This problem has been bothering me for a long time, and I come back to it every couple of months or so but can never seem to make any progress:
Let $f(x)=x^2-3/2$. Consider the function $$f^{\circ n}(x)=\overbrace{f(f(...f}^{n}(x)...)$$ What can be said about the number of real zeroes of $f^{\circ n}(x)$ for large values of $n$?
I’m sure that the number of zeroes grows exponentially $\sim a\cdot b^n$ for some $a,b$, but I can’t seem to calculate $b$. I can’t even come up with very satisfying bounds for $b$, even when I try looking for some self-similarity in the function $f$ that I might exploit.
Help?