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\begin{document}

\begin{center}
    \bf preREU - Last week projects
\end{center}


\begin{problem}[Expanding maps]
This problem will use the following as a black box:

\vspace{.5cm}

\noindent {\bf Theorem.} Let $f : \R / \Z \to \R / \Z$ be a continuous function. Then there exists a unique continuous function $F : \R \to \R$ and integer $d \in \Z$ such that  $0 \le F(0) < 1$, $f([(x)]) = [F(x)]$ (ie, $f \of p = p \of F$) and $F(x +1) = F(x) + d$ for all $x \in \R$. The number $d$ is called the {\it degree} of $f$, and denoted $d = \deg(f)$ and $F$ is called the {\it lift} of $f$.

\vspace{.5cm}

The goal of this project is to build an invertible map $h : \R / \Z \to \R / \Z$ such that 

\begin{equation}\label{eq:conjugacy}
h \of f = L_d \of h \tag{$\clubsuit$}
\end{equation}

 when $f$ has degree $d$ and satisfies a certain condition.

\begin{enumerate}[{\bf A)}]
\item Show that if $h : \R / \Z \to \R / \Z$ is invertible, then $\deg(h) = \pm 1$.
\item A map $f : \R / \Z \to \R / \Z$ is called {\it expading} if there exists $\lambda > 1$ such that $\abs{F'(x)} \ge \lambda$ for all $x \in \R$. Show that if $f$ is expanding, then $\abs{\deg(f)} > 1$.
\item Show that if $H : \R \to \R$ is a function such that

\[ H(x) = \frac{1}{d} H(F(x)) \qquad \mbox{and} \qquad H(x+1) = H(x) + 1 \mbox{ for all } x \in \R,\]

then $h([x]) := [H(x)]$ is well-defined and satisfies equation \eqref{eq:conjugacy}.
 \item Define $\mathcal{F}(H)(x) = \frac{1}{d} H(F(x))$ as a dynamical system on continuous functions. Show that $\mathcal{F}$ preserves the set of functions $H$ such that $H(x+1) = H(x) + 1$, and that there exists $\mu < 1$ such that for every pair of functions $H_1$ and $H_2$ and $x \in \R$, 
 
 \[ \abs{\mathcal{F}(H_1)(x) - \mathcal{F}(H_2)(x)} \le \mu \abs{H_1(x) - H_2(x)}.\]
 
 \item Look up the {\it contraction mapping principle}, and use it to show that the conjugating map $h$ exists and is unique.
    %- Conclude that there is f is semiconjugated to E_k.
    %- Show that \mathcal{F} is never a contraction on C^1. This is evidence that a C^1 (semi)conjugacy is rare!
\end{enumerate}
\end{problem}


\begin{problem}[Linear flows]
The {\it two-dimensional flat torus} is the set of equivalence classes of $\R^2 / \Z^2$. That is, we define the equivalence class

\[ \left[ \begin{pmatrix}x \\ y \end{pmatrix} \right] = \set{\begin{pmatrix} x + m \\ y +n \end{pmatrix} : m,n \in \Z}.\]

If $v = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} \in \R^2$ is a nonzero vector, the {\it flow} generated by $v$ is the dynamical system

\[ \varphi_t\left(\left[\begin{pmatrix}x \\ y \end{pmatrix} \right]\right) = \left[\begin{pmatrix}x + tv_1 \\ y + tv_2 \end{pmatrix} \right] \]

