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| In [[mathematics]], the '''Denjoy–Wolff theorem''' is a theorem in [[complex analysis]] and [[dynamical systems]] concerning fixed points and iterations of [[holomorphic mapping]]s of the [[unit disc]] in the [[complex numbers]] into itself. The result was proved independently in 1926 by the French mathematician [[Arnaud Denjoy]] and the Dutch mathematician [[Julius Wolff (mathematician)|Julius Wolff]].
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| ==Statement==
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| '''Theorem.''' Let ''D'' be the open unit disk in '''C''' and let ''f'' be a holomorphic function mapping ''D'' into ''D'' which is not an automorphism of ''D'' (i.e. a [[Möbius transformation]]). Then there is a unique point ''z'' in the closure of ''D'' such that the iterates of ''f'' tend to ''z'' uniformly on compact subsets of ''D''. If ''z'' lies in ''D'', it is the unique fixed point of ''f''. The mapping ''f'' leaves invariant [[Poincaré_metric#Metric_and_volume_element_on_the_Poincar.C3.A9_disk|hyperbolic disks]] centered on ''z'', if ''z'' lies in ''D'', and disks tangent to the unit circle at ''z'', if ''z'' lies on the boundary of ''D''.
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| When the fixed point is at ''z'' = 0, the hyperbolic disks centred at ''z'' are just the Euclidean disks with centre 0. Otherwise ''f'' can be conjugated by a Möbius transformation so that the fixed point is zero. An elementary proof of the theorem is given below, taken from {{harvtxt|Shapiro|1993}} and {{harvtxt|Burckel|1981}}. Two other short proofs can be found in {{harvtxt|Carleson|Gamelin|1993}}. | |
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| ==Proof of theorem==
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| ===Fixed point in the disk===
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| If ''f'' has a fixed point ''z'' in ''D'' then, after conjugating by a Möbius transformation, it can be assumed that ''z'' = 0. Let ''M''(''r'') be the maximum modulus of ''f'' on ''|z|'' = ''r'' < 1. By the [[Schwarz lemma]]<ref>{{harvnb|Shapiro|1992|p=79}}</ref>
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| :<math> |f(z)|\le \delta(r) |z|,</math>
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| for |''z''| ≤ ''r'', where | |
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| :<math>\delta(r)={M(r)\over r} < 1.</math>
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| It follows by iteration that
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| :<math>|f^n(z)|\le \delta(r)^n</math>
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| for |''z''| ≤ ''r''. These two inequalities imply the result in this case. | |
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| ===No fixed points===
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| When ''f'' acts in ''D'' without fixed points, Wolff showed that there is a point ''z'' on the boundary such that the iterates of ''f'' leave invariant each disk tangent to the boundary at that point. | |
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| Take a sequence <math>r_k</math> increasing to 1 and set<ref>{{harvnb|Burckel|1981}}</ref><ref>{{harvnb|Steinmetz|1993|pp=43–44}}</ref>
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| :<math>f_k(z)=r_kf(z).</math>
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| By applying [[Rouché's theorem]] to <math>f_k(z) - z</math> and <math>g(z)=z</math>, <math>f_k </math> has exactly one zero <math>z_k</math> in ''D''.
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| Passing to a subsequence if necessary, it can be assumed that <math>z_k\rightarrow w.</math> The point ''z'' cannot lie in ''D'', because,
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| by passing to the limit, ''z'' would have to be a fixed point. The result for the case of fixed points implies that the maps <math>f_k </math> leave invariant Euclidean disks. Explicit computation shows that, as ''k'' increases, disks can be chosen appropriately so that they tend to any given disk tangent to the boundary at ''z''. By continuity, ''f'' leaves each such disk Δ invariant. | |
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| To see that <math>f^n</math> converges uniformly on compacta to the constant ''z'', it is enough to show that the same is true for any subsequence <math>f^{n_k}</math>, convergent in the same sense to ''g'', say. Such limits exist by Montel's theorem, and if
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| ''g'' is non-constant, it can also be assumed that <math>f^{n_{k+1}-n_k}</math> has a limit, ''h'' say. But then | |
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| :<math>h(g(w))=g(w), \,</math>
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| for ''w'' in ''D''.
