When faced with a recurrence of the form $x_{n+1} = f(x_n)$, my toolbelt for proving convergence is very limited: if $f$ isn't $k$-lipschitzian with $k<1$, and/or if I can't find some complete set that $f$ maps into itself, I'm pretty much at a loss. I'd like to expand my arsenal a bit.

Are there any books, or at least chapters of books, which cover general, practical results for analysing the convergence, limits and basins of attraction of recurrences of this form? I'm only really interested in results that can be applied to functions $f$ over the real numbers.

  • $\begingroup$ Perhaps some discrete form of Lyapunov function? $\endgroup$ – copper.hat Dec 28 '15 at 20:39
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    $\begingroup$ This is the beginning of the theory of discrete dynamical systems. If you can find an introductory textbook on dynamical systems, you will find an exhaustive treatment of problems like this. $\endgroup$ – Lukas Geyer Dec 28 '15 at 22:08

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