Namespaces
Variants
Actions

Difference between revisions of "Denjoy-Wolff theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (AUTOMATIC EDIT (latexlist): Replaced 158 formulas out of 158 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 +
<!--This article has been texified automatically. Since there was no Nroff source code for this article,
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
 +
 +
Out of 158 formulas, 158 were replaced by TEX code.-->
 +
 +
{{TEX|semi-auto}}{{TEX|done}}
 
''Wolff–Denjoy theorem''
 
''Wolff–Denjoy theorem''
  
For a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d1300801.png" /> in a complex [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d1300802.png" /> one denotes by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d1300803.png" /> the set of all holomorphic self-mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d1300804.png" /> (cf. also [[Analytic function|Analytic function]]).
+
For a domain $\mathcal{D}$ in a complex [[Banach space|Banach space]] $X$ one denotes by $\operatorname {Hol}( \mathcal{D} )$ the set of all holomorphic self-mappings of $\mathcal{D}$ (cf. also [[Analytic function|Analytic function]]).
  
The classical Denjoy–Wolff theorem is the following one-dimensional result: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d1300805.png" /> be the open unit disc in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d1300806.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d1300807.png" /> is not the identity and is not an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d1300808.png" /> with exactly one fixed point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d1300809.png" />, then there is a unique point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008010.png" /> in the closed unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008011.png" /> such that the iterates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008013.png" /> converge to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008014.png" />, uniformly on compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008015.png" />.
+
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 [[#References|[a9]]], [[#References|[a33]]], [[#References|[a34]]], [[#References|[a35]]], [[#References|[a36]]].
 
This result is, in fact, a summary of the following three assertions A)–C) due to A. Denjoy and J. Wolff [[#References|[a9]]], [[#References|[a33]]], [[#References|[a34]]], [[#References|[a35]]], [[#References|[a36]]].
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008017.png" />, the set
+
For $\xi \in \partial \Delta$ and $R &gt; 0$, the set
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
\begin{equation} \tag{a1} D _ { \xi } = D ( \xi , R ) : = \left\{ z \in \Delta : \frac { | 1 - z \overline { \xi } | ^ { 2 } } { 1 - | z | ^ { 2 } } &lt; R \right\} \end{equation}
  
is called a horocycle at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008019.png" /> with radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008020.png" />. This set is a disc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008021.png" /> which is internally tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008022.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008023.png" /> (cf. also [[Horocycle|Horocycle]]).
+
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|Horocycle]]).
  
A) The Wolff–Schwarz lemma: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008024.png" /> has no fixed point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008025.png" />, then there is a unique unimodular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008026.png" /> such that every horocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008027.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008028.png" />, internally tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008029.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008030.png" />, is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008031.png" />-invariant, i.e.,
+
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.,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\begin{equation} \tag{a2} F ( D _ { a } ) \subset D _ { a } \end{equation}
  
 
This assertion is a natural complement of the [[Julia–Wolff–Carathéodory theorem|Julia–Wolff–Carathéodory theorem]] [[#References|[a6]]].
 
This assertion is a natural complement of the [[Julia–Wolff–Carathéodory theorem|Julia–Wolff–Carathéodory theorem]] [[#References|[a6]]].
  
B) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008033.png" /> has no fixed point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008034.png" />, then there is a unique unimodular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008035.png" /> such that the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008036.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008037.png" />, uniformly on compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008038.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008039.png" /> is not an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008040.png" /> but has a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008041.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008042.png" />, then this point is unique in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008043.png" />, and the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008044.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008045.png" /> uniformly on compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008046.png" />. The limit point in B) is sometimes called the Denjoy–Wolff point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008047.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008048.png" /> in A) and the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008049.png" /> in B) are one and the same. However, this is not always the case in higher-dimensional situations.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008050.png" /> in A) is usually called the sink point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008051.png" />. So, the sink point is the Denjoy–Wolff point if it is also attractive.
+
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|Schwarz lemma]], assertion C) can be rephrased as follows:
 
By using the [[Schwarz lemma|Schwarz lemma]], assertion C) can be rephrased as follows:
  
D) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008052.png" /> have a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008053.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008054.png" /> is not the identity, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008055.png" /> is power convergent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008056.png" />.
+
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 $| F ^ { \prime } ( c ) | &lt; 1$.
  
 
Since 1926, these results have been developed in several directions. For a nice exposition of the one-dimensional case see [[#References|[a5]]].
 
