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\begin{document}


\title{Title here}
\subtitle{Subtitle optional}
\author{Author 1 \and Author 2}

\institute{University here}
\date{Equidistribution and Arithmetic Dynamics\\ Oklahoma State University\\ June 2022}

\begin{frame}
\titlepage
\thispagestyle{empty}
\end{frame}

\begin{frame}{Slide 1}
 A \alert{dynamical system} is simply a function $f: X \ra X$ which we \alert{iterate} on a given set $X$.

\vskip10pt

\pause Basic example: How can we compute a numerical value for $\sqrt{2}$? We can use \alert{Newton's root-finding method} to find a root of $f(x)=x^2-2$. Given a guess $x_0$, we let
\[
 \alert{N(x) = x - \frac{f(x)}{f'(x)}} = x - \frac{x^2 - 2}{2x} = \frac{x^2+2}{2x}.
\]
We \alert{iterate} $N:\bP^1\ra \bP^1$ to find a sequence of approximations:
\begin{align*}
 x_1 &= N(x_0) \\
 x_2 &= N(x_1) = N(N(x_0)) = N^2(x_0) \\
 x_3 &= N(x_2) = N(N(N(x_0)))=N^3(x_0) \\
 &\ \ \vdots 
\end{align*}
 
\end{frame}

\begin{frame}{Dynamical systems}

\begin{columns}[T]
 \begin{column}{0.32\textwidth}
 \begin{block}{Text block}
  % Code for a picture:
  %\begin{figure}[h]
  %\includegraphics[scale=0.5]{newton.pdf}
  %\end{figure}
 \end{block}
 \end{column}
 \begin{column}{0.62\textwidth}
    \begin{block}{Why does Newton's method work?}
    If the sequence $x_n$ converges to \emph{some} $x$, what must $x$ be?
    \begin{align*}
    \alert{N(x)} &= N\left(\lim_{n\ra \infty} x_n\right)\\
    &= \lim_{n\ra \infty} N(x_n)\\
    &= \lim_{n\ra \infty} x_{n+1}\\
    & \alert{= x},
    \end{align*}
    so $x$ is a \alert{fixed point} of $N$. Further, this means $N(x)=x - f(x)/f'(x)=x$, so if $f'(x)\neq 0$, we must have \alert{a root $f(x)=0$}.
    \end{block}
 \end{column}
 \end{columns}
\end{frame}

\begin{frame}{Thank you for listening!}
 Thanks!
\end{frame}



\end{document}
