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\centerline{\Large {\bf Dynamical Systems, MATH 374}}
\vspace{.15in}
\centerline{\Large {\bf Homework Assignment \#4}}
\vspace{.15in}
\centerline{\large {\bf DUE DATE: Wed., Oct. 19th, Start of class.}}
\vspace{.15in}

You should write up solutions neatly to all problems, making sure to show all your work.  
A nonempty subset will be graded. 
You are strongly encouraged to work on these problems with other classmates, although the 
solutions you turn in should be your own work.

{\bf Note:}  Please list the names of any students or faculty who you worked with on the assignment.

\vspace{.15in}

\centerline{ {\bf Problems on Topological Conjugacy} }

\begin{enumerate}
\item   Suppose that $f$ and $g$ are topologically conjugate with conjugacy $h$, that 
is $h \circ f = g \circ h$.
Show that $h$ is a conjugacy between $f^n$ and $g^n$ for any $n \in \mathbb{N}$.  In other words,
show that 
\begin{equation}
h \circ f^n \; = \; g^n \circ h  \quad \forall n \in \mathbb{N}.
\label{eq}
\end{equation}
{\it Hint:}  Use the fact that $h$ is invertible.

\item  Use equation~(\ref{eq}) to show that, if $f$ and $g$ are topologically conjugate, 
then $h$ maps period $n$ points of $f$ to period $n$ points of $g$.  Specifically, show 
that $p$ is a period $n$ point for $f$ if and only if $h(p)$ is a period $n$ point for $g$.

\item  Suppose that $f$ and $g$ are topologically conjugate and that $p$ is an attracting 
fixed point for $f$.  Prove that
if $x$ is asymptotically attracted to $p$ under $f$, then $h(x)$ is asymptotically attracted 
to $h(p)$ under iteration of $g$.
Conclude that $h$ maps the basin of attraction of $p$ to the basin of attraction of $h(p)$.
(The {\it basin of attraction} of a fixed point is the set of all points attracted to the fixed
point under iteration.)

\item  Find a linear conjugacy $h(x) = \alpha x + \beta$ between the quadratic map 
$Q_c(x) = x^2 + c$ and the logistic map
$F_\lambda(x) = \lambda x (1-x)$.  Give the values of $\alpha$ and $\beta$ and the 
relationship between the parameters
$c$ and $\lambda$.  

\item   Using your conjugacy from the previous question, for what values of $\lambda$ 
should we expect to see a period-doubling bifurcation occur for the logistic map.  

\item  Notice that $Q_{1/4}$ is conjugate to $F_1$ via the conjugacy from Problem 4.  
However, while $Q_c$ undergoes a saddle-node bifurcation at $c = 1/4$, $F_\lambda$ has 
a {\bf different} bifurcation at $\lambda = 1$ (see Exercise 8, Ch. 6 from HW \#3.)
Why are the bifurcations for these two conjugate families different?  By examining 
the relationship between the parameters
$c$ and $\lambda$ given by the linear conjugacy, state precisely the values of the 
parameters for which $Q_c$ is conjugate to
$F_\lambda$.

\end{enumerate}


\centerline{ {\bf Problems on Symbolic Dynamics} }

\vspace{.1in}

\noindent {\bf Chapter 9 Exercises} (pp. 111 - 113) 

\noindent  Problems: 1, 2, 7, 8, 18a, 18b, 18f

\vspace{.1in}

\noindent {\bf Note:} For the continuity problems in \#18, the goal is to find a $\delta$
that works in the definition of continuity or convince yourself no such $\delta$
exists.  For starters, go over the proof that the shift map is continuous from class.

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