Let $\{x_n\}_{n\ge 0}$ be a sequence of reals such that $x_{n+1}=g(x_n)$, where $g:\mathbb{R}\to \mathbb{R}$ is a continuous function such that $0$ is a fixed point of $g$. My question is the following:
If for an initial condition $x_0$, we have $x_n\to 0$ as $n\to \infty$, how should one proceed to find the rate of convergence of the sequence $x_n$?
I know that if $g$ is a contraction, then this rate of convergence is linear. However,
how should one proceed if $g$ is not a contraction?
For example, let $g(x) = x^2$. If $x_0<1$, then we have $x_n = x_0^{2^n}\stackrel{n\to \infty}{\to} 0$, and the rate of convergence is quadratic.
Note that if $g$ is Lipschitz, then I can always find an initial condition suitably so that the sequence converges to $0$ linearly. However, this estimated rate would only be an upper bound on the actual rate, as it was seen for the $x^2$ example, the actual rate is quadratic and is thus very fast.
Please provide some references to relevant literature. Thanks in advance.
