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\def\classtitle{Dynamical Systems}
\def\classnumber{Math A4500}

\def\assignment{Homework 4}

\begin{document}
\begin{center}
{\Large \bf \classnumber: \classtitle: \assignment}\\
\end{center}

\noindent
{\bf Name:} INSERT YOUR NAME HERE

\begin{questions}
\question {\bf Devaney 3e: I.6 \#4:}
Let $\Sigma'$ consist of all sequences in $\Sigma_2$ satisfying: if $s_j = 0$ then $s_{j+1} = 1$. In
other words, $\Sigma'$ consists of only those sequences in $\Sigma_2$ which never have two consecutive zeros.
\begin{parts}
\part
Show that $\sigma$ preserves $\Sigma'$ and that $\Sigma'$ is a closed subset of $\Sigma_2$.
\begin{solution}
Answer part 1 here!
\end{solution}

\part
Show that periodic points of $\sigma$ are dense in $\Sigma'$.
\begin{solution}
Answer part 2 here!
\end{solution}

\part
Show that there is a dense orbit in $\Sigma'$.
\begin{solution}
Answer part 3 here!
\end{solution}

\part
How many ﬁxed points are there for $\sigma$, $\sigma^2$, $\sigma^3$ in $\Sigma'$?
\begin{solution}
Answer part 4 here!
\end{solution}

\part
Find a recursive formula for the number of ﬁxed points of $\sigma^n$ in
terms of the number of ﬁxed points of $\sigma^{n-1}$ and $\sigma^{n-2}$.
\begin{solution}
Answer part 5 here!
\end{solution}

\end{parts}

\question {\bf Devaney 3e: I.7 \#2:}
A point $p$ is a {\em non-wandering point} for $f$, if, for any open interval $J$ containing $p$, there exists $x \in J$ and $n >0$ such that $f^n(x) \in J$. Note that we do not require that $p$ itself return to $J$. Let $\Omega(f)$ denote the set of non-wandering points of $f$.
\begin{parts}
\part
Prove that $\Omega(f)$ is a closed set.
\begin{solution}
Answer part 1 here!
\end{solution}

\part
If $F_\mu$ is the quadratic map with $\mu>2+\sqrt{5}$, show that $\Omega(F_\mu)=\Lambda$.
\begin{solution}
Answer part 2 here!
\end{solution}

\part
Identify $\Omega(F_\mu)$ for each $\mu$ satisfying $0<\mu\leq 3$.
\begin{solution}
Answer part 3 here!
\end{solution}

\end{parts}

\question {\bf Devaney 3e: I.7 \#5:}
Let $f : X \to X$ and $g : Y \to Y$ and suppose $h : X \to Y$ is either a
conjugacy or a semi-conjugacy between these maps. Prove that if the orbit of
$x \in X$ under $f$ is dense in $X$, then the orbit of $h(x)$ under $g$ is dense in $Y$.
\begin{solution}
Answer here!
\end{solution}


\question {\bf Devaney 3e: I.9 \#6:}
Prove that $S(x) = \sin(x)$ is not structurally stable.
\begin{solution}
Answer here!
\end{solution}


\question {\bf Devaney 3e: I.9 \#7:}
Prove that, if $f \sim g$ via $h$ and $f$ has a local maximum at $x_0$, then $g$ has
either a local maximum or minimum at $h(x_0)$.
\begin{solution}
Answer here!
\end{solution}


\question {\bf Devaney 3e: I.9 \#9:}
Let $S_\lambda(x) = \lambda \sin(x)$. If $0 < \lambda_1 < \lambda_2 < 1$, prove that $S_{\lambda_1} \sim S_{\lambda_2}$.
\begin{solution}
Answer here!
\end{solution}


\end{questions}
\end{document}
