TITLE: Supremum of $ a_n = a_{n-1}^3 - a_{n-2} $
QUESTION [13 upvotes]: Let $a_1=0$ and let $ - \ln(2) < a_2 < \ln(2) $ 
Define
$$ a_n = a_{n-1}^3 - a_{n-2} $$
Then 
$$ \sup_{n>2} a_n = a_2 $$
And
$$ \inf_{n>2} a_n = - a_2 $$
How to prove that ?

REPLY [6 votes]: I encountered a similar problem with the iteration $u_n=\sqrt{u_{n-1}^2+1}-u_{n-2}$, where there is fundamentally a $9$-cycle in the shape of a maple leaf
(replace the +1 by 0 to see this). I asked an expert 20 years ago who told me that using the
KAM (Kolmogorov-Arnold-Moser) theorem, one could prove that the "curve" drawn
above by R. Israel is not a curve at all, but a very thin strip of width 0.07
or something. Don't ask me for the proof, I have no idea.