Denjoy-Wolff theorem

Wolff–Denjoy theorem

For a domain $\mathcal{D}$ in a complex Banach space $X$ one denotes by $\operatorname {Hol}( \mathcal{D} )$ the set of all holomorphic self-mappings of $\mathcal{D}$ (cf. also Analytic function).

The classical Denjoy–Wolff theorem is the following one-dimensional result: Let $\Delta$ be the open unit disc in the complex plane $\mathbf{C}$. If $F \in \operatorname { Hol } ( \Delta )$ is not the identity and is not an automorphism of $\Delta$ with exactly one fixed point in $\Delta$, then there is a unique point $a$ in the closed unit disc $\overline{\Delta}$ such that the iterates $\{ F ^ { n } \} _ { n = 1 } ^ { \infty } $ of $F$ converge to $a$, uniformly on compact subsets of $\Delta$.

This result is, in fact, a summary of the following three assertions A)–C) due to A. Denjoy and J. Wolff [a9], [a33], [a34], [a35], [a36].

For $\xi \in \partial \Delta$ and $R > 0$, the set

\begin{equation} \tag{a1} D _ { \xi } = D ( \xi , R ) : = \left\{ z \in \Delta : \frac { | 1 - z \overline { \xi } | ^ { 2 } } { 1 - | z | ^ { 2 } } < R \right\} \end{equation}

is called a horocycle at $\xi $ with radius $R$. This set is a disc in $\Delta$ which is internally tangent to $\partial \Delta$ at $\xi $ (cf. also Horocycle).

A) The Wolff–Schwarz lemma: If $F \in \operatorname { Hol } ( \Delta )$ has no fixed point in $\Delta$, then there is a unique unimodular point $a \in \partial \Delta$ such that every horocycle $D _ { a }$ in $\Delta$, internally tangent to $\partial \Delta$ at $a$, is $F$-invariant, i.e.,

\begin{equation} \tag{a2} F ( D _ { a } ) \subset D _ { a } \end{equation}

This assertion is a natural complement of the Julia–Wolff–Carathéodory theorem [a6].

B) If $F \in \operatorname { Hol } ( \Delta )$ has no fixed point in $\Delta$, then there is a unique unimodular point $b \in \partial \Delta$ such that the sequence $\{ F ^ { n } \} _ { n = 1 } ^ { \infty } $ converges to $b$, uniformly on compact subsets of $\Delta$.

C) If $F \in \operatorname { Hol } ( \Delta )$ is not an automorphism of $\Delta$ but has a fixed point $c$ in $\Delta$, then this point is unique in $\Delta$, and the sequence $\{ F ^ { n } \} _ { n = 1 } ^ { \infty } $ converges to $c$ uniformly on compact subsets of $\Delta$. The limit point in B) is sometimes called the Denjoy–Wolff point of $F$.

The point $a$ in A) and the point $b$ 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 $a$ in A) is usually called the sink point of $F$. 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 $F \in \operatorname { Hol } ( \Delta )$ have a fixed point $c \in \Delta$. If $F$ is not the identity, then $F$ is power convergent if and only if $| F ^ { \prime } ( c ) | < 1$.

Since 1926, these results have been developed in several directions. For a nice exposition of the one-dimensional case see [a5].

When $X = \mathcal{H}$ is a complex Hilbert space with the inner product $( \, . \, , \, . \, )$, and $\mathbf{B}$ is its open unit ball, the following generalization of the Wolff–Schwarz lemma is due to K. Goebel [a13]: If $F \in \operatorname { Hol } ( \bf B )$ has no fixed point, then there exists a unique point $a \in \partial \bf B$ such that for each $0 < R < \infty$ the set

\begin{equation} \tag{a3} E ( a , R ) = \left\{ x \in \mathbf{B} : \frac { | 1 - ( x , a ) | ^ { 2 } } { 1 - \| x \| ^ { 2 } } < R \right\} \end{equation}

is $F$-invariant.

Geometrically, the set $E ( a , R )$ is an ellipsoid the closure of which intersects the unit sphere $\partial \bf B$ at the point $a$. It is a natural analogue of the horocycle $D ( a , R )$.

In the finite-dimensional case, $\mathcal{H} = \mathbf{C} ^ { n }$, the sink point $a$ is also the Denjoy–Wolff point of $F$; see [a14], [a24], [a20], [a7].

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 $\operatorname{Hol} ( \mathbf{B} )$ lead to a generalization of assertion B). In particular, C.-H. Chu and P. Mellon [a8] showed that if $F \in \operatorname { Hol } ( \bf B )$ is a compact mapping with no fixed point in $\mathbf{B}$, then the sink point $a$ in (a3) is attractive in the topology of locally uniform convergence. For weak convergence results see, for example, [a12].

In 1941, M.H. Heins [a15] extended the Denjoy–Wolff theorem to a finitely connected domain bounded by Jordan curves in $\mathbf{C}$ (cf. also Jordan curve). His approach is specific to the one-dimensional case.