Since 1926, these results have been developed in several directions. For a nice exposition of the one-dimensional case see [[#References|[a5]]].
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008057.png" /> is a complex [[Hilbert space|Hilbert space]] with the [[Inner product|inner product]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008058.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008059.png" /> is its open unit ball, the following generalization of the Wolff–Schwarz lemma is due to K. Goebel [[#References|[a13]]]: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008060.png" /> has no fixed point, then there exists a unique point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008061.png" /> such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008062.png" /> the set
+
When $X = \mathcal{H}$ is a complex [[Hilbert space|Hilbert space]] with the [[Inner product|inner product]] $( \, . \, , \, . \, )$, and $\mathbf{B}$ is its open unit ball, the following generalization of the Wolff–Schwarz lemma is due to K. Goebel [[#References|[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 $0 &lt; R &lt; \infty$ the set
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008063.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
\begin{equation} \tag{a3} E ( a , R ) = \left\{ x \in \mathbf{B} : \frac { | 1 - ( x , a ) | ^ { 2 } } { 1 - \| x \| ^ { 2 } } &lt; R \right\} \end{equation}
  
is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008064.png" />-invariant.
+
is $F$-invariant.
  
Geometrically, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008065.png" /> is an ellipsoid the closure of which intersects the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008066.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008067.png" />. It is a natural analogue of the horocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008068.png" />.
+
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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008069.png" />, the sink point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008070.png" /> is also the Denjoy–Wolff point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008071.png" />; see [[#References|[a14]]], [[#References|[a24]]], [[#References|[a20]]], [[#References|[a7]]].
+
In the finite-dimensional case, $\mathcal{H} = \mathbf{C} ^ { n }$, the sink point $a$ is also the Denjoy–Wolff point of $F$; see [[#References|[a14]]], [[#References|[a24]]], [[#References|[a20]]], [[#References|[a7]]].
  
 
For infinite-dimensional Hilbert balls, A. Stachura [[#References|[a30]]] has given a counterexample to show that the convergence result fails even for biholomorphic self-mappings.
 
For infinite-dimensional Hilbert balls, A. Stachura [[#References|[a30]]] has given a counterexample to show that the convergence result fails even for biholomorphic self-mappings.
  
Nevertheless, some restrictions on a mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008072.png" /> lead to a generalization of assertion B). In particular, C.-H. Chu and P. Mellon [[#References|[a8]]] showed that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008073.png" /> is a [[Compact mapping|compact mapping]] with no fixed point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008074.png" />, then the sink point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008075.png" /> in (a3) is attractive in the topology of locally [[Uniform convergence|uniform convergence]]. For weak convergence results see, for example, [[#References|[a12]]].
+
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 [[#References|[a8]]] showed that if $F \in \operatorname { Hol } ( \bf B )$ is a [[Compact mapping|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|uniform convergence]]. For weak convergence results see, for example, [[#References|[a12]]].
  
In 1941, M.H. Heins [[#References|[a15]]] extended the Denjoy–Wolff theorem to a finitely connected domain bounded by Jordan curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008076.png" /> (cf. also [[Jordan curve|Jordan curve]]). His approach is specific to the one-dimensional case.
+
In 1941, M.H. Heins [[#References|[a15]]] extended the Denjoy–Wolff theorem to a finitely connected domain bounded by Jordan curves in $\mathbf{C}$ (cf. also [[Jordan curve|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 [[#References|[a38]]] concerning a characterization of the horocycle in terms of the Poincaré hyperbolic metric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008077.png" /> (cf. also [[Poincaré model|Poincaré model]]). More precisely, he established the following formula:
+
Another look at the Denjoy–Wolff theorem is provided by a useful result of P. Yang [[#References|[a38]]] concerning a characterization of the horocycle in terms of the Poincaré hyperbolic metric in $\Delta$ (cf. also [[Poincaré model|Poincaré model]]). More precisely, he established the following formula:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008078.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008079.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008080.png" /> can be described by the formula
+
So, in these terms the horocycle $D _ { a }$ in $\Delta$ can be described by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008081.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
+
\begin{equation} \tag{a5} D ( a , R ) = \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008082.png" /></td> </tr></table>
+
\begin{equation*} = \left\{ z \in \Delta : \operatorname { lim } _ { \omega \rightarrow a } [ \rho ( z , \omega ) - \rho ( 0 , \omega ) ] &lt; \frac { 1 } { 2 } \operatorname { log } R \right\}. \end{equation*}
  
Since a hyperbolic metric can be defined in each bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008083.png" />, one can try to extend this formula and use it as a definition of the horosphere in a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008084.png" />. Unfortunately, in general the limit in (a4) does not exist.
+
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 [[#References|[a1]]] introduced two kinds of horospheres. More precisely, he defined the small horosphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008085.png" /> of centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008086.png" />, pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008087.png" /> and radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008088.png" /> by the formula
+
To overcome this difficulty, M. Abate [[#References|[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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008089.png" /></td> </tr></table>
+
\begin{equation*} E _ { z _ { 0 } } ( x , R ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008090.png" /></td> </tr></table>
+
\begin{equation*} = \left\{ z \in \mathcal{D} : \operatorname { limsup } _ { w \rightarrow x } [ K _ { \mathcal{D} } ( z , w ) - K _ { \mathcal{D} } ( z _ { 0 } , w ) ] &lt; \frac { 1 } { 2 } \operatorname { log } R \right\}, \end{equation*}
  
