Fractional Iterates and Oscillatory Convergence

TL;DR

This paper explores fractional iterates of algorithms related to continued fractions for Golden and Silver means, and examines the behavior of cosine and logistic maps within a specific parameter range.

Contribution

It introduces the concept of fractional iterates for continued fraction algorithms and analyzes their properties, which has not been previously published.

Findings

Fractional iterates for Golden and Silver mean algorithms are developed.

Analysis of cosine and logistic maps with parameters between 2 and 3.

New insights into oscillatory convergence behaviors.

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 $2 < \lambda < 3$).