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.
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)[1] and Burckel (1981).
[2] Two other short proofs can be found in Carleson & Gamelin (1993).
[3] If f has a fixed point z in D then, after conjugating by a Möbius transformation, it can be assumed that z = 0.
By the Schwarz lemma[4] for |z| ≤ r, where It follows by iteration that for |z| ≤ r. These two inequalities imply the result in this case.
increasing to 1 and set[5][6] By applying Rouché's theorem to
The result for the case of fixed points implies that the maps
leave invariant all Euclidean disks whose hyperbolic center is located at
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.
converges uniformly on compacta to the constant z, it is enough to show that the same is true for any subsequence
Such limits exist by Montel's theorem, and if g is non-constant, it can also be assumed that
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.
converges uniformly on compacta to the constant z.