and the big horosphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008091.png" /> of centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008092.png" />, pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008093.png" /> and radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008094.png" /> by the formula
+
and the big horosphere $F _ { z _ { 0 } } ( x , R )$ of centre $x$, pole $z_0$ and radius $R$ by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008095.png" /></td> </tr></table>
+
\begin{equation*} F _ { z _ { 0 } } ( x , R ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008096.png" /></td> </tr></table>
+
\begin{equation*} = \{ z \in {\cal D} : \operatorname { lim\,inf } _ { w \rightarrow x } [ K _ {\cal D } ( z , w ) - K _ {\cal D } ( z_0 , w ) ] &lt; \frac { 1 } { 2 } \operatorname { log } R \}, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008097.png" /> is a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d13008099.png" /> is its Kobayashi metric (cf. [[Hyperbolic metric|Hyperbolic metric]]). For the Euclidean ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080101.png" />.
+
where $\mathcal{D}$ is a bounded domain in $\mathbf{C} ^ { n }$ and $K _ { \mathcal{D} }$ is its Kobayashi metric (cf. [[Hyperbolic metric|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080102.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080103.png" /> the existence of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080104.png" /> such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
+
\begin{equation} \tag{a6} F ^ { n } ( E _ { z } ( a , R ) ) \subset F _ { z } ( a , R ) \end{equation}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080109.png" /> is a generalization of the Wolff–Schwarz lemma. This is true, for example, for a bounded convex domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080110.png" /> [[#References|[a1]]]. However, in this case B) does not hold in general. Nevertheless, the convergence result does hold for bounded strongly convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080111.png" /> domains, and for strongly pseudo-convex hyperbolic domains with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080112.png" /> boundary [[#References|[a1]]], [[#References|[a2]]].
+
for all $z \in \mathcal{D}$, $R &gt; 0$, $F \in \operatorname{Hol} ( {\cal D} )$ and $n = 1,2 , \dots$ is a generalization of the Wolff–Schwarz lemma. This is true, for example, for a bounded convex domain in $\mathbf{C} ^ { n }$ [[#References|[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 [[#References|[a1]]], [[#References|[a2]]].
  
Assertion A) can be generalized to the operator ball over a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080113.png" /> and, more generally, to the open unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080114.png" /> of a so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080116.png" />-algebra (see [[#References|[a37]]], [[#References|[a25]]] and the references there), while B) fails in general even if the compactness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080117.png" /> is assumed [[#References|[a8]]].
+
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 [[#References|[a37]]], [[#References|[a25]]] and the references there), while B) fails in general even if the compactness of $F$ is assumed [[#References|[a8]]].
  
For the particular case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080118.png" /> is defined by the Riesz–Dunford integral in the sense of the [[Functional calculus|functional calculus]] (cf. also [[Dunford integral|Dunford integral]]), a full analogue of the Denjoy–Wolff theorem is due to K. Fan [[#References|[a10]]], [[#References|[a11]]].
+
For the particular case when $F$ is defined by the Riesz–Dunford integral in the sense of the [[Functional calculus|functional calculus]] (cf. also [[Dunford integral|Dunford integral]]), a full analogue of the Denjoy–Wolff theorem is due to K. Fan [[#References|[a10]]], [[#References|[a11]]].
  
 
For general Banach spaces there are examples showing that there are situations where even a sink point does not exist [[#References|[a22]]].
 
For general Banach spaces there are examples showing that there are situations where even a sink point does not exist [[#References|[a22]]].
  
However, the question is still open (as of 2000) for reflexive Banach spaces. Moreover, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080119.png" /> is the open unit ball of a strictly convex Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080121.png" /> is compact or, more generally, condensing (cf. also [[Contraction operator|Contraction operator]]), then the analogue of B) is valid [[#References|[a16]]], [[#References|[a17]]], [[#References|[a18]]].
+
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|Contraction operator]]), then the analogue of B) is valid [[#References|[a16]]], [[#References|[a17]]], [[#References|[a18]]].
  
The situation is more fully understood when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080122.png" /> has a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080123.png" /> inside a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080124.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080125.png" /> to be an attractive fixed point, even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080126.png" /> is not an automorphism. Nevertheless, rephrasing C) in the form D), one can show (see, for example, [[#References|[a22]]]) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080127.png" /> is an attractive fixed point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080128.png" /> if and only if the spectral radius of the [[Fréchet derivative|Fréchet derivative]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080129.png" /> is strictly less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080130.png" />.
+
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, [[#References|[a22]]]) that $c$ is an attractive fixed point of $F$ if and only if the spectral radius of the [[Fréchet derivative|Fréchet derivative]] $F ^ { \prime } ( c )$ is strictly less than $1$.
  
