Tutte seminar - Ron Mullin, distinguished professor emeritus

A Set-Theoretic Approach to Fractional Iterates

Abstract:

This talk relates to an early excursion of Bill Tutte into professional level mathematics. The connection will be explained in the talk.

Let f(x) be an invertible function defined on the real line, and let fⁿ denote the function f composed with itself n times. By convention, f⁰(x)=x for all x. Clearly such iterates satisfy f^m(fⁿ)= f^m+n for all natural numbers m,n.

This talk addresses the following two questions about the iterates of such functions.

(1) Functional square roots: Given f, does there exist f^1/2 such that f^1/2(f^1/2) = f? This has been generalized to include other specific rational functional roots.

(2) Fractional Iterates (also called Flows).

Given f, does there exist a family of real functions F^r, for all real numbers r, such that, for all non-negative real numbers r,s, F^r(F^s)=F^r+s and, for all natural numbers n, Fⁿ=fⁿ.
This has been generalized to include the case of for all integers n and all real numbers r and s (Complete real roots.)

These questions date back to at least 1815 and have been investigated for both mathematical interest and their applications in connection with dynamical systems. Although they have been studied by many prominent mathematicians such as J. Hadamard, G.H. Hardy, J. Aczel and W.T. Tutte, the subject seems to have gained even more prominence with the study of fractals and chaos theory.

Let G be a group of cardinality c that contains the integers as a normal subgroup. A G-generalized set of iterates for f is a set of functions F^a, for each a in G, such that, for every a,b in G, F^a(F^b)=F^a+b and, for every integer n, Fⁿ=fⁿ.

In this talk it will be shown that such iterates exist for all such G if and only if f is invertible, in particular, there exists a set of complete real roots (and for that matter, complex roots) if and only if f is invertible.

The result is extended to give an answer to a question that Tutte posed.