Why does this definition of an attractor work, and where does it come from?

Question

I recetly saw an attractor of a function being defind as the point $(x,f(x))$ such that $$|f(f(x))'|<1$$ Where does this definition come from, both litterally, as in where can I read more about it, and why does it work?

Edit: Here's the paper where the definition was used, page 13: http://www.csun.edu/~vcmth02i/Collatz.pdf

Answer 1

The cited paper is talking about an attracting 2-cycle, $(c_1, c_2)$, where $c_2=f(c_1)$ and $c_1=f(c_2)$. Then $c_1$ and $c_2$ are attracting fixed points of $f\circ f$.

The condition on the derivative is $|(f\circ f)'(c_i)|<1$, not $|f(f(c_i))'|<1$, which does not make much sense.

Note that $(f\circ f)'(c_i)=f'(f(c_i))f'(c_i)=f'(c_1)f'(c_2)$.