In the one-dimensional case, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080131.png" /> is not the identity, has an interior fixed point and is power convergent, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080132.png" /> is unique. However, this is no longer true in higher dimensions.
+
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 [[#References|[a32]]]: Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080133.png" /> has a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080134.png" />, and denote the spectrum of the linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080135.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080136.png" /> (cf. also [[Spectrum of an operator|Spectrum of an operator]]). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080137.png" /> is power convergent if and only if the following two conditions hold:
+
A full description of such a situation was obtained by E. Vesentini [[#References|[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|Spectrum of an operator]]). Then $F$ is power convergent if and only if the following two conditions hold:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080138.png" />; and
+
i) $\sigma ( F ^ { \prime } ( c ) ) \subset \Delta \cup \{ 1 \}$; and
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080139.png" /> is a pole of the resolvent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080140.png" /> of order at most one.
+
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
 
Condition ii) is actually equivalent to the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080141.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { Ker } ( I - F ^ { \prime } ( c ) ) \bigoplus \operatorname { Im } ( I - F ^ { \prime } ( c ) ) = X \end{equation*}
  
(see, for example, [[#References|[a23]]]). It is also known that conditions i) and ii) are equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080142.png" /> being power-convergent to a projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080143.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080144.png" />. So, if the retraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080145.png" /> is the limit point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080146.png" /> under these conditions, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080147.png" /> is constant if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080148.png" />.
+
(see, for example, [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080149.png" /> of iterates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080150.png" /> can be considered a one-parameter discrete sub-semi-group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080151.png" /> (cf. also [[Semi-group of holomorphic mappings|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080152.png" />. This approach has been used to study the asymptotic behaviour of solutions to Cauchy problems (see, for example, [[#References|[a4]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a21]]], [[#References|[a26]]], [[#References|[a27]]], and [[#References|[a18]]]). E. Berkson and H. Porta [[#References|[a4]]] have applied their continuous analogue of the Denjoy–Wolff theorem to the study of the eigenvalue problem for composition operators on Hardy spaces.
+
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|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, [[#References|[a4]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a21]]], [[#References|[a26]]], [[#References|[a27]]], and [[#References|[a18]]]). E. Berkson and H. Porta [[#References|[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, [[#References|[a19]]], [[#References|[a28]]], [[#References|[a29]]], [[#References|[a31]]].
 
