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Square root of a function (in the sense of composition)

I'm interested in solving equations of the form $f(f(x))=g(x)$ for $x\in\mathbb{R}$ where $g(x)$ is a known function.

For example: if $g(x)=x$ then we $f(f(x))=x$ so one solution is $f(x)=x$, $\forall x\in\mathbb{R}$. However, another is

$f(x)=x+1$ if $x\in(0,1]$, $f(x)=x-1$ if $x\in(1,2]$ and $f(x)=x$ otherwise.

Hence there are infinitely many solutions to this equation when $g(x)=x$.

Problem: Find an $f$ where $f(f(x))=e^x$.

Extra: Are there extra constraints that could be placed on $f$ so the solution is unique?

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    The links provided here were also linked to the math overflow site where I have found the answers I was looking for.2010-09-19

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