# How often can we renormalize unimodal maps?

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 $$f^n(J)\subseteq J$$.

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