\begin{enumerate}[{\bf A)}]
\item Find a fundamental domain for $\R^2 / \Z^2$.
\item Show that \begin{equation}\label{eq:flow-eq}\varphi_t \of \varphi_s = \varphi_{t+s} \tag{$\varheartsuit$}\end{equation} for all $t,s \in \R$. Equation \eqref{eq:flow-eq} is called the {\it flow equation}.
\item Show that if $\varphi_t$ is the flow generated by $\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$ and $\psi_t$ is the flow generated by $\begin{pmatrix} v_2 \\ v_1 \end{pmatrix}$, then $\varphi_t$ is conjugated to $\psi_t$.
\item Let $v = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$ have $v_2 > 0$ and $Y = \set{\left[\begin{pmatrix} x \\ 0 \end{pmatrix} \right] : x \in \R}$. Find the first return map and first return time to $Y$.
\item Describe the orbits of $\varphi_t$ based on the vector $v$. Can you make sense of these questions: When are the orbits periodic? When are the orbits dense?
\end{enumerate}
\end{problem}

\begin{problem}[Convergents and continued fractions]
Let $\alpha \in (0,1)$ have continued fraction expansion

\[ \alpha = \dfrac{1}{a_1 + \dfrac{1}{a_2 + \dfrac{1}{a_3 + \dfrac{1}{\ddots}}}}\]

and $p_k/q_k$ be the convergent

\[ \dfrac{p_k}{q_k} = \dfrac{1}{a_1 + \dfrac{1}{a_2 + \dfrac{1}{a_3 + \dfrac{1}{\ddots+\dfrac{1}{a_k}}}}}\]

\begin{enumerate}[{\bf A)}]
\item Define the vector $v_k = \begin{pmatrix} p_k \\ q_k \end{pmatrix}$ and the matrix $B(a) = \begin{pmatrix} 0 & 1 \\1 & a \end{pmatrix}$. Show that

\begin{equation}
\label{eq:vk-recursion}v_k = B(a_1)B(a_2)\dots B(a_k) \begin{pmatrix}1 \\ 0 \end{pmatrix} \tag{$\spadesuit$}
\end{equation}

\item Show that if $\alpha$ has periodic continued fraction expansion, with repeating digits $a_1,\dots,a_k$, then $v_0 = \begin{pmatrix}\alpha \\ 1 \end{pmatrix}$ is an eigenvalue of 

\[ A_k = B(a_1)B(a_2)\dots B(a_k). \]
\item Use the formula \eqref{eq:vk-recursion} to find a recursive definition of $p_k$ and $q_k$, and prove that your formula holds by induction. [{\it Hint}: You will need a 2-step induction]
\item Show by induction that the trace of $A_k$ is an integer than 1 and that the determinant of $A_k$ is equal to $\pm 1$.
\item Show that any matrix of the form $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ of determinant 1 such that $0 \le a \le c$, $a \le b$, and $0 \le b \le d$, then $A$ is a product of matrices of the form $B(a_1) \dots B(a_k)$.
\end{enumerate}
\end{problem}


\begin{problem}[Local dynamics and Morse-Smale Systems]
Let $(a,b) \subset \R$ be an interval, $x \in (a,b)$ and $f : (a,b) \to \R$ be a continuously differentiable function such that $f(x) = x$. 

\begin{enumerate}[{\bf A)}]
\item Show that if $f'(x) < 1$, then there exists $\ve >0$ such that if $\abs{y-x} < \ve$, then $\displaystyle\lim_{n \to \infty} f^n(y) = x$.
\item Let $f : \R / \Z \to \R / \Z$ be a dynamical system, and assume that some interval $[a,b]$ is invariant under $f$. Show that $f$ has a fixed point in $[a,b]$
\item A dynamical system on $\R / \Z$ is called {\it Morse-Smale} if it is invertible, there exists finitely many fixed points, and at each fixed point $p$, $f'(p) >0$ and $f'(p) \not= 1$. Find examples of Morse-Smale dynamical systems.
\item Show that it is impossible for a Morse-Smale system to have only one fixed point. [{\it Hint}: Use {\bf B)} to find another fixed point]
\item Show that if $f$ and $g$ are Morse-Smale dynamical systems on $\R / \Z$ with exactly two fixed points, then they are conjugated. (This type of system is said to have {\it north-south pole dynamics})  
\end{enumerate}
\end{problem}

\end{document}