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 $\Delta$ (cf. also Poincaré model). More precisely, he established the following formula:

\begin{equation} \tag{a4} \operatorname { lim } _ { \mu \rightarrow \alpha } [ \rho ( \lambda , \mu ) - \rho ( 0 , \mu ) ] = \frac { 1 } { 2 } \operatorname { log } \frac { | 1 - \lambda \overline { a }| ^ { 2 } } { 1 - | \lambda | ^ { 2 } }. \end{equation}

So, in these terms the horocycle $D _ { a }$ in $\Delta$ can be described by the formula

\begin{equation} \tag{a5} D ( a , R ) = \end{equation}

\begin{equation*} = \left\{ z \in \Delta : \operatorname { lim } _ { \omega \rightarrow a } [ \rho ( z , \omega ) - \rho ( 0 , \omega ) ] < \frac { 1 } { 2 } \operatorname { log } R \right\}. \end{equation*}

Since a hyperbolic metric can be defined in each bounded domain in $\mathbf{C} ^ { n }$, one can try to extend this formula and use it as a definition of the horosphere in a domain in $\mathbf{C} ^ { n }$. 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 $E _ { z _ { 0 } } ( x , R )$ of centre $x$, pole $z_0$ and radius $R$ by the formula

\begin{equation*} E _ { z _ { 0 } } ( x , R ) = \end{equation*}

\begin{equation*} = \left\{ z \in \mathcal{D} : \operatorname { limsup } _ { w \rightarrow x } [ K _ { \mathcal{D} } ( z , w ) - K _ { \mathcal{D} } ( z _ { 0 } , w ) ] < \frac { 1 } { 2 } \operatorname { log } R \right\}, \end{equation*}

and the big horosphere $F _ { z _ { 0 } } ( x , R )$ of centre $x$, pole $z_0$ and radius $R$ by the formula

\begin{equation*} F _ { z _ { 0 } } ( x , R ) = \end{equation*}

\begin{equation*} = \{ z \in {\cal D} : \operatorname { lim\,inf } _ { w \rightarrow x } [ K _ {\cal D } ( z , w ) - K _ {\cal D } ( z_0 , w ) ] < \frac { 1 } { 2 } \operatorname { log } R \}, \end{equation*}

where $\mathcal{D}$ is a bounded domain in $\mathbf{C} ^ { n }$ and $K _ { \mathcal{D} }$ is its Kobayashi metric (cf. Hyperbolic metric). For the Euclidean ball in $\mathbf{C} ^ { n }$, $E _ { z _ { 0 } } ( x , R ) = F _ { z _ { 0 } } ( x , R )$.

Thus, each assertion which states for a domain $\mathcal{D}$ in $\mathbf{C} ^ { n }$ the existence of a point $a \in \partial \mathcal{D}$ such that

\begin{equation} \tag{a6} F ^ { n } ( E _ { z } ( a , R ) ) \subset F _ { z } ( a , R ) \end{equation}

for all $z \in \mathcal{D}$, $R > 0$, $F \in \operatorname{Hol} ( {\cal D} )$ and $n = 1,2 , \dots$ is a generalization of the Wolff–Schwarz lemma. This is true, for example, for a bounded convex domain in $\mathbf{C} ^ { n }$ [a1]. However, in this case B) does not hold in general. Nevertheless, the convergence result does hold for bounded strongly convex $C ^ { 2 }$ domains, and for strongly pseudo-convex hyperbolic domains with a $C ^ { 2 }$ boundary [a1], [a2].

Assertion A) can be generalized to the operator ball over a Hilbert space $\mathcal{H}$ and, more generally, to the open unit ball $U$ of a so-called $J ^ { * }$-algebra (see [a37], [a25] and the references there), while B) fails in general even if the compactness of $F$ is assumed [a8].

For the particular case when $F$ 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 $\mathcal{D}$ is the open unit ball of a strictly convex Banach space $X$ and $F$ 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 $F$ has a fixed point $c$ inside a bounded domain $\mathcal{D} \subset X$.

Simple examples show that one cannot always expect $c$ to be an attractive fixed point, even if $F$ is not an automorphism. Nevertheless, rephrasing C) in the form D), one can show (see, for example, [a22]) that $c$ is an attractive fixed point of $F$ if and only if the spectral radius of the Fréchet derivative $F ^ { \prime } ( c )$ is strictly less than $1$.

In the one-dimensional case, if $F \in \operatorname{Hol} ( {\cal D} )$ is not the identity, has an interior fixed point and is power convergent, then $c$ 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 $F$ has a fixed point $c \in \mathcal{D}$, and denote the spectrum of the linear operator $F ^ { \prime } ( c )$ by $\sigma ( F ^ { \prime } ( c ) )$ (cf. also Spectrum of an operator). Then $F$ is power convergent if and only if the following two conditions hold:

i) $\sigma ( F ^ { \prime } ( c ) ) \subset \Delta \cup \{ 1 \}$; and

ii) $1$ is a pole of the resolvent of $F ^ { \prime } ( c )$ of order at most one.

Condition ii) is actually equivalent to the condition

\begin{equation*} \operatorname { Ker } ( I - F ^ { \prime } ( c ) ) \bigoplus \operatorname { Im } ( I - F ^ { \prime } ( c ) ) = X \end{equation*}

(see, for example, [a23]). It is also known that conditions i) and ii) are equivalent to $F ^ { \prime } ( c )$ being power-convergent to a projection $P$ onto $\operatorname { Ker } ( I - F ^ { \prime } ( c ) )$. So, if the retraction $R \in \operatorname { Hol } ( \mathcal{D} )$ is the limit point of $\{ F ^ { n } \}$ under these conditions, then $R = c$ is constant if and only if $P = 0$.

The family $\{ F ^ { n } \} _ { n = 1 } ^ { \infty } $ of iterates of $F \in \operatorname{Hol} ( {\cal D} )$ can be considered a one-parameter discrete sub-semi-group of $\operatorname {Hol}( \mathcal{D} )$ (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 $D$. 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].

References

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