Mixing
Example of a homeomorphism of a plane which fixes unit circle point wise
0
$begingroup$
What would a non trivial example of a homeomorphism of a plane which fixes unit circle point wise. If I take
begin{equation}
h(x,y)=begin{cases}
(x,y)& text{$(x,y)in S^1$}\
(x+2,y+2) & text{elsewhere}
end{cases}
end{equation}
Is this a homeomorphism?
real-analysis general-topology geometric-topology geometric-group-theory
asked Jan 19 at 16:09
ershersh
423113
$endgroup$
|
show 3 more comments
0
$begingroup$
What would a non trivial example of a homeomorphism of a plane which fixes unit circle point wise. If I take
begin{equation}
h(x,y)=begin{cases}
(x,y)& text{$(x,y)in S^1$}\
(x+2,y+2) & text{elsewhere}
end{cases}
end{equation}
Is this a homeomorphism?
real-analysis general-topology geometric-topology geometric-group-theory
asked Jan 19 at 16:09
ershersh
423113
$endgroup$
3
$begingroup$
It's not continuous.
$endgroup$
– Lord Shark the Unknown
Jan 19 at 16:11
$begingroup$
Yes Right!! I am not able to think of such homeomorphisms, any help is appreciated.
$endgroup$
– ersh
Jan 19 at 16:13
1
$begingroup$
How about defining a suitable function $g:Bbb R^2to Bbb R^+$ which is constant on circles centred at the origin and consider $f(x,y)=g(x,y)(x,y)$.
$endgroup$
– Lord Shark the Unknown
Jan 19 at 16:15
1
$begingroup$
Take the flow of a bounded smooth vector field that vanishes on the circle.
$endgroup$
– YCor
Jan 19 at 23:29
3
$begingroup$
@WilliamElliot no. His example $(x,y)mapsto (x^2+y^2)^{-1/4}(x,y)$ works.
$endgroup$
– YCor
Jan 19 at 23:38
|
show 3 more comments
0
0
0
$begingroup$
What would a non trivial example of a homeomorphism of a plane which fixes unit circle point wise. If I take
begin{equation}
h(x,y)=begin{cases}
(x,y)& text{$(x,y)in S^1$}\
(x+2,y+2) & text{elsewhere}
end{cases}
end{equation}
Is this a homeomorphism?
real-analysis general-topology geometric-topology geometric-group-theory
asked Jan 19 at 16:09
ershersh
423113
$endgroup$
What would a non trivial example of a homeomorphism of a plane which fixes unit circle point wise. If I take
begin{equation}
h(x,y)=begin{cases}
(x,y)& text{$(x,y)in S^1$}\
(x+2,y+2) & text{elsewhere}
end{cases}
end{equation}
Is this a homeomorphism?
real-analysis general-topology geometric-topology geometric-group-theory
real-analysis general-topology geometric-topology geometric-group-theory
asked Jan 19 at 16:09
ershersh
423113
asked Jan 19 at 16:09
ershersh
423113
asked Jan 19 at 16:09
ershersh
423113
asked Jan 19 at 16:09
ershersh
423113
asked Jan 19 at 16:09
ershersh
423113
423113
3
$begingroup$
It's not continuous.
$endgroup$
– Lord Shark the Unknown
Jan 19 at 16:11
$begingroup$
Yes Right!! I am not able to think of such homeomorphisms, any help is appreciated.
$endgroup$
– ersh
Jan 19 at 16:13
1
$begingroup$
How about defining a suitable function $g:Bbb R^2to Bbb R^+$ which is constant on circles centred at the origin and consider $f(x,y)=g(x,y)(x,y)$.
$endgroup$
– Lord Shark the Unknown
Jan 19 at 16:15
1
$begingroup$
Take the flow of a bounded smooth vector field that vanishes on the circle.
