I am being asked to find fixed points of further iterations of the Cat map, which is defined as:
$$C(x,y) = (2x+y , x+y) \pmod 1 $$ for $(x,y)$ an element of $[0,1)$
Hence the equation to solve the fixed points for the 2nd iteration of this cat map would look like this: $$ \begin{cases} x = 5x + 3y \pmod 1 \\ y = 3x + 2y \pmod 1 \end {cases} $$
I have no idea how to solve these, short of having to solve ~ 7 x 4 simultaneous equations (as the domain limits mod(1) from $0 \to 7$). I've attempted solving the first few and all that happens is I end up throwing out most solutions as they don't provide an answer in the domain. Any help would be appreciated.