TITLE: Could the orbit of $0$ through a polynomial be dense in $\mathbb R$?
QUESTION [9 upvotes]: Could we construct a polynomial $P(x)\in\mathbb R[x]$ such that the sequence $(x_n)$ which is given by $x_0=P(0)$ and $x_{n+1}=P(x_n)$ for any $n\ge0$, is dense in $\mathbb R$ ?. 
I guess the answer is no, but I could not prove it.

REPLY [8 votes]: The theory of iteration of polynomial functions is well understood. It is usually studied in the complex plane, but its methods and results apply to the real line.
To answer your question:

No orbit under the iteration of a polynomial is dense in the real line.

Indeed:

An orbit of a point under the iteration of a polynomial $P(x)$ either remains bounded forever or escapes to infinity.
The set of starting points whose orbit is bounded form a compact set, called the filled Julia set.

Explicitly, if $P(x)=a_d x^d + \cdots +a_1 x + a_0$, then all orbits starting outside the circle of radius $R = \dfrac{1+|a_d|+\cdots+|a_0|}{|a_d|}$ centered at the origin escape to infinity, a bound given by Douady.
Therefore, an orbit can be dense at most inside a compact interval. This does happen: $P(x)=x^2-2$ with $x_0=0$ is an example.
All this is the elementary part of the theory.
For this and much more, see the classic books The Beauty of Fractals and The Science of Fractal Images, and the surveys by Blanchard, Milnor, and Devaney and Keen.