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, [[#References|[a19]]], [[#References|[a28]]], [[#References|[a29]]], [[#References|[a31]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Abate,  "Horospheres and iterates of holomorphic maps"  ''Math. Z.'' , '''198'''  (1988)  pp. 225–238</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Abate,  "Converging semigroups of holomorphic maps"  ''Atti Accad. Naz. Lincei'' , '''82'''  (1988)  pp. 223–227</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Abate,  "The infinitesimal generators of semigroups of holomorphic maps"  ''Ann. Mat. Pura Appl.'' , '''161'''  (1992)  pp. 167–180</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Berkson,  H. Porta,  "Semigroups of analytic functions and composition operators"  ''Michigan Math. J.'' , '''25'''  (1978)  pp. 101–115</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.B. Burckel,  "Iterating analytic self-maps of discs"  ''Amer. Math. Monthly'' , '''88'''  (1981)  pp. 396–407</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C.C. Cowen,  B.D. MacCluer,  "Composition operators on spaces of analytic functions" , CRC  (1995)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  G.N. Chen,  "Iteration for holomorphic maps of the open unit ball and the generalized upper half-plane of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080153.png" />"  ''J. Math. Anal. Appl.'' , '''98'''  (1984)  pp. 305–313</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  C.-H. Chu,  P. Mellon,  "Iteration of compact holomorphic maps on a Hilbert ball"  ''Proc. Amer. Math. Soc.'' , '''125'''  (1997)  pp. 1771–1777</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A. Denjoy,  "Sur l'itération des fonctions analytiques"  ''C.R. Acad. Sci. Paris'' , '''182'''  (1926)  pp. 255–257</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  K. Fan,  "Iteration of analytic functions of operators I"  ''Math. Z.'' , '''179'''  (1982)  pp. 293–298</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  K. Fan,  "Iteration of analytic functions of operators II"  ''Linear and Multilinear Algebra'' , '''12'''  (1983)  pp. 295–304</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  K. Goebel,  S. Reich,  "Uniform convexity, hyperbolic geometry and nonexpansive mappings" , M. Dekker  (1984)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  K. Goebel,  "Fixed points and invariant domains of holomorphic mappings of the Hilbert ball"  ''Nonlin. Anal.'' , '''6'''  (1982)  pp. 1327–1334</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  M. Hervé,  "Quelques propriétés des applications analytiques d'une boule à <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080154.png" /> dimensions dans elle-même"  ''J. Math. Pures Appl.'' , '''42'''  (1963)  pp. 117–147</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  M.H. Heins,  "On the iteration of functions which are analytic and single-valued in a given multiply-connected region"  ''Amer. J. Math.'' , '''63'''  (1941)  pp. 461–480</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  J. Kapeluszny,  T. Kuczumow,  S. Reich,  "The Denjoy–Wolff theorem in the open unit ball of a strictly convex Banach space"  ''Adv. Math.'' , '''143'''  (1999)  pp. 111–123</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  J. Kapeluszny,  T. Kuczumow,  S. Reich,  "The Denjoy–Wolff theorem for condensing holomorphic mappings"  ''J. Funct. Anal.'' , '''167'''  (1999)  pp. 79–93</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  T. Kuczumow,  S. Reich,  D. Shoikhet,  "The existence and non-existence of common fixed points for commuting families of holomorphic mappings"  ''Nonlin. Anal.''  (in press)</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  T. Kuczumow,  A. Stachura,  "Iterates of holomorphic and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080155.png" />-nonexpansive mappings in convex domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080156.png" />"  ''Adv. Math.'' , '''81'''  (1990)  pp. 90–98</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  Y. Kubota,  "Iteration of holomorphic maps of the unit ball into itself"  ''Proc. Amer. Math. Soc.'' , '''88'''  (1983)  pp. 476–480</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  V. Khatskevich,  S. Reich,  D. Shoikhet,  "Asymptotic behavior of solutions of evolution equations and the construction of holomorphic retractions"  ''Math. Nachr.'' , '''189'''  (1998)  pp. 171–178</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top">  V. Khatskevich,  D. Shoikhet,  "Differentiable operators and nonlinear equations" , Birkhäuser  (1994)</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top">  Yu. Lyubich,  J. Zemanek,  "Precompactness in the uniform ergodic theory"  ''Studia Math.'' , '''112'''  (1994)  pp. 89–97</TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top">  B.D. MacCluer,  "Iterates of holomorphic self-maps of the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080157.png" />"  ''Michigan Math. J.'' , '''30'''  (1983)  pp. 97–106</TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top">  