How can we locate the points belonging to an ncycle? 
For example, consider the 2cycle consisting of the points x = a
and x = b. 
From the graph it is easy to see T(a) = b and T(b) = a.
Consequently, 
T(T(a)) = T(b) = a  and  T(T(b)) = T(a) = b 
 That is,  
T^{2}(a) = a  and  T^{2}(b) = b 

In other words, if x = a and x = b belong to a 2cycle
for T(x), both are fixed points for T^{2}(x) = T(T(x)). 
This is useful, because we know how to locate graphically the fixed points of any function: look
for the intersections of the graph of T^{2}(x) (purple below) with the diagonal. 

Click the picture to animate. 
Note the fixed points of T(x) also are fixed points of T^{2}(x). 
This is hardly a surprise: 
if T(c) = c, then T^{2}(c) = T(T(c)) = T(c) = c. 
This observation has obvious generalizations. For example,
the points of a 4cycle of T(x) are fixed points of T^{4}(x). But the fixed points of
T^{4}(x) include also fixed points of T(x) and the points of 2cycles
of T(x). 
Do you see the general pattern? 