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.

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).

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.

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.

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