\documentclass[letterpaper,12pt]{exam}
% Visit this page for info on the exam class
% http://www-math.mit.edu/~psh/exam/examdoc.pdf 

% Print Answers?
\printanswers
%\noprintanswers

\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{verbatim}
\usepackage{graphicx}
\usepackage{color}
\usepackage[normalem]{ulem}

% Macros:
\newcommand{\R}{{\mathbb R}}% The real numbers


% Edit This:
\def\classtitle{Dynamical Systems}
%\def\classnumber{Math 323}
\def\examtitle{Problem Set 1}

\def\skippage{{\ifprintanswers \else \newpage \fi}} % new page if not printing answers


%\def\skippage{{\ifprintanswers \else \newpage \fi}} % new page if not printing answers

\addpoints
%\usepackage[left=0.75in,top=0.75in,right=0.75in,bottom=0.75in,nohead,nofoot]{geometry}
\pagestyle{head} %If you want every page to have a header but no footer, give the command

\ifprintanswers % No header when printing answers:
\header{}{}{Page \thepage}
\runningheadrule % line under header
\firstpageheader{}{}{}
\else % Header of test:
\header{\classnumber}{\examtitle}{Page \thepage\ of \numpages}
\runningheadrule % line under header
\firstpageheader{}{}{}
\fi
%\extrawidth{1in} % decrease left and right margins by 1/2 inch each

\date{}



\begin{document}
\begin{center}
%{\large \bf \classnumber: \classtitle: \examtitle}\\
{\large \bf \classtitle: \examtitle
\\}
\end{center}

\noindent
{\bf Student:} Insert your name here!
\vspace{1em}

This problem set concerns the quadratic maps of $\R$. A {\em quadratic map} is a map of the form
$$Q(x) = ax^2 + bx + c,$$
where $a,b,c \in \R$ are constants with $a \neq 0$. Note that $Q(x)$ is not invertible.

The graph of $Q(x)$ is a parabola. The {\em critical point} of $Q$ is the point $x_c$ for which $Q'(x_c)=0$.

You should be able to answer the following questions using a combinations of explicit calculations and cobweb plots for orbits.


\begin{questions}
\question The function $Q$ is either concave up (the case $\lim_{x \to \pm \infty} Q(x)=+\infty$). Or concave down (the case $\lim_{x \to \pm \infty} Q(x)=-\infty$).  Let $F(x)=-x$. Show that if $Q$ is concave up, then the topologically conjugate quadratic map $F \circ Q \circ F^{-1}$ is concave down. (This says that to understand quadratic maps, it is enough to understand those that are concave down.)
\begin{solution}
Insert answer here.
\end{solution}

\question Suppose that $R(x)$ is a concave down quadratic map with no fixed points. Describe the long term behavior of orbits: For any $x$, what is $\lim_{n \to +\infty} R^n(x)$?
\begin{solution}
Insert answer here.
\end{solution}

\question Now suppose that $Q$ is a concave up quadratic map with no fixed points. Explain how the two previous parts can be used to understand the long term behavior of orbits of $Q$ without redoing your analysis.
\begin{solution}
Insert answer here.
\end{solution}

\question Now consider the case when $Q(x)$ is concave down and has only one fixed point. Call the fixed point $p$. For any $x$, what is $\lim_{n \to +\infty} Q^n(x)$? ({\em Hint:} Let $q$ be the point not equal to $p$ so that $Q(q)=p$. How does the behavior of points in the interval $[p,q]$ differ from those outside?
\begin{solution}
Insert answer here.
\end{solution}

\question 
\label{q:L}
The dynamics maps $Q$ with two fixed points can be quite complex as we will see later in the course. We will content ourselves with putting $Q$ into a standard form. A map of the form $L(x)=dx + e$ with $d \neq 0$ is called an {\em affine linear map}. Let $G(y)=L \circ Q \circ L^{-1}(y)$. Show:
\begin{parts}
\part The map $G$ is quadratic. 
\begin{solution}
Insert answer here.
\end{solution}
\part If $y=L(x)$, then $G'(y)=Q'(x)$. 
\begin{solution}
Insert answer here.
\end{solution}
\part If $y=L(x)$, then $x$ is a fixed point of $Q$ if and only if $y$ is a fixed point of $G$.
\begin{solution}
Insert answer here.
\end{solution}
\end{parts}

\question Suppose $F$ is a quadratic map with two fixed points, with a critical point at $\frac{1}{2}$ (i.e., $F'(\frac{1}{2})=0$). Also suppose that zero is fixed and the other fixed point is closer to the critical point. Prove that $F'(0)>1$. Set $\mu=F'(0)>1$. Find an explicit formula for $F(x)$ in terms of $x$ and $\mu$.
\begin{solution}
Insert answer here.
\end{solution}


\question Use the previous two parts to prove that any quadratic map $Q(x)$ with two fixed points is conjugate by an affine linear map to a map of the form $F_\mu$ for some $\mu>1$. 
\begin{solution}
Insert answer here.
\end{solution}


\end{questions}
\end{document}
