# Discrete dynamical systems fixed points I'm trying to understand this question which asked to find the fixed points of this tent map. However, I was under the impression that the fixed points are the points that hit the y=x line.

However, why does $T^{2}$ have fixed points of period 2? I just don't see how you can deduce what is below. Particularly, $T^{2}$. Surely you just have four fixed point at the intersection of y=x? ## 1 Answer

$T^2$ has, as you say, $4$ fixed points, which are the values of $x$ where the graphs of $T^2(x)$ and $x$ intersect. Two of those are the fixed points of $T$ (any fixed point of $T$ is a fixed point of $T^k$ for positive integers $k$). The other two are not fixed points of $T$, so they are periodic points of $T$ with period $2$ (i.e. they are points $x$ such that $T(x) \ne x$ but $T^2(x) = x$; we then have $T^3(x) = T(x)$, $T^4(x) = x$, etc.)

• Oh wait, I can see it. It's talking about the fixed points in $T$ and not $T^{2}$. Oh thanks for that. I can see how it works now. – simplicity May 23 '12 at 19:46