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.