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| Since ''h'' is holomorphic and ''g''(''D'') open,
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| :<math>h(w) = w</math>
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| for all ''w''.
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| Setting <math> m_k= n_{k+1} - n_k</math>, it can also be assumed that <math>f^{m_k-1}</math> is convergent to ''F'' say.
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| But then ''f''(''F''(''w'')) = ''w'' = ''f''(''F''(''w'')), contradicting the fact that ''f'' is not an automorphism.
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| Hence every subsequence tends to some constant uniformly on compacta in ''D''.
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| The invariance of Δ implies each such constant lies in the closure of each disk Δ, and hence their intersection, the single point ''z''. By Montel's theorem, it follows that <math>f^n</math> converges uniformly on compacta to the constant ''z''.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation|last=Beardon|first= A. F.|title=Iteration of contractions and analytic maps|journal=
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| J. London Math. Soc.|volume=41|year= 1990|pages= 141–150}}
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| *{{citation|last=Burckel|first=R. B.|title=Iterating analytic self-maps of discs|journal=Amer. Math. Monthly|volume= 88 |year=1981|pages= 396–407}}
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| *{{citation|last=Carleson|first=L.|last2= Gamelin|first2= T. D. W.|title=Complex dynamics|series=
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| Universitext: Tracts in Mathematics|publisher= Springer-Verlag|year=1993|isbn=0-387-97942-5}}
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| *{{citation|first=A.|last= Denjoy|title= Sur l’itération des fonctions analytiques|journal= C. R. Acad. Scie.|volume= 182|year=1926|pages= 255–257}}
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| *{{citation|last=Shapiro|first=J. H.|title=Composition operators and classical function theory|series=Universitext: Tracts in Mathematics|publisher= Springer-Verlag|year= 1993|isbn=0-387-94067-7}}
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| *{{citation|last=Shoikhet|first=D.|title=Semigroups in geometrical function theory|publisher= Kluwer Academic Publishers|year= 2001|isbn=0-7923-7111-9 }}
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| *{{citation|last=Steinmetz|first= Norbert|title=Rational iteration. Complex analytic dynamical systems|series =de Gruyter Studies in Mathematics|volume=16|publisher=Walter de Gruyter & Co.|year=1993|isbn= 3-11-013765-8}}
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| *{{citation|first=J.|last= Wolff|title= Sur l’itération des fonctions holomorphes dans une région, et dont les valeurs appartiennent a cette région|journal= C. R. Acad. Sci.|volume= 182 |year=1926|pages= 42–43}}
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| *{{citation|first=J. |last=Wolff|title= Sur l’itération des fonctions bornées|journal= C. R. Acad. Sci. |volume=182 |year=1926|pages=200–201}}
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| *{{citation|first=J.|last= Wolff|title=Sur une généralisation d’un théorème de Schwarz|journal= C. R. Acad. Sci.|volume=
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| 182 |year=1926|pages=918–920}}
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| {{DEFAULTSORT:Denjoy-Wolff theorem}}
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| [[Category:Theorems in dynamical systems]]
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| [[Category:Theorems in complex analysis]]
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Hikers have it good. They are working shape enough to get up and down a hill, or possibly mountain with relative ease and discomfort. I was envious, until I kicked myself in the ass, and made myself do it. Now I love it, going at least twice a 1 week. But when I got my new hiking boots, I thought I was for you to scream. Nobody told me how to break these puppies within just. So I put my head to the grindstone, and this is exactly what I found. Hope you use them prior to going out in new boots. They helps save a blister or five.
Fifth, choose whether we want to make the leather parts out of real leather or house. If the budget permits, the real leather ugg news is frequently more ideal as some kind of leather have a nice and also texture.
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