Iterative Roots and Fractional Iteration

References collected by Lars Kindermann

A solution φ of the functional equation φ(φ(x))=f(x) or φ2=f is called an iterative root of the given function f. More general, solutions of φn=fm are the fractional iterates of f: φ=fm/n. Extending this notation towards even real exponents ft defines the continous iteration (semi)group of the function f. The "standard interpretation" of this procedure is the embedding of a discrete time dynamical system into a continuous flow. While this problem is investigated since 1815, Charles Babbage was the first to look for the "roots of identity" φ(φ(x))=x, it remains an often extremely difficult task to find the iterative roots of even very simple functions. Many questions about their existence, uniqueness and properties are still unsolved.

The most cited reference to this topic is the book Iterative functional equations by Marek Kuczma, Bogdan Choczewski & Roman Ger (1990) together with the update article Recent results on functional equations in a single variable, perspectives and open problems by Karol Baron & Witold Jarczyk (2001). Another good book with some more emphasis on applications is Topics in Iteration Theory (1981) with the update Progress of Iteration Theory since 1981 (1995) by György Targonski.

Some basic introduction and mathematical classification can be found online on the webpage by D. Rusin: Difference and functional equations. In: The mathematical Atlas at


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