Convergence of Newton's method 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.
 A: 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.
A: For Newton's method (and more general iterative methods) for finding roots of complex polynomials, you may want to look at Curt McMullen's paper:
Families of Rational Maps and Iterative Root-Finding Algorithms,
Curt McMullen,
Annals of Mathematics,
Second Series, Vol. 125, No. 3 (May, 1987), pp. 467-493 
From the abstract: "In this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root-finding algorithms. We answer a question of Smale's by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more. We settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics; and we give a classification of all such algorithms."
A: I don't think your initial assertion is accurate. Consider, for example, $f(z)=z^5-z-1$. If you iterate the Newton's method function $N(z) = z-f(z)/f'(z)$ from $z_0=0$, you'll quickly find an attractive orbit of period 3. The basin of attraction of that orbit is a positive measure set with no point converging to a root of $f$. The standard Newton method picture looks like so:

Those black regions are exactly where your assertion fails. Notice, also, the five regions converging to five simple roots.