P. Mellon,  "Another look at results of Wolff and Julia type for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080158.png" />-algebras"  ''J. Math. Anal. Appl.'' , '''198'''  (1996)  pp. 444–457</TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top">  S. Reich,  D. Shoikhet,  "Semigroups and generators on convex domains with the hyperbolic metric"  ''Atti Accad. Naz. Lincei'' , '''8'''  (1997)  pp. 231–250</TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top">  S. Reich,  D. Shoikhet,  "The Denjoy–Wolff theorem"  ''Ann. Univ. Mariae Curie–Skłodowska'' , '''51'''  (1997)  pp. 219–240</TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top">  S. Reich,  "Averaged mappings in the Hilbert ball"  ''J. Math. Anal. Appl.'' , '''109'''  (1985)  pp. 199–206</TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top">  S. Reich,  "The asymptotic behavior of a class of nonlinear semigroups in the Hilbert ball"  ''J. Math. Anal. Appl.'' , '''157'''  (1991)  pp. 237–242</TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top">  A. Stachura,  "Iterates of holomorphic self-maps of the unit ball in Hilbert space"  ''Proc. Amer. Math. Soc.'' , '''93'''  (1985)  pp. 88–90</TD></TR><TR><TD valign="top">[a31]</TD> <TD valign="top">  R. Sine,  "Behavior of iterates in the Poincaré metric"  ''Houston J. Math.'' , '''15'''  (1989)  pp. 273–289</TD></TR><TR><TD valign="top">[a32]</TD> <TD valign="top">  E. Vesentini,  "Su un teorema di Wolff e Denjoy"  ''Rend. Sem. Mat. Fis. Milano'' , '''53'''  (1983)  pp. 17–26</TD></TR><TR><TD valign="top">[a33]</TD> <TD valign="top">  J. Wolff,  "Sur l'itération des fonctions holomorphes dans une région, et dont les valeurs appartiennent à cette région"  ''C.R. Acad. Sci. Paris'' , '''182'''  (1926)  pp. 42–43</TD></TR><TR><TD valign="top">[a34]</TD> <TD valign="top">  J. Wolff,  "Sur l'itération des fonctions bornées"  ''C.R. Acad. Sci. Paris'' , '''182'''  (1926)  pp. 200–201</TD></TR><TR><TD valign="top">[a35]</TD> <TD valign="top">  J. Wolff,  "Sur une généralisation d'un théorème de Schwarz"  ''C.R. Acad. Sci. Paris'' , '''182'''  (1926)  pp. 918–920</TD></TR><TR><TD valign="top">[a36]</TD> <TD valign="top">  J. Wolff,  "Sur une généralisation d'un théorème de Schwarz"  ''C.R. Acad. Sci. Paris'' , '''183'''  (1926)  pp. 500–502</TD></TR><TR><TD valign="top">[a37]</TD> <TD valign="top">  K. Wlodarczyk,  "Julia's lemma and Wolff's theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130080/d130080159.png" />-algebras"  ''Proc. Amer. Math. Soc.'' , '''99'''  (1987)  pp. 472–476</TD></TR><TR><TD valign="top">[a38]</TD> <TD valign="top">  P. Yang,  "Holomorphic curves and boundary regularity of biholomorphic maps of pseudoconvex domains"  ''preprint''  (1978)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  M. Abate,  "Horospheres and iterates of holomorphic maps"  ''Math. Z.'' , '''198'''  (1988)  pp. 225–238</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  M. Abate,  "Converging semigroups of holomorphic maps"  ''Atti Accad. Naz. Lincei'' , '''82'''  (1988)  pp. 223–227</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M. Abate,  "The infinitesimal generators of semigroups of holomorphic maps"  ''Ann. Mat. Pura Appl.'' , '''161'''  (1992)  pp. 167–180</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  E. Berkson,  H. Porta,  "Semigroups of analytic functions and composition operators"  ''Michigan Math. J.'' , '''25'''  (1978)  pp. 101–115</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  R.B. Burckel,  "Iterating analytic self-maps of discs"  ''Amer. Math. Monthly'' , '''88'''  (1981)  pp. 396–407</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  C.C. Cowen,  B.D. MacCluer,  "Composition operators on spaces of analytic functions" , CRC  (1995)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  G.N. Chen,  "Iteration for holomorphic maps of the open unit ball and the generalized upper half-plane of $\mathbf{C} ^ { n }$"  ''J. Math. Anal. Appl.'' , '''98'''  (1984)  pp. 305–313</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  C.-H. Chu,  P. Mellon,  "Iteration of compact holomorphic maps on a Hilbert ball"  ''Proc. Amer. Math. Soc.'' , '''125'''  (1997)  pp. 1771–1777</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  A. Denjoy,  "Sur l'itération des fonctions analytiques"  ''C.R. Acad. Sci. Paris'' , '''182'''  (1926)  pp. 255–257</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  K. Fan,  "Iteration of analytic functions of operators I"  ''Math. Z.'' , '''179'''  (1982)  pp. 293–298</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  K. Fan,  "Iteration of analytic functions of operators II"  ''Linear and Multilinear Algebra'' , '''12'''  (1983)  pp. 295–304</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  K. Goebel,  S. Reich,  "Uniform convexity, hyperbolic geometry and nonexpansive mappings" , M. Dekker  (1984)</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  K. Goebel,  "Fixed points and invariant domains of holomorphic mappings of the Hilbert ball"  ''Nonlin. Anal.'' , '''6'''  (1982)  pp. 1327–1334</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  M. Hervé,  "Quelques propriétés des applications analytiques d'une boule à $m$ dimensions dans elle-même" ''J. Math. Pures Appl.'' , '''42'''  (1963)  pp. 117–147</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  M.H. Heins,  "On the iteration of functions which are analytic and single-valued in a given multiply-connected region"  ''Amer. J. Math.'' , '''63'''  (1941)  pp. 461–480</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  J. Kapeluszny,  T. Kuczumow,  S. Reich,  "The Denjoy–Wolff theorem in the open unit ball of a strictly convex Banach space"  ''Adv. Math.'' , '''143'''  (1999)  pp. 111–123</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  J. Kapeluszny,  T. Kuczumow,  S. Reich,  "The Denjoy–Wolff theorem for condensing holomorphic mappings"  ''J. Funct. Anal.'' , '''167'''  (1999)  pp. 79–93</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  T. Kuczumow,  S. Reich,  D. Shoikhet,  "The existence and non-existence of common fixed points for commuting families of holomorphic mappings"  ''Nonlin. Anal.''  (in press)</td></tr><tr><td valign="top">[a19]</td> <td valign="top">  T. Kuczumow,  A. Stachura,  "Iterates of holomorphic and $k _ { D }$-nonexpansive mappings in convex domains in $\mathbf{C} ^ { n }$" ''Adv. Math.'' , '''81'''  (1990)  pp. 90–98</td></tr><tr><td valign="top">[a20]</td> <td valign="top">  Y. Kubota,  "Iteration of holomorphic maps of the unit ball into itself"  ''Proc. Amer. Math. Soc.'' , '''88'''  (1983)  pp. 476–480</td></tr><tr><td valign="top">[a21]</td> <td valign="top">  V. Khatskevich,  S. Reich,  D. Shoikhet,  "Asymptotic behavior of solutions of evolution equations and the construction of holomorphic retractions"  ''Math. Nachr.'' , '''189'''  (1998)  pp. 171–178</td></tr><tr><td valign="top">[a22]</td> <td valign="top">  V. Khatskevich,  D. Shoikhet,  "Differentiable operators and nonlinear equations" , Birkhäuser  (1994)</td></tr><tr><td valign="top">[a23]</td> <td valign="top">  Yu. Lyubich,  J. Zemanek,  "Precompactness in the uniform ergodic theory"  ''Studia Math.'' , '''112'''  (1994)  pp. 89–97</td></tr><tr><td valign="top">[a24]</td> <td valign="top">  B.D. MacCluer,  "Iterates of holomorphic self-maps of the unit ball in $\mathbf{C} ^ { n }$" ''Michigan Math. J.'' , '''30'''  (1983)  pp. 97–106</td></tr><tr><td valign="top">[a25]</td> <td valign="top">  P. Mellon,  "Another look at results of Wolff and Julia type for $J ^ { * }$-algebras"  ''J. Math. Anal. Appl.'' , '''198'''  (1996)  pp. 444–457</td></tr><tr><td valign="top">[a26]</td> <td valign="top">  S. Reich,  D. Shoikhet,  "Semigroups and generators on convex domains with the hyperbolic metric"  ''Atti Accad. Naz. Lincei'' , '''8'''  (1997)  pp. 231–250</td></tr><tr><td valign="top">[a27]</td> <td valign="top">  S. Reich,  D. Shoikhet,  "The Denjoy–Wolff theorem"  ''Ann. Univ. Mariae Curie–Skłodowska'' , '''51'''  (1997)  pp. 219–240</td></tr><tr><td valign="top">[a28]</td> <td valign="top">  S. Reich,  "Averaged mappings in the Hilbert ball"  ''J. Math. Anal. Appl.'' , '''109'''  (1985)  pp. 199–206</td></tr><tr><td valign="top">[a29]</td> <td valign="top">  S. Reich,  "The asymptotic behavior of a class of nonlinear semigroups in the Hilbert ball"  ''J. Math. Anal. Appl.'' , '''157'''  (1991)  pp. 237–242</td></tr><tr><td valign="top">[a30]</td> <td valign="top">  A. Stachura,  "Iterates of holomorphic self-maps of the unit ball in Hilbert space"  ''Proc. Amer. Math. Soc.'' , '''93'''  (1985)  pp. 88–90</td></tr><tr><td valign="top">[a31]</td> <td valign="top">  R. Sine,  "Behavior of iterates in the Poincaré metric"  ''Houston J. Math.'' , '''15'''  (1989)  pp. 273–289</td></tr><tr><td valign="top">[a32]</td> <td valign="top">  E. Vesentini,  "Su un teorema di Wolff e Denjoy"  ''Rend. Sem. Mat. Fis. Milano'' , '''53'''  (1983)  pp. 17–26</td></tr><tr><td valign="top">[a33]</td> <td valign="top">  J. Wolff,  "Sur l'itération des fonctions holomorphes dans une région, et dont les valeurs appartiennent à cette région"  ''C.R. Acad. Sci. Paris'' , '''182'''  (1926)  pp. 42–43</td></tr><tr><td valign="top">[a34]</td> <td valign="top">  J. Wolff,  "Sur l'itération des fonctions bornées"  ''C.R. Acad. Sci. Paris'' , '''182'''  (1926)  pp. 200–201</td></tr><tr><td valign="top">[a35]</td> <td valign="top">  J. Wolff,  "Sur une généralisation d'un théorème de Schwarz"  ''C.R. Acad. Sci. Paris'' , '''182'''  (1926)  pp. 918–920</td></tr><tr><td valign="top">[a36]</td> <td valign="top">  J. Wolff,  "Sur une généralisation d'un théorème de Schwarz"  ''C.R. Acad. Sci. Paris'' , '''183'''  (1926)  pp. 500–502</td></tr><tr><td valign="top">[a37]</td> <td valign="top">  K. Wlodarczyk,  "Julia's lemma and Wolff's theorem for $J ^ { * }$-algebras"  ''Proc. Amer. Math. Soc.'' , '''99'''  (1987)  pp. 472–476</td></tr><tr><td valign="top">[a38]</td> <td valign="top">  P. Yang,  "Holomorphic curves and boundary regularity of biholomorphic maps of pseudoconvex domains"  ''preprint''  (1978)</td></tr></table>

