TITLE: Convergence of Newton's method
QUESTION [14 upvotes]: For a polynomial $P$ of degree $n$ with real coefficients and with $n$ distinct real roots, the Newton's method $z_{n+1} = z_n - {P(z_n) \over P'(z_n)}$ converges for almost all initial values $z_0$ in $\mathbb R$ (or almost all $z_0$ in $\bf C$ with respect to the area measure) to a root of $P$. This is a result due to M. Lyubich (~ 1984).
I think I remember that for a polynomial with complex coefficients, almost all initial values $z_0$ has an orbit that converges to a periodic orbit in ${\mathbb C} \cup \{\infty\}$, but there are examples where that orbit is not a root of $P$.
Unfortunately, I can't remember who is the author of that result and I would like to find a reference.
EDIT: the result is actually false. There are polynomials whose Newton's method has a periodic Siegel disk, see e.g. this answer. In that case,  there is an open set of points whose orbit's $\omega$-limit set is a circle.

REPLY [7 votes]: Your statement that iterates of the Newton method converge to a cycle almost everywhere is equivalent to the statement that for every polynomial $f$
the Julia set of the rational function $z-f(z)/f'(z)$ has zero area.
This is unlikely to be true, but I do not know a published counterexample.
For the state of the art on Newton Method for polynomials, I recommend these papers:
MR1859017 
J. Hubbard, D. Schleicher,  S. Sutherland, 
How to find all roots of complex polynomials by Newton's method. 
Invent. Math. 146 (2001), no. 1, 1–33.
MR3659421 D. Schleicher, R. Stoll, Newton's method in practice: Finding all roots of polynomials of degree one million efficiently. Theoret. Comput. Sci. 681 (2017), 146–166.