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\begin{document}
\begin{center}
{\bf \large Math 701 Problem Set 6} \vs \\
{\bf due Friday 10/25/2013} \vs \vs
\end{center}  
The letter $C$ will be reserved for the standard middle-thirds Cantor set in $\RR$. \\

\noindent
{\bf Problem 1.} A non-empty compact, perfect, totally disconnected metric space is called a {\bit Cantor space}. All Cantor spaces are homeomorphic to the standard Cantor set $C$. Show that if $X$ is a Cantor space and $Y \neq \es$ is a clopen (=closed and open) subset of $X$, then $Y$ is a Cantor space, so $X$ and $Y$ are homeomorphic. \vs 

\noindent
{\bf Problem 2.} Show that $C$ is homeomorphic to $C \times C$. (Hint: Check that $C \times C$ is a Cantor space.) \vs

\noindent
{\bf Problem 3.} Prove that the arithmetical difference set
$$
C-C= \{ x-y: x, y \in C \}
$$
is the interval $[-1,1]$. (Hint: For each $t \in [-1,1]$, use geometry to verify that the line $y=x-t$ intersects $C_n \times C_n$ for every $n$. Here $C_n$ is the level-$n$ approximation of $C$ consisting of $2^n$ intervals of length $1/3^n$.) \vs

\noindent
{\bf Problem 4.} Show that $C$ is homogeneous in the following sense: Given any pair $p,q \in C$ there exists a homeomorphism $f:C \to C$ such that $f(p)=q$. This shows in particular that the ``endpoints'' of $C$ (such as $0$, $1$, $1/3$ or $8/9$) enjoy no topological distinction whatsoever. \vs

\noindent
{\bf Problem 5.} Let $\varphi: C \to \Sigma_2$ be the homeomorphism which assigns to each $x \in C$ its unique dyadic address $\varphi(x)=s_1s_2s_3 \cdots$ determined by  
$$
\bigcap_{n \geq 1} I_{s_1 \cdots s_n} = \{ x \}.
$$ 
Show that if $0<\alpha<\log2/\log 3$, the map $\varphi$ is H\"{o}lder of exponent $\alpha$. That is, there exists an $M>0$ such that 
$$
d(\varphi(x),\varphi(y)) \leq M |x-y|^{\alpha} \qquad \text{if} \ x,y \in C. \vs 
$$

\noindent 
{\bf Problem 6.} Let $\sigma: \Sigma_2 \to \Sigma_2$ be the {\bit shift map} defined by 
$$
\sigma(s_1s_2s_3\cdots)=s_2s_3s_4\cdots
$$
\begin{enumerate}
\item[(i)]
Verify that $\sigma$ is continuous and $2$-to-$1$. \vs

\item[(ii)]
How many fixed points does $\sigma$ have? How many periodic points of period $2$? Of period $3$? Of a prime period? Of a general period? \vs
 
\item[(iii)]
Show that periodic points of $\sigma$ are dense in $\Sigma_2$, that is, given $s \in \Sigma_2$ and $\ve>0$ there is a $t \in \Sigma_2$ which is fixed by some iterate $\sigma^{\circ k}$, such that $d(s,t)<\ve$. \vs

\item[(iv)]
Show that there are points $s \in \Sigma_2$ whose orbit $\{ s, \sigma(s), \sigma^{\circ 2}(s), \sigma^{\circ 3}(s), \ldots \}$ is dense in $\Sigma_2$. \vs

\item[(v)]
Under the homeomorphism $\varphi: C \to \Sigma_2$, the conjugate map $f=\varphi^{-1} \circ \sigma \circ \varphi: C \to C$ has similar orbit properties as $\sigma$. Can you find an explicit formula for $f$?  
\end{enumerate}

\end{document}

    