Latest revision as of 15:30, 1 July 2020

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

[a1] M. Abate, "Horospheres and iterates of holomorphic maps" Math. Z. , 198 (1988) pp. 225–238
[a2] M. Abate, "Converging semigroups of holomorphic maps" Atti Accad. Naz. Lincei , 82 (1988) pp. 223–227
[a3] M. Abate, "The infinitesimal generators of semigroups of holomorphic maps" Ann. Mat. Pura Appl. , 161 (1992) pp. 167–180
[a4] E. Berkson, H. Porta, "Semigroups of analytic functions and composition operators" Michigan Math. J. , 25 (1978) pp. 101–115
[a5] R.B. Burckel, "Iterating analytic self-maps of discs" Amer. Math. Monthly , 88 (1981) pp. 396–407
[a6] C.C. Cowen, B.D. MacCluer, "Composition operators on spaces of analytic functions" , CRC (1995)
[a7] G.N. Chen, "Iteration for holomorphic maps of the open unit ball and the generalized upper half-plane of $\mathbf{C} ^ { n }$" J. Math. Anal. Appl. , 98 (1984) pp. 305–313
[a8] C.-H. Chu, P. Mellon, "Iteration of compact holomorphic maps on a Hilbert ball" Proc. Amer. Math. Soc. , 125 (1997) pp. 1771–1777
[a9] A. Denjoy, "Sur l'itération des fonctions analytiques" C.R. Acad. Sci. Paris , 182 (1926) pp. 255–257
[a10] K. Fan, "Iteration of analytic functions of operators I" Math. Z. , 179 (1982) pp. 293–298
[a11] K. Fan, "Iteration of analytic functions of operators II" Linear and Multilinear Algebra , 12 (1983) pp. 295–304
[a12] K. Goebel, S. Reich, "Uniform convexity, hyperbolic geometry and nonexpansive mappings" , M. Dekker (1984)
[a13] K. Goebel, "Fixed points and invariant domains of holomorphic mappings of the Hilbert ball" Nonlin. Anal. , 6 (1982) pp. 1327–1334
[a14] M. Hervé, "Quelques propriétés des applications analytiques d'une boule à $m$ dimensions dans elle-même" J. Math. Pures Appl. , 42 (1963) pp. 117–147
[a15] M.H. Heins, "On the iteration of functions which are analytic and single-valued in a given multiply-connected region" Amer. J. Math. , 63 (1941) pp. 461–480
[a16] J. Kapeluszny, T. Kuczumow, S. Reich, "The Denjoy–Wolff theorem in the open unit ball of a strictly convex Banach space" Adv. Math. , 143 (1999) pp. 111–123
[a17] J. Kapeluszny, T. Kuczumow, S. Reich, "The Denjoy–Wolff theorem for condensing holomorphic mappings" J. Funct. Anal. , 167 (1999) pp. 79–93
[a18] T. Kuczumow, S. Reich, D. Shoikhet, "The existence and non-existence of common fixed points for commuting families of holomorphic mappings" Nonlin. Anal. (in press)
[a19] T. Kuczumow, A. Stachura, "Iterates of holomorphic and $k _ { D }$-nonexpansive mappings in convex domains in $\mathbf{C} ^ { n }$" Adv. Math. , 81 (1990) pp. 90–98
[a20] Y. Kubota, "Iteration of holomorphic maps of the unit ball into itself" Proc. Amer. Math. Soc. , 88 (1983) pp. 476–480
[a21] V. Khatskevich, S. Reich, D. Shoikhet, "Asymptotic behavior of solutions of evolution equations and the construction of holomorphic retractions" Math. Nachr. , 189 (1998) pp. 171–178
[a22] V. Khatskevich, D. Shoikhet, "Differentiable operators and nonlinear equations" , Birkhäuser (1994)
[a23] Yu. Lyubich, J. Zemanek, "Precompactness in the uniform ergodic theory" Studia Math. , 112 (1994) pp. 89–97
[a24] B.D. MacCluer, "Iterates of holomorphic self-maps of the unit ball in $\mathbf{C} ^ { n }$" Michigan Math. J. , 30 (1983) pp. 97–106
[a25] P. Mellon, "Another look at results of Wolff and Julia type for $J ^ { * }$-algebras" J. Math. Anal. Appl. , 198 (1996) pp. 444–457
[a26] S. Reich, D. Shoikhet, "Semigroups and generators on convex domains with the hyperbolic metric" Atti Accad. Naz. Lincei , 8 (1997) pp. 231–250
[a27] S. Reich, D. Shoikhet, "The Denjoy–Wolff theorem" Ann. Univ. Mariae Curie–Skłodowska , 51 (1997) pp. 219–240
[a28] S. Reich, "Averaged mappings in the Hilbert ball" J. Math. Anal. Appl. , 109 (1985) pp. 199–206
[a29] S. Reich, "The asymptotic behavior of a class of nonlinear semigroups in the Hilbert ball" J. Math. Anal. Appl. , 157 (1991) pp. 237–242
[a30] A. Stachura, "Iterates of holomorphic self-maps of the unit ball in Hilbert space" Proc. Amer. Math. Soc. , 93 (1985) pp. 88–90
[a31] R. Sine, "Behavior of iterates in the Poincaré metric" Houston J. Math. , 15 (1989) pp. 273–289
[a32] E. Vesentini, "Su un teorema di Wolff e Denjoy" Rend. Sem. Mat. Fis. Milano , 53 (1983) pp. 17–26
[a33] J. Wolff, "Sur l'itération des fonctions holomorphes dans une région, et dont les valeurs appartiennent à cette région" C.R. Acad. Sci. Paris , 182 (1926) pp. 42–43
[a34] J. Wolff, "Sur l'itération des fonctions bornées" C.R. Acad. Sci. Paris , 182 (1926) pp. 200–201
[a35] J. Wolff, "Sur une généralisation d'un théorème de Schwarz" C.R. Acad. Sci. Paris , 182 (1926) pp. 918–920
[a36] J. Wolff, "Sur une généralisation d'un théorème de Schwarz" C.R. Acad. Sci. Paris , 183 (1926) pp. 500–502
[a37] K. Wlodarczyk, "Julia's lemma and Wolff's theorem for $J ^ { * }$-algebras" Proc. Amer. Math. Soc. , 99 (1987) pp. 472–476
[a38] P. Yang, "Holomorphic curves and boundary regularity of biholomorphic maps of pseudoconvex domains" preprint (1978)
How to Cite This Entry:
Denjoy-Wolff theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Denjoy-Wolff_theorem&oldid=18384
This article was adapted from an original article by Simeon ReichDavid Shoikhet (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article