Mixing
Automorphisms of the Unit Disc Fixing a Point
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For each $w in B(0,1)$ and $theta in mathbbR$, define $phi_w,theta(z)=e^i theta dfracz-w1- bar wz$.Let $Aut_0(B(0,1))$ denote the analytic automorphisms of $B(0,1)$ fixing $0$ and $Aut(B(0,1))$ denote all the analytic automorphisms of $B(0,1)$ .Consider a subgroup $G$ of $Aut(B(0,1))$ containing
$Aut_0(B(0,1))$. Show that if $phi_w_0,theta_0 in G$, then $phi_w,theta in G$ for each $w$ satisfying $|w|=|w_0|$ and each $theta in mathbbR$.Further if $G neq Aut_0(B(0,1))$, then show that $G = Aut(B(0,1))$. I was trying to characterize the elements of $Aut_0(B(0,1))$ via the map $phi_w,theta$, but it seems that this approach is not working. Please help me with this. Thanks for any help.
complex-analysis
asked Jul 16 at 17:31
Ester
8791925
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up vote
0
down vote
favorite
For each $w in B(0,1)$ and $theta in mathbbR$, define $phi_w,theta(z)=e^i theta dfracz-w1- bar wz$.Let $Aut_0(B(0,1))$ denote the analytic automorphisms of $B(0,1)$ fixing $0$ and $Aut(B(0,1))$ denote all the analytic automorphisms of $B(0,1)$ .Consider a subgroup $G$ of $Aut(B(0,1))$ containing
$Aut_0(B(0,1))$. Show that if $phi_w_0,theta_0 in G$, then $phi_w,theta in G$ for each $w$ satisfying $|w|=|w_0|$ and each $theta in mathbbR$.Further if $G neq Aut_0(B(0,1))$, then show that $G = Aut(B(0,1))$. I was trying to characterize the elements of $Aut_0(B(0,1))$ via the map $phi_w,theta$, but it seems that this approach is not working. Please help me with this. Thanks for any help.
complex-analysis
asked Jul 16 at 17:31
Ester
8791925
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
For each $w in B(0,1)$ and $theta in mathbbR$, define $phi_w,theta(z)=e^i theta dfracz-w1- bar wz$.Let $Aut_0(B(0,1))$ denote the analytic automorphisms of $B(0,1)$ fixing $0$ and $Aut(B(0,1))$ denote all the analytic automorphisms of $B(0,1)$ .Consider a subgroup $G$ of $Aut(B(0,1))$ containing
$Aut_0(B(0,1))$. Show that if $phi_w_0,theta_0 in G$, then $phi_w,theta in G$ for each $w$ satisfying $|w|=|w_0|$ and each $theta in mathbbR$.Further if $G neq Aut_0(B(0,1))$, then show that $G = Aut(B(0,1))$. I was trying to characterize the elements of $Aut_0(B(0,1))$ via the map $phi_w,theta$, but it seems that this approach is not working. Please help me with this. Thanks for any help.
complex-analysis
asked Jul 16 at 17:31
Ester
8791925
For each $w in B(0,1)$ and $theta in mathbbR$, define $phi_w,theta(z)=e^i theta dfracz-w1- bar wz$.Let $Aut_0(B(0,1))$ denote the analytic automorphisms of $B(0,1)$ fixing $0$ and $Aut(B(0,1))$ denote all the analytic automorphisms of $B(0,1)$ .Consider a subgroup $G$ of $Aut(B(0,1))$ containing
$Aut_0(B(0,1))$. Show that if $phi_w_0,theta_0 in G$, then $phi_w,theta in G$ for each $w$ satisfying $|w|=|w_0|$ and each $theta in mathbbR$.Further if $G neq Aut_0(B(0,1))$, then show that $G = Aut(B(0,1))$. I was trying to characterize the elements of $Aut_0(B(0,1))$ via the map $phi_w,theta$, but it seems that this approach is not working. Please help me with this. Thanks for any help.
complex-analysis
asked Jul 16 at 17:31
Ester
8791925
edited Jul 16 at 17:36
asked Jul 16 at 17:31
Ester
8791925
asked Jul 16 at 17:31
Ester
8791925
asked Jul 16 at 17:31
Ester
8791925
8791925
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add a comment |Â
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