lovely.

Well, not surjectivity at least.

You can use $f_t:[0,1]\to [0,1]$ , f_t(x)=x^3-tx

and there is no small delted nbhd of 0 where f_t

is onto. We're measuring smallness in that way.

This gives me hope that injections are also unstable, and similarly smooth homeomorphisms.

Oops, yes, you want injections not immersions. Just patch these together.

I'm not sure what you mean, the domain has to be compact, so I don't think a patch job is helpful, unless I'm misunderstanding you (which is entirely possible). If that was another misunderstanding, please comment to that effect, otherwise I'll try and think about what you said to figure out how to work it.

outermeasure · **Posted:** Thu, 8 Mar 2012 06:25:23 UTC

Shadow wrote:

I'm not sure what you mean, the domain has to be compact, so I don't think a patch job is helpful, unless I'm misunderstanding you (which is entirely possible). If that was another misunderstanding, please comment to that effect, otherwise I'll try and think about what you said to figure out how to work it.

The problem with $f_t(x)=x^3-tx;\quad\mathbb{R}\to\mathbb{R}$ is that it fails to inject when t>0, and it manifests in the example I gave in my previous post, namely: the map $f\colon\mathbb{RP}^1\to\mathbb{RP}^1;\quad x\mapsto x^3$ near 0 (and infinity) is f_0(x)=x^3

, and so if you try to pull apart with nonzero derivative at 0 and infinity, you end up with this same problem.

Of course, you can measure your "smallness" in a different norm that accounts for that (essentially, C^1 with respect to another differential structure such that your original map is an immersion).

Shadow · **Posted:** Thu, 8 Mar 2012 06:35:18 UTC

outermeasure wrote:

Shadow wrote:

I'm not sure what you mean, the domain has to be compact, so I don't think a patch job is helpful, unless I'm misunderstanding you (which is entirely possible). If that was another misunderstanding, please comment to that effect, otherwise I'll try and think about what you said to figure out how to work it.

The problem with $f_t(x)=x^3-tx;\quad\mathbb{R}\to\mathbb{R}$ is that it fails to inject when t>0, and it manifests in the example I gave in my previous post, namely: the map $f\colon\mathbb{RP}^1\to\mathbb{RP}^1;\quad x\mapsto x^3$ near 0 (and infinity) is f_0(x)=x^3

, and so if you try to pull apart with nonzero derivative at 0 and infinity, you end up with this same problem.

Of course, you can measure your "smallness" in a different norm that accounts for that (essentially, C^1 with respect to another differential structure such that your original map is an immersion).

Right, the issue is that we were supposed to have a compact domain, and ostensibly this might not be a problem there.

I made a small error before, though, my domain and codomain should be [-1,1], and then I can break all three at once I think, since x^3

is a smooth homeomorphism on [-1,1]

and for small t, I know that I can guarantee that the f_t

still has the right codomain (the extreme values on the interior happen at $\pm\sqrt{t\over 3}$ , and give outputs which are within the right range to still land in [-1,1]. Since I break 1-1 AND onto, clearly the homeomorphism property is also broken.

I like the projective space version as well, it gives me a boundary-less version, which is really nice, because I could not think of one before that.

Thanks for the help outermeasure, and for the expanded ideas/examples--they're quite understandable, and nice to have in my box of things to draw on!

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Stability of injection

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