Nonlinear recursions on the reals and a problem of Graham

Abstract

We study sequences $(x_n)_{n=1}^{\infty}$ of reals given by $x_{n+1} = f(x)$ where $$f(x) = x - \sum_{i=1}^{m} \frac{\alpha_i}{x - \beta_i},$$ where $\alpha_1, \dots, \alpha_m \in \mathbb{R}_{>0}$ and $\beta_1, \dots, \beta_m \in \mathbb{R}$ are arbitrary. A special case is $x_{n+1} = x_n - 1/x_n$ due to Ronald Graham for which Chamberland \& Martelli showed that the dynamics is chaotic (topologically conjugate to the doubling map). We prove that the general nonlinear recursion, despite being potentially chaotic, is effective at ensuring that most iterates end up close to one of the poles $\beta_i$ relatively quickly. More precisely, for a positive proportion of initial values $x \in \mathbb{R}$, the sequence gets very close (distance $\lesssim |x|^{-1}$) to one of the poles $\beta_i$ within a relatively small ($\lesssim x^2$) number of iteration steps.