A continuous map $f : A \longrightarrow B$ between two metric spaces $( A , d _ { A } )$, $( B , d _ { B } )$ is said to be a bilipschitz map if there exists a real number $\lambda > 0$ such that $( 1 / \lambda ) d _ { A } \left( a _ { 0 } , a _ { 1 } \right) \leq d _ { B } \left( f \left( a _ { 0 } \right) , f \left( a _ { 1 } \right) \right) \leq \lambda d _ { A } \left( a _ { 0 } , a _ { 1 } \right)$ for each $a _ { 0 } , a _ { 1 } \in A$. Let $X = \mathbb { R } ^ { 2 } \backslash \{ 0 \}$ and $Y = \mathbb { S } ^ { 1 } \times \mathbb { R } = \left\{ ( x , y , z ) \in \mathbb { R } ^ { 3 } \mid x ^ { 2 } + y ^ { 2 } = 1 \right\}$. Let $d _ { X }$ (respectively, $d _ { Y }$ ) be the euclidean metric on $X$ induced from $\mathbb { R } ^ { 2 }$ (respectively, on $Y$ induced from $\mathbb { R } ^ { 3 }$ ). Let $f : X \longrightarrow Y$ be a bilipschitz map. (A) Let $R > 0$ and $C _ { R } \subseteq X$ the circle of radius $R$ with centre at 0. Let $\bar { f } : X \longrightarrow \mathbb { R }$ be the composite of $f$ and the projection from $Y = \mathbb { S } ^ { 1 } \times \mathbb { R }$ to its second factor $\mathbb { R }$. Let $L _ { R }$ be the length of the interval $\bar { f } \left( C _ { R } \right) \subseteq \mathbb { R }$. Let $a , b \in X$ be such that $\bar { f } ( b ) = \bar { f } ( a ) + L _ { R }$. Show that $d _ { X } ( a , b ) \geq ( 2 R - 2 \lambda ) / \lambda ^ { 2 }$. (B) Let $C _ { 1 }$ and $C _ { 2 }$ be the two arcs of $C _ { R }$, joining $a$ and $b$. Show that there exists $x _ { 1 } \in C _ { 1 }$ and $x _ { 2 } \in C _ { 2 }$ such that $\bar { f } \left( x _ { 1 } \right) = \bar { f } \left( x _ { 2 } \right) = \frac { f ( a ) + f ( b ) } { 2 }$. Show that $d _ { Y } \left( f \left( x _ { 1 } \right) , f \left( x _ { 2 } \right) \right) \leq 2$. (C) Show that for all sufficiently large $R , d _ { Y } \left( f \left( x _ { 1 } \right) , f \left( x _ { 2 } \right) \right) > 2$. (Hint: Let $a _ { i } \in C _ { i }$ be such that $d _ { X } \left( a , a _ { i } \right) = R / \lambda ^ { 2 }$; show that $d _ { X } \left( x _ { 1 } , x _ { 2 } \right) \geq d _ { X } \left( a _ { 1 } , a _ { 2 } \right)$.) (D) What is your conclusion about $f$?
A continuous map $f : A \longrightarrow B$ between two metric spaces $( A , d _ { A } )$, $( B , d _ { B } )$ is said to be a bilipschitz map if there exists a real number $\lambda > 0$ such that $( 1 / \lambda ) d _ { A } \left( a _ { 0 } , a _ { 1 } \right) \leq d _ { B } \left( f \left( a _ { 0 } \right) , f \left( a _ { 1 } \right) \right) \leq \lambda d _ { A } \left( a _ { 0 } , a _ { 1 } \right)$ for each $a _ { 0 } , a _ { 1 } \in A$.
Let $X = \mathbb { R } ^ { 2 } \backslash \{ 0 \}$ and $Y = \mathbb { S } ^ { 1 } \times \mathbb { R } = \left\{ ( x , y , z ) \in \mathbb { R } ^ { 3 } \mid x ^ { 2 } + y ^ { 2 } = 1 \right\}$. Let $d _ { X }$ (respectively, $d _ { Y }$ ) be the euclidean metric on $X$ induced from $\mathbb { R } ^ { 2 }$ (respectively, on $Y$ induced from $\mathbb { R } ^ { 3 }$ ). Let $f : X \longrightarrow Y$ be a bilipschitz map.\\
(A) Let $R > 0$ and $C _ { R } \subseteq X$ the circle of radius $R$ with centre at 0. Let $\bar { f } : X \longrightarrow \mathbb { R }$ be the composite of $f$ and the projection from $Y = \mathbb { S } ^ { 1 } \times \mathbb { R }$ to its second factor $\mathbb { R }$. Let $L _ { R }$ be the length of the interval $\bar { f } \left( C _ { R } \right) \subseteq \mathbb { R }$. Let $a , b \in X$ be such that $\bar { f } ( b ) = \bar { f } ( a ) + L _ { R }$. Show that $d _ { X } ( a , b ) \geq ( 2 R - 2 \lambda ) / \lambda ^ { 2 }$.\\
(B) Let $C _ { 1 }$ and $C _ { 2 }$ be the two arcs of $C _ { R }$, joining $a$ and $b$. Show that there exists $x _ { 1 } \in C _ { 1 }$ and $x _ { 2 } \in C _ { 2 }$ such that $\bar { f } \left( x _ { 1 } \right) = \bar { f } \left( x _ { 2 } \right) = \frac { f ( a ) + f ( b ) } { 2 }$. Show that $d _ { Y } \left( f \left( x _ { 1 } \right) , f \left( x _ { 2 } \right) \right) \leq 2$.\\
(C) Show that for all sufficiently large $R , d _ { Y } \left( f \left( x _ { 1 } \right) , f \left( x _ { 2 } \right) \right) > 2$. (Hint: Let $a _ { i } \in C _ { i }$ be such that $d _ { X } \left( a , a _ { i } \right) = R / \lambda ^ { 2 }$; show that $d _ { X } \left( x _ { 1 } , x _ { 2 } \right) \geq d _ { X } \left( a _ { 1 } , a _ { 2 } \right)$.)\\
(D) What is your conclusion about $f$?