Denjoy–Wolff theorem
In mathematics, the Denjoy–Wolff theorem is a theorem in complex analysis and dynamical systems concerning fixed points and iterations of holomorphic mappings of the unit disc in the complex numbers into itself. The result was proved independently in 1926 by the French mathematician Arnaud Denjoy and the Dutch mathematician Julius Wolff. Statement. Theorem. Let "D" be the open unit disk in C and let "f" be a holomorphic function mapping "D" into "D" which is not an automorphism of "D" (i.e. a Möbius transformation). Then there is a unique point "z" in the closure of "D  such that the iterates of f" tend to "z" uniformly on compact subsets of "D". If "z" lies in "D", it is the unique fixed point of "f". The mapping "f" leaves invariant hyperbolic disks centered on "z", if "z" lies in "D", and disks tangent to the unit circle at "z", if "z" lies on the boundary of "D". When the fixed point is at "z" = 0, the hyperbolic disks centred at "z" are just the Euclidean disks with centre 0. Otherwise "f" can be conjugated by a Möbius transformation so that the fixed point is zero. An  elementary proof of the theorem is given below, taken from Shapiro (1993) and Burckel (1981). Two other short proofs can be found in Carleson & Gamelin (1993). Proof of theorem. Fixed point in the disk. If "f" has a fixed point "z" in "D" then, after conjugating by a Möbius transformation, it can be assumed that "z" = 0. Let "M"("r") be the maximum modulus of "f" on "|z|" = "r" < 1. By the Schwarz lemma for |"z"| ≤ "r", where It follows by iteration that for |"z"| ≤ "r". These two inequalities imply the result in this case. No  fixed points. When "f" acts in "D" without fixed points, Wolff showed that there is a point "z" on the boundary such that the iterates of "f" leave invariant each disk tangent to the boundary at that point. Take a sequence formula_4 increasing to 1 and set By applying Rouché's theorem to formula_6 and formula_7, formula_8 has exactly one zero formula_9 in "D". Passing to a subsequence if necessary, it can be assumed that formula_10 The point "z" cannot lie in "D", because, by passing to the limit, "z" would have to be a fixed point. The result  for the case of fixed points implies that the maps formula_8 leave invariant all Euclidean disks whose hyperbolic center is located at formula_9. Explicit computations show that, as "k" increases, one can choose such disks so that they tend to any given disk tangent to the boundary at "z". By continuity, "f" leaves each such disk Δ invariant. To see that formula_13 converges uniformly on compacta to the constant "z", it is enough to show that the same is true for any subsequence formula_14, convergent in the same sense to "g", say. Such limits exist by Montel's theorem, and  if "g" is non-constant, it can also be assumed that formula_15 has a limit, "h" say. But then for "w" in "D". Since "h" is holomorphic and "g"("D") open, for all "w". Setting formula_18, it can also be assumed that formula_19 is convergent to "F" say. But then "f"("F"("w")) = "w" = "f"("F"("w")), contradicting the fact that "f" is not an automorphism. Hence every subsequence tends to some constant uniformly on compacta in "D". The invariance of Δ implies each such constant lies in the closure of each disk Δ, and hence their intersection, the single point "z". By  Montel's theorem, it follows that formula_13 converges uniformly on compacta to the constant "z".