Exercises in Iterational Asymptotics III

TL;DR

This work develops iterational asymptotics for nonlinear recurrences linked to simple continued fractions, notably those converging to the Golden mean $φ$ and Silver mean $ψ$, and analyzes Abel's equation in a parametric cubic family. It derives precise asymptotic constants for error sequences and demonstrates geometric convergence rates tied to $φ$ and $ψ$, with explicit limits for transformed error terms such as $(1+φ)^{2k}u_k$ and $(1+2ψ)^{2k}u_k$. The paper combines analytic error-decomposition methods with high-precision, machine-assisted computations (ML-exc3F3-exc3) to obtain constants like $C(2)$ and its derivatives, while also highlighting potential nonuniform convergence in the Abel-graph setting. Collectively, these results deepen understanding of iterational convergence in nonlinear recurrences and illuminate connections between continued fractions, Abel's equation, and asymptotic expansions.

Abstract

The nonlinear recurrences we consider here include simple continued fractions for the Golden & Silver means and a parametric family of cubics in connection with Abel's functional equation.

Figures (6)

• Figure 1: Blue curve is $10f_{a}(x)$, scaled for visibility; green curve is $F_{a}(x)$; parameter $a=3/2$. $f_{a}$ is increasing; $F_{a}$ is decreasing. Distance between vertical notches is four times that for horizontal.
• Figure 2: Parameter $a=5/3$. Both curves remain monotone; gentle undulations start to appear in $F_{a}$.
• Figure 3: Parameter $a=\sqrt{3}$. Undulations become more pronounced. $F_{a}$-critical points are at $x=0.577,1.304,1.613,1.701,\ldots.$
• Figure 4: Parameter $a=9/5$. Curves are no longer monotone. $F_{a}$-minimum points appear at $x=0.436,1.286,1.663,1.766\ldots$; $F_{a}$-maximum points appear at $x=0.763,1.472,1.717,1.780\ldots.$
• Figure 5: Parameter $a=19/10$. Violent waves dominate the graph. $F_{a}$-minimum points appear at $x=0.372,1.368,1.771,1.871\ldots$; $F_{a}$-maximum points appear at $x=0.893,1.623,1.836,1.886\ldots.$