Let $f : A \to A \subseteq \mathbb R$ be a real function with a fixed point $a_0 = f(a_0)$ which is not attractive.

Let $f^k = f \circ f \circ ... \circ f$ be the $k^{th}$ iterate of $f$ (with the usual convention that $f^0 = \text{id}_A$).

Let $A_n(a) = \frac{1}{n}\sum_{k=0}^{n-1} f^k(a)$ be the average of the first $n$ iterates of $f$ starting at $a \in A$.

Are there known sufficient conditions for the limit to exist $lim_{n \to \infty} A_n(a) = a_0$?

The above quite obviously holds if $x_0$ is an attractive fixed point, or if $f^k(a)$ can be otherwise shown to converge. But in cases where $f^k(a)$ diverges, it is still possible that $A_n$ converges.

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The question is related to a post on MSE the other day (deleted by its author since) that noted that for $f(x) = \ln(|x|)$ the average of the first $n$ iterates appeared to converge to a constant limit, which (not unexpectedly) turned out to be $-\Omega = -W(1)$ the fixed point where $log(-x) = x$.