Let's call a real-valued continuous function an n-th identity root when it generates the identity function after exactly n compositions with itself. For example f(x)=-x is a 2nd identity root because f(f(x))=x.

1. The function f(x)=1-1/x is a 3rd identity root. Unfortunately it is undefined at x=0. Are there identity roots for n>2 which are defined for all real numbers?

2. For a given real number c and n>1, give an example of an n-th identity root which is defined for all real numbers except c. How many such roots exist?

The linear fractional transformation f(x)=(Ax+B)/(Cx+D) with A=D=cos(2pi/n) and B=-C=sin(2pi/n) works for part 2. when c=cot(2pi/n).

I don't see how to get one like this for a general value of c, but I suspect that for each c there are infinitely many linear fractional transformations that work for each n.

Posted by Richard on 2006-11-08 23:48:05