Recall the fixed points of a function occur at the intersection
of the graph of the function and
the diagonal. 
Consequently, each funcion f_{n} has n fixed points. For example, 

It is not hard to see that the graph of f_{n}^{m} consists
of line segments having slope n^{m}. Here is a simple, sample calculation, illuminated by this
graph. 

f_{2}^{2}(x) = f_{2}(f_{2}(x)) = 
2⋅f_{2}(x)  for 0 ≤ f_{2}(x) < 1/2 
2⋅f_{2}(x)  1  for 1/2 ≤ f_{2}(x) ≤ 1 

= 
2⋅f_{2}(x)  for 0 ≤ x < 1/4 and 1/2 ≤ x < 3/4 
2⋅f_{2}(x)  1  for 1/4 ≤ x < 1/2 and 3/4 ≤ x ≤ 1 

  = 
2⋅2x = 4x  for 0 ≤ x < 1/4 
2⋅2x  1 = 4x  1  for 1/4 ≤ x < 1/2 
2⋅(2x  1) 4x  2  for 1/2 ≤ x < 3/4 
2⋅(2x  1)f  1 = 4x  3  for 3/4 ≤ x ≤ 1 


Because the segments of the graph of f_{n}^{m} consists of
segments of slope n^{m}, and the leftmost starts at (0, 0), the graph of
f_{n}^{m} consists of n^{m} straight line segments, each going
from x = 0 to x = 1. 
Consequently, f_{n}^{m} has n^{m} fixed points. 