Situation: $$ x_{n+1}=f_{\mu}(x_n) ~ , ~ x_0 \in I \subset \mathbb{R}$$ Here, $f_{\mu}$ is a family of self-mapping functions from $I$ to $I$. We further know that we get period-doubling (for example by looking at the bifurcation diagram).
Now, Feigenbaum's universality tells us that the doubling rate must converge to a speed of $\delta \approx 4.669$.
But what exactly are the prerequisites to $f$ for making that statement? Is smoothness enough? Does it need to be a unimodal mapping?