# Fixed point property.

• May 2nd 2011, 12:15 PM
MathSucker
Fixed point property.
I'm trying to find if the following has the fixed point property:

$\displaystyle S= \{ (x,y) \, : \, x^2+y^2=1 \}$

Is it enough to show:

Let $\displaystyle f \, : S \rightarrow S$

$\displaystyle 0^2 + (-1)^2=1 \,$ so $\displaystyle (0,-1)$ belongs to $\displaystyle S$ but $\displaystyle f(0,-1) = (0,1)$

Thus, $\displaystyle S$ does not have the fixed point property.
• May 2nd 2011, 12:35 PM
Ackbeet
Not sure I understand. Typically, you think of a function as having the fixed point property. What is your function f from S to S?
• May 2nd 2011, 02:11 PM
Drexel28
Quote:

Originally Posted by MathSucker
I'm trying to find if the following has the fixed point property:

$\displaystyle S= \{ (x,y) \, : \, x^2+y^2=1 \}$

Is it enough to show:

Let $\displaystyle f \, : S \rightarrow S$

$\displaystyle 0^2 + (-1)^2=1 \,$ so $\displaystyle (0,-1)$ belongs to $\displaystyle S$ but $\displaystyle f(0,-1) = (0,1)$

Thus, $\displaystyle S$ does not have the fixed point property.

What's your mapping? What about $\displaystyle e^{i\theta}\mapsto e^{i\theta+\frac{\pi}{2}}$ this is evidently continuous and doesn't have a fixed point.
• May 2nd 2011, 02:13 PM
Drexel28
Quote:

Originally Posted by Ackbeet
Not sure I understand. Typically, you think of a function as having the fixed point property. What is your function f from S to S?

Ackbeet, generally a structure $\displaystyle X$ is said to have the 'fixed point property' if every morphism $\displaystyle X\to X$ has a fixed point.
• May 3rd 2011, 09:50 AM
MathSucker
Quote:

Originally Posted by Drexel28
What's your mapping? What about $\displaystyle e^{i\theta}\mapsto e^{i\theta+\frac{\pi}{2}}$ this is evidently continuous and doesn't have a fixed point.

Okay, forget the second part. I was just trying to find out if and why/why not $\displaystyle S$ had the fixed point property...
• May 3rd 2011, 10:31 AM
Deveno
Quote:

Originally Posted by Drexel28
Ackbeet, generally a structure $\displaystyle X$ is said to have the 'fixed point property' if every morphism $\displaystyle X\to X$ has a fixed point.

then this would depend on the structure imposed on S. for example, if one is regarding S as a topological group, then sure, because then (1,0) has to be a fixed point. but if one is just regarding S as a topological space, then no, we can pick any rotation by an angle that is not an integral multiple of $\displaystyle 2\pi$
• May 3rd 2011, 11:44 AM
Drexel28
Quote:

Originally Posted by Deveno
then this would depend on the structure imposed on S. for example, if one is regarding S as a topological group, then sure, because then (1,0) has to be a fixed point. but if one is just regarding S as a topological space, then no, we can pick any rotation by an angle that is not an integral multiple of $\displaystyle 2\pi$

Thank you, I am aware of this. But I think it's clear from the OPs question that we are talking about top. spaces and not top. groups.
• May 3rd 2011, 11:47 AM
MathSucker
Now that I look at it again, the question has it as $\displaystyle S^1$

Does that change things?
• May 3rd 2011, 11:52 AM
Drexel28
Quote:

Originally Posted by MathSucker
Now that I look at it again, the question has it as $\displaystyle S^1$

Does that change things?

Now, $\displaystyle \mathbb{S}^1$ is just the unit circle and any rotation by $\displaystyle 0<\theta<2\pi$ is a continuous map without a fixed point.