The classical Denjoy–Wolff theorem is the following one-dimensional result: Let be the open unit disc in the complex plane . If is not the identity and is not an automorphism of with exactly one fixed point in , then there is a unique point in the closed unit disc such that the iterates of converge to , uniformly on compact subsets of .
For and , the set
is called a horocycle at with radius . This set is a disc in which is internally tangent to at (cf. also Horocycle).
A) The Wolff–Schwarz lemma: If has no fixed point in , then there is a unique unimodular point such that every horocycle in , internally tangent to at , is -invariant, i.e.,
B) If has no fixed point in , then there is a unique unimodular point such that the sequence converges to , uniformly on compact subsets of .
C) If is not an automorphism of but has a fixed point in , then this point is unique in , and the sequence converges to uniformly on compact subsets of . The limit point in B) is sometimes called the Denjoy–Wolff point of .
The point in A) and the point in B) are one and the same. However, this is not always the case in higher-dimensional situations.
Therefore, in the general case, the point in A) is usually called the sink point of . So, the sink point is the Denjoy–Wolff point if it is also attractive.
By using the Schwarz lemma, assertion C) can be rephrased as follows:
D) Let have a fixed point . If is not the identity, then is power convergent if and only if .
Since 1926, these results have been developed in several directions. For a nice exposition of the one-dimensional case see [a5].
When is a complex Hilbert space with the inner product , and is its open unit ball, the following generalization of the Wolff–Schwarz lemma is due to K. Goebel [a13]: If has no fixed point, then there exists a unique point such that for each the set
Geometrically, the set is an ellipsoid the closure of which intersects the unit sphere at the point . It is a natural analogue of the horocycle .
For infinite-dimensional Hilbert balls, A. Stachura [a30] has given a counterexample to show that the convergence result fails even for biholomorphic self-mappings.
Nevertheless, some restrictions on a mapping from lead to a generalization of assertion B). In particular, C.-H. Chu and P. Mellon [a8] showed that if is a compact mapping with no fixed point in , then the sink point in (a3) is attractive in the topology of locally uniform convergence. For weak convergence results see, for example, [a12].
Another look at the Denjoy–Wolff theorem is provided by a useful result of P. Yang [a38] concerning a characterization of the horocycle in terms of the Poincaré hyperbolic metric in (cf. also Poincaré model). More precisely, he established the following formula:
So, in these terms the horocycle in can be described by the formula
Since a hyperbolic metric can be defined in each bounded domain in , one can try to extend this formula and use it as a definition of the horosphere in a domain in . Unfortunately, in general the limit in (a4) does not exist.
To overcome this difficulty, M. Abate [a1] introduced two kinds of horospheres. More precisely, he defined the small horosphere of centre , pole and radius by the formula
and the big horosphere of centre , pole and radius by the formula
where is a bounded domain in and is its Kobayashi metric (cf. Hyperbolic metric). For the Euclidean ball in , .
Thus, each assertion which states for a domain in the existence of a point such that
for all , , and is a generalization of the Wolff–Schwarz lemma. This is true, for example, for a bounded convex domain in [a1]. However, in this case B) does not hold in general. Nevertheless, the convergence result does hold for bounded strongly convex domains, and for strongly pseudo-convex hyperbolic domains with a boundary [a1], [a2].
Assertion A) can be generalized to the operator ball over a Hilbert space and, more generally, to the open unit ball of a so-called -algebra (see [a37], [a25] and the references there), while B) fails in general even if the compactness of is assumed [a8].
For the particular case when is defined by the Riesz–Dunford integral in the sense of the functional calculus (cf. also Dunford integral), a full analogue of the Denjoy–Wolff theorem is due to K. Fan [a10], [a11].
For general Banach spaces there are examples showing that there are situations where even a sink point does not exist [a22].
However, the question is still open (as of 2000) for reflexive Banach spaces. Moreover, when is the open unit ball of a strictly convex Banach space and is compact or, more generally, condensing (cf. also Contraction operator), then the analogue of B) is valid [a16], [a17], [a18].
The situation is more fully understood when has a fixed point inside a bounded domain .
Simple examples show that one cannot always expect to be an attractive fixed point, even if is not an automorphism. Nevertheless, rephrasing C) in the form D), one can show (see, for example, [a22]) that is an attractive fixed point of if and only if the spectral radius of the Fréchet derivative is strictly less than .
In the one-dimensional case, if is not the identity, has an interior fixed point and is power convergent, then is unique. However, this is no longer true in higher dimensions.
A full description of such a situation was obtained by E. Vesentini [a32]: Suppose that has a fixed point , and denote the spectrum of the linear operator by (cf. also Spectrum of an operator). Then is power convergent if and only if the following two conditions hold:
i) ; and
ii) is a pole of the resolvent of of order at most one.
Condition ii) is actually equivalent to the condition
(see, for example, [a23]). It is also known that conditions i) and ii) are equivalent to being power-convergent to a projection onto . So, if the retraction is the limit point of under these conditions, then is constant if and only if .
The family of iterates of can be considered a one-parameter discrete sub-semi-group of (cf. also Semi-group of holomorphic mappings). Therefore, another direction is concerned with analogues of the Denjoy–Wolff theorem for continuous semi-groups of holomorphic self-mappings of . This approach has been used to study the asymptotic behaviour of solutions to Cauchy problems (see, for example, [a4], [a2], [a3], [a21], [a26], [a27], and [a18]). E. Berkson and H. Porta [a4] have applied their continuous analogue of the Denjoy–Wolff theorem to the study of the eigenvalue problem for composition operators on Hardy spaces.
For results on the asymptotic behaviour (in the spirit of the Denjoy–Wolff theorem) of (not necessarily holomorphic) mappings and semi-groups that are non-expansive with respect to hyperbolic metrics, see, for example, [a19], [a28], [a29], [a31].
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Denjoy-Wolff theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Denjoy-Wolff_theorem&oldid=22349