Brouwer fixed-point theorem: Difference between revisions

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He studied a question analogous to that of the surface movement in a cup of coffee. What can we say, in general, about the trajectories on a surface animated by a constant [[flow (mathematics)|flow]]?<ref>This question was studied in: {{cite journal |first=H. |last=Poincaré |title=Sur les courbes définies par les équations différentielles |journal=[[Journal de Mathématiques Pures et Appliquées]] |volume=2 |issue=4 |pages=167–244 |year=1886 }}</ref> Poincaré discovered that the answer can be found in what we now call the [[topology|topological]] properties in the area containing the trajectory. If this area is [[compact space|compact]], i.e. both [[closed set|closed]] and [[bounded set|bounded]], then the trajectory either becomes stationary, or it approaches a [[limit cycle]].<ref>This follows from the [[Poincaré–Bendixson theorem]].</ref> Poincaré went further; if the area is of the same kind as a disk, as is the case for the cup of coffee, there must necessarily be a fixed point. This fixed point is invariant under all functions which associate to each point of the original surface its position after a short time interval&nbsp;''t''. If the area is a circular band, or if it is not closed,<ref>Multiplication by {{sfrac|1|2}} on ]0,&nbsp;1[² has no fixed point.</ref> then this is not necessarily the case.

To understand differential equations better, a new branch of mathematics was born. Poincaré called it ''analysis situs''. The French [[Encyclopædia Universalis]] defines it as the branch which "treats the properties of an object that are invariant if it is deformed in any continuous way, without tearing".<ref>"concerne les propriétés invariantes d'une figure lorsqu’on la déforme de manière continue quelconque, sans déchirure (par exemple, dans le cas de la déformation de la sphère, les propriétés corrélatives des objets tracés sur sa surface". From C. Houzel M. Paty ''[http://www.scientiaestudia.org.br/associac/paty/pdf/Paty,M_1997g-PoincareEU.pdf Poincaré, Henri (1854–1912)] {{webarchive|url=https://web.archive.org/web/20101008232932/http://www.scientiaestudia.org.br/associac/paty/pdf/Paty%2CM_1997g-PoincareEU.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.scientiaestudia.org.br/associac/paty/pdf/Paty%2CM_1997g-PoincareEU.pdf |archive-date=2022-10-09 |url-status=live |date=2010-10-08 }}'' Encyclopædia Universalis Albin Michel, Paris, 1999, p.&nbsp;696–706</ref> In 1886, Poincaré proved a result that is equivalent to Brouwer's fixed-point theorem,<ref>Poincaré's theorem is stated in: V. I. Istratescu ''Fixed Point Theory an Introduction'' Kluwer Academic Publishers (réédition de 2001) p 113 {{isbn|1-4020-0301-3}}</ref> although the connection with the subject of this article was not yet apparent.<ref>{{SpringerEOM|title=Brouwer theorem |first=M.I. |last=Voitsekhovskii |isbn=1-4020-0609-8}}</ref> A little later, he developed one of the fundamental tools for better understanding the analysis situs, now known as the [[fundamental group]] or sometimes the Poincaré group.<ref>{{cite book |first=Jean |last=Dieudonné |author-link=Jean Dieudonné |title=A History of Algebraic and Differential Topology, 1900–1960 |___location=Boston |publisher=Birkhäuser |year=1989 |isbn=978-0-8176-3388-2 |pages=[https://archive.org/details/historyofalgebra0000dieu_g9a3/page/17 17–24] |url=https://archive.org/details/historyofalgebra0000dieu_g9a3/page/17 }}</ref> This method can be used for a very compact proof of the theorem under discussion.<!-- fr.wikipedia has it in its article on the fundamental group, we don't -->

Poincaré's method was analogous to that of [[Charles Émile Picard|Émile Picard]], a contemporary mathematician who generalized the [[Cauchy–Lipschitz theorem]].<ref>See for example: [[Charles Émile Picard|Émile Picard]] ''[http://portail.mathdoc.fr/JMPA/PDF/JMPA_1893_4_9_A4_0.pdf Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires] {{Webarchive|url=https://web.archive.org/web/20110716055143/http://portail.mathdoc.fr/JMPA/PDF/JMPA_1893_4_9_A4_0.pdf# |date=2011-07-16 }}'' Journal de Mathématiques p 217 (1893)</ref> Picard's approach is based on a result that would later be formalised by [[Banach fixed-point theorem|another fixed-point theorem]], named after [[Stefan Banach|Banach]]. Instead of the topological properties of the ___domain, this theorem uses the fact that the function in question is a [[contraction mapping|contraction]].

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===A proof using degree===

Brouwer's original 1911 proof relied on the notion of the [[degree of a continuous mapping]], stemming from ideas in [[differential topology]]. Several modern accounts of the proof can be found in the literature, notably {{harvtxt|Milnor|1965}}.<ref name="Milnor">{{harvnb|Milnor|1965|pages=1–19}}</ref><ref>{{cite book | last = Teschl| first = Gerald| author-link = Gerald Teschl| title = Topics in Linear and Nonlinear Functional Analysis|url=https://www.mat.univie.ac.at/~gerald/ftp/book-fa/fa.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.mat.univie.ac.at/~gerald/ftp/book-fa/fa.pdf |archive-date=2022-10-09 |url-status=live|chapter=10. The Brouwer mapping degree|access-date=1 February 2022|year=2019|publisher=[[American Mathematical Society]]|series=Graduate Studies in Mathematics}}</ref>

Let <math>K=\overline{B(0)}</math> denote the closed unit ball in <math>\mathbb R^n</math> centered at the origin. Suppose for simplicity that <math>f:K\to K</math> is continuously differentiable. A [[regular value]] of <math>f</math> is a point <math>p\in B(0)</math> such that the [[Jacobian matrix and determinant|Jacobian]] of <math>f</math> is non-singular at every point of the preimage of <math>p</math>. In particular, by the [[inverse function theorem]], every point of the preimage of <math>f</math> lies in <math>B(0)</math> (the interior of <math>K</math>). The degree of <math>f</math> at a regular value <math>p\in B(0)</math> is defined as the sum of the signs of the [[Jacobian determinant]] of <math>f</math> over the preimages of <math>p</math> under <math>f</math>:

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|url=https://archive.org/details/topologyfromdiff0000miln}}

*{{cite journal|mr=0505523|last=Milnor|first= John W.|title=Analytic proofs of the "hairy ball theorem'' and the Brouwer fixed-point theorem|journal=[[Amer. Math. Monthly]]|volume=85 |year=1978|issue= 7|pages=521–524

|url=https://people.ucsc.edu/~lewis/Math208/hairyball.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://people.ucsc.edu/~lewis/Math208/hairyball.pdf |archive-date=2022-10-09 |url-status=live|jstor=2320860}}

*{{springer | title=Brouwer theorem | id=B/b017670 | last=Sobolev | first=Vladimir I. | author-link=<!--Vladimir Ivanovich Sobolev-->}}

*{{cite book|last=Spanier|first= Edwin H.|title=Algebraic topology|publisher= McGraw-Hill |___location= New York-Toronto-London|year= 1966}}