In my previous post I asked the following question.
What -values satisfy the equation
More generally, let and . Let be the composition of with itself times. Similarly, let be the composition of with itself times.
What are the solutions to ?
First, let us look at some properties of the iterates of and .
is periodic with period . The maxima occur at and minima occur at . (The first few iterates are shown below.)
When , is periodic with period . When is odd the maxima occur at and the minima occur at . When is even the minima occur at and the maxima occur at . (The first few iterates are shown below.)
Now we will address the question at hand by examining some specific cases.
m=2 and n=2
We will show that there are no solutions. In particular, we will show that and hence for all .
First notice that . Then see that
Here are the graphs of and .
m=3 and n=3
As we see in the graph below, and cross twice in each interval of length .
m=4 and n=4 (the original question)
Above we showed that for all . Since and is strictly increasing in , is increasing in . So,
for all .
Thus there are no solutions.
m=5 and n=5
The maximum value of is and the minimum value of is Thus the two graphs are disjoint. We see the graphs of and below.
m>5 and n>5
It appears that the graphs of and flatten out as gets larger. Indeed this is the case. It turns out that as , the graph of limits on the line , and the graph of limits on the line . The distances to those lines decrease with each iteration, thus the two graphs never cross again. We justify this below.
A value is a fixed point of a function if . Graphically, we can identify fixed points by finding the points of intersection of the line and the graph . Our functions have one fixed point each: has a fixed point at and has a fixed point at (i.e., the unique solution to ).
A fixed point is attracting if, whenever is close to the sequence (or orbit, using the terminology of dynamical systems) limits upon ; that is, . It is globally attracting if the orbit of every point limits upon .
As we can see in the cobweb plots below, is an attracting fixed point for ; we see that it attracts all points in the interval , and since , is a global attractor.
Similarly, as we see below, the fixed point for is a global attractor.
Thus, since for all and for all , the graphs of and limit on the lines and , respectively.
For all the other cases I used graphing software to find the number of points of intersection of and . The values along the top correspond to and the values along the side correspond to . The values in the table are the number of points of intersection in each interval of length (the arrows mean that the last given value repeats indefinitely).
[I found the and proofs here.]