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?

(In reply to Curious by JLo)

Chalk mine up to experience.  I have had some exposure to homogenous coordinates and the projective closure of the real line.  The linear fractional transformations become linear in homogeneous coordinates -- f(x)=(Ax+B)/(Cx+D) is represented in homogeneous coordinates by the 2x2 matrix with rows (A,B),(C,D).  Also, the matrices (A,B),(-B,A) multiply just like the complex numbers A+iB. This is all related to your circle idea in some way, I'm sure, but I only vaguely see (more like feel, actually) the relationship at the moment.  Anyway, you have brushed up against some geometry here of a type which unfortunately does not seem to be studied very much anymore.

Your f(x-s)+s was a surprise to me, but it checks out. Thanks for letting me in on a nice fact!

 

Edited on November 10, 2006, 12:18 am

Posted by Richard on 2006-11-09 18:15:56