A unimodal map is function $f:[0,1]\rightarrow [0,1]$ such that there exist $c\in (0,1)$ such that $f$ is strictly increasing on $[0,c)$ and strictly decreasing on $(c,1]$.

A unimodal map f is renormalizable if there is a proper subinterval $J$ of $[0,1]$ and an integer $n>1$ such that $f^n|_J$ is itself (topologically conjugate to) a unimodal map.

Of course we can ask ourselves whether we can renormalize the renormalized function again. My question is:

Can we determine the maximal number of renormalizations we can perform for a given unimodal function?

In case this is too difficult, is it known for the logistic family $f_\lambda(x)= \lambda x (1-x)$?

Added: I forgot to mention that $f^n(J)\subseteq J$.

  • $\begingroup$ Yes, sorry, forgot to write that. @PietroMajer $\endgroup$ Dec 8, 2019 at 20:35


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