Function Spaces: Self-Composition?

Question

Let H be the set of all functions h for which there exists a continuously differentiable real function f such that h(x) = f(f(x)).  Such functions include h(x) = x, h(x)=e^(e^x), etc.  (H is clearly infinite in size).

What can we derive in general about h?  Are there any non-trivial operations under which H is closed?  What are we describing if we extend H to transformations in R^n?
Has H been studied at all?  Or is it just a cramped, relatively pointless set of functions?

I am not looking for any specific answers here, just some discussion with luck some enlightening deductions.  However, for those who fear these kinds of questions, here are some more specific questions (more food for thought than anything.  These have relatively trivial proofs):

a)  Prove that if h is bounded if f is bounded, but f unbounded does not force h unbounded
b)  Prove that f is not unique, even when h(x) constant
c)  Prove that h(x) = -x is not a member of H

String · Accepted Answer

Here are some light considerations on the topic (not much but at least something):

CONSIDERATION NR. 1
If f(x) = ax+b then h(x) = a²x+(a+1)b = αx+β. The system of equations:

α = a²
β = (a+1)b

always has two solutions (a,b) - except when α = 0 or when α = 1 and β ≠ 0. In the first case (a,b) = (0,β) is the only solution. In the latter case a=-1 cannot be used as it forces β = 0. But then the system at least still has one solution!

CONCLUSION NR. 1
Any line h(x)=αx+β of positive slope α is a member of H.

CONSIDERATION NR. 2
If f(x) = ax²+bx+c, where we require a ≠ 0 since otherwise we are back to my first consideration, then h(x) = αx^4+βx³+γx²+δx+ε where:

α = a³
β = 2a²b
γ = 2a²c+ab²+ab
δ = 2abc+b²
ε = ac²+bc+c

The first equation uniquely determines a. When a is determined the second equation uniquely determines b (remember that a is assumed non-zero). When b is determined the third equation uniquely determines c. When a, b and c are found so they (uniquely) solve the first three equations, then δ and ε are also determined and has to be equal to their respective expressions in a, b and c in order for the entire system to be solved. That means that if we take the first three parameters to be freely chosen then the last two are uniquely determined from that.

CONCLUSION NR. 2
If h(x) = αx^4+βx³+γx²+δx+ε where α ≠ 0 then a solution f(x) = ax²+bx+c can only be found if the last two coefficients fit uniquely with the first three.

fullem · Answer

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Function Spaces: Self-Composition?

2 Answers