<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
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\begin{document}

{\bf Question}

Find \underline{three} 2-cycles and \underline{three} 3-cycles for
the hyperbolic toral automorphism given by the matrix
$A=\left(\begin{array}{cc} 3 &amp; 7\\ 2 &amp; 5
\end{array} \right)$. (There are many more!)

[\underline{Hint}: find explicit solutions $v$ to $A^nv=v$ mod 1,
for $n=2,\ 3$.]
\medskip

{\bf Answer}

$(A^2-I)^{-1}=\ds\frac{1}{16} \left(\begin{array}{cc} -12 &amp; 8\\ 8
&amp; -4 \end{array} \right)$ We apply this matrix to
$\left(\begin{array}{c} k\\l \end{array} \right)$, $k,l \in
\bf{Z}$. $\left(\rm{drop\ the}\ \ds\frac{1}{30}\right)$

$\left(\begin{array}{c} k\\l \end{array}
\right)=\left(\begin{array}{c} 1\\0 \end{array} \right)$:
\underline{$\left(\begin{array}{c} -19\\8 \end{array} \right)
\mapsto \left(\begin{array}{c} -1\\2 \end{array} \right) \mapsto
\left(\begin{array}{c} -19\\8 \end{array} \right)$}

$\left(\begin{array}{c} k\\l \end{array}
\right)=\left(\begin{array}{c} 0\\1 \end{array} \right)$:
\underline{$\left(\begin{array}{c} 28\\-11 \end{array} \right)
\mapsto \left(\begin{array}{c} 7\\1 \end{array} \right) \mapsto
\left(\begin{array}{c} 28\\-11 \end{array} \right)$}.

Add: \underline{$\left(\begin{array}{c} 9\\-3 \end{array} \right)
\mapsto \left(\begin{array}{c} 6\\3 \end{array} \right) \mapsto
\left(\begin{array}{c} 9\\-3 \end{array} \right)$}.

$(A^3-I)^{-1}=\ds\frac{1}{54} \left(\begin{array}{cc} -34 &amp; 14\\
49 &amp; -20 \end{array} \right)$:

points on \underline{distinct} 3-cycles are $\ds\frac{1}{54}
\left(\begin{array} {r} -34 \\ 49
\end{array} \right),\ \left(\begin{array} {r} 14 \\ -20
\end{array} \right),\ \left(\begin{array} {r} -20 \\ 29
\end{array} \right)$
\end{document}
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