$endgroup$
– YCor
Jan 19 at 23:29
3
$begingroup$
@WilliamElliot no. His example $(x,y)mapsto (x^2+y^2)^{-1/4}(x,y)$ works.
$endgroup$
– YCor
Jan 19 at 23:38
|
show 3 more comments
3
$begingroup$
It's not continuous.
$endgroup$
– Lord Shark the Unknown
Jan 19 at 16:11
$begingroup$
Yes Right!! I am not able to think of such homeomorphisms, any help is appreciated.
$endgroup$
– ersh
Jan 19 at 16:13
1
$begingroup$
How about defining a suitable function $g:Bbb R^2to Bbb R^+$ which is constant on circles centred at the origin and consider $f(x,y)=g(x,y)(x,y)$.
$endgroup$
– Lord Shark the Unknown
Jan 19 at 16:15
1
$begingroup$
Take the flow of a bounded smooth vector field that vanishes on the circle.
$endgroup$
– YCor
Jan 19 at 23:29
3
$begingroup$
@WilliamElliot no. His example $(x,y)mapsto (x^2+y^2)^{-1/4}(x,y)$ works.
$endgroup$
– YCor
Jan 19 at 23:38
3
3
$begingroup$
It's not continuous.
$endgroup$
– Lord Shark the Unknown
Jan 19 at 16:11
$begingroup$
It's not continuous.
$endgroup$
– Lord Shark the Unknown
Jan 19 at 16:11
$begingroup$
Yes Right!! I am not able to think of such homeomorphisms, any help is appreciated.
$endgroup$
– ersh
Jan 19 at 16:13
$begingroup$
Yes Right!! I am not able to think of such homeomorphisms, any help is appreciated.
$endgroup$
– ersh
Jan 19 at 16:13
1
1
$begingroup$
How about defining a suitable function $g:Bbb R^2to Bbb R^+$ which is constant on circles centred at the origin and consider $f(x,y)=g(x,y)(x,y)$.
$endgroup$
– Lord Shark the Unknown
Jan 19 at 16:15
$begingroup$
How about defining a suitable function $g:Bbb R^2to Bbb R^+$ which is constant on circles centred at the origin and consider $f(x,y)=g(x,y)(x,y)$.
$endgroup$
– Lord Shark the Unknown
Jan 19 at 16:15
1
1
$begingroup$
Take the flow of a bounded smooth vector field that vanishes on the circle.
$endgroup$
– YCor
Jan 19 at 23:29
$begingroup$
Take the flow of a bounded smooth vector field that vanishes on the circle.
$endgroup$
– YCor
Jan 19 at 23:29
3
3
$begingroup$
@WilliamElliot no. His example $(x,y)mapsto (x^2+y^2)^{-1/4}(x,y)$ works.
$endgroup$
– YCor
Jan 19 at 23:38
$begingroup$
@WilliamElliot no. His example $(x,y)mapsto (x^2+y^2)^{-1/4}(x,y)$ works.
$endgroup$
– YCor
Jan 19 at 23:38
|
show 3 more comments
0
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3
$begingroup$
It's not continuous.
$endgroup$
– Lord Shark the Unknown
Jan 19 at 16:11
$begingroup$
Yes Right!! I am not able to think of such homeomorphisms, any help is appreciated.
$endgroup$
– ersh
Jan 19 at 16:13
1
$begingroup$
How about defining a suitable function $g:Bbb R^2to Bbb R^+$ which is constant on circles centred at the origin and consider $f(x,y)=g(x,y)(x,y)$.
$endgroup$
– Lord Shark the Unknown
Jan 19 at 16:15
1
$begingroup$
Take the flow of a bounded smooth vector field that vanishes on the circle.
$endgroup$
– YCor
Jan 19 at 23:29
3
$begingroup$
@WilliamElliot no. His example $(x,y)mapsto (x^2+y^2)^{-1/4}(x,y)$ works.
$endgroup$
– YCor
Jan 19 at 23:38