“The orbit of a point x in [0,1] for the tent map T is
orbit(x) = {Tn(x): n = 0,1,2,3,...}”
This makes sense to me as a definition, suppose our tent maps
μ = 2, than for n = 1, orbit(1/2) = {1,0}
The elements of the set created by the orbital function are the values
of the tent map at the 0th and 1st iterate of the tent function for the input of
the orbital function.
For me the confusion arises in the claim that
"The points with finite orbit are just the periodic points of T"
I'm not seeing how the finite orbital sets elements are the periodic points of T
Suppose you have a finite orbit $S$ of size $n$. Choose any $x \in S$. Then $S$ must be just the points $x, Tx, T^2x, \ldots, T^{n-1}x$ and $T^nx = x$. So $T$ cyclically permutes the elements of $S$.