Fractional Iterates and Oscillatory Convergence

TL;DR

This work tackles the problem of real fractional iteration for maps exhibiting oscillatory convergence, extending Koenig's framework with Abelian and Modified-Abel constructions to produce half- and higher-order iterates. It showcases complex half-iterates for the simple map 1+1/x via Schröder and Abel conjugacies and develops a Modified Abel method that yields real-valued, though discretized, fractional iterates (often forming disjoint curve segments). The paper also treats the related maps 2+1/x, cos(x), and the logistic family λx(1−x), providing numerical half-iterate values, slow-converging series, and open questions, including the nonexistence of simple real half-iterates in the golden/silver mean cases. A corrigendum clarifies these real-iteration limitations and underscores persistent challenges in achieving a continuous real fractional interpolation for oscillatory convergence. The results illuminate both the potential and the limits of fractional iteration for classical nonlinear maps and motivate further refinement of oscillatory-convergence techniques.

Abstract

The simple continued fractions for the Golden & Silver means are well-known. It is astonishing that, as far as we know, no one has published half-iterates (let alone quarter-iterates) for the corresponding algorithms. We also examine the cosine and logistic maps (with parameter $2 < λ< 3$).