Brouwer fixed-point theorem: Difference between revisions

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Consider the function

:<math>f(x) = \frac{x+1}{2},</math>

which is a continuous function from the open interval <math>(−1-1,1)</math> to itself. Since xthe =point <math>x=1</math> is not part of the interval, there is notno apoint fixedin pointthe of___domain such that <math>f(x) = x</math>. The spaceset <math>(−1-1,1)</math> is convex and bounded, but not closed. On the other hand, the function ''<math>f''</math> {{em|does}} have a fixed point forin the ''closed'' interval <math>[−1-1,1]</math>, namely ''f''<math>x=1</math>. The closed interval <math>[-1,1]</math> is compact, the open interval <math>(-1,1)</math> =is 1not.

===Convexity===

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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# |archive-url=https://web.archive.org/web/20110716055143/http://portail.mathdoc.fr/JMPA/PDF/JMPA_1893_4_9_A4_0.pdf |archive-date=2011-07-16 |url-status=live |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]].

===First proofs===

[[Image:Hadamard2.jpg|thumb|right|[[Jacques Hadamard]] helped Brouwer to formalize his ideas.]]

At the dawn of the 20th century, the interest in analysis situs did not stay unnoticed. However, the necessity of a theorem equivalent to the one discussed in this article was not yet evident. [[Piers Bohl]], a [[Latvia]]n mathematician, applied topological methods to the study of differential equations.<ref>J. J. O'Connor E. F. Robertson ''[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bohl.html Piers Bohl]''</ref> In 1904 he proved the three-dimensional case of our theorem,<ref name="Bohl1904" /> but his publication was not noticed.<ref>{{cite journal |first1=A. D. |last1=Myskis |first2=I. M. |last2=Rabinovic |title=Первое доказательство теоремы о неподвижной точке при непрерывном отображении шара в себя, данное латышским математиком П.Г.Болем |trans-title=The first proof of a fixed-point theorem for a continuous mapping of a sphere into itself, given by the Latvian mathematician P. G. Bohl |language=ru |journal=Успехи математических наук |volume=10 |issue=3 |year=1955 |pages=188–192 |url=http://mi.mathnet.ru/eng/umn/v10/i3/p179 }}</ref>

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:<math>\operatorname{deg}_p(f) = \sum_{x\in f^{-1}(p)} \operatorname{sign}\,\det (df_x).</math>

The degree is, roughly speaking, the number of "sheets" of the preimage ''f'' lying over a small [[open set]] around ''p'', with sheets counted oppositely if they are oppositely oriented. This is thus a generalization of [[winding number]] to higher dimensions.

The degree satisfies the property of ''homotopy invariance'': let <math>f</math> and <math>g</math> be two continuously differentiable functions, and <math>H_t(x)=tf+(1-t)g</math> for <math>0\le t\le 1</math>. Suppose that the point <math>p</math> is a regular value of <math>H_t</math> for all ''t''. Then <math>\deg_p f = \deg_p g</math>.

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<math>H(t,x) = \frac{x-tf(x)}{\sup_{y\in K}\left|y-tf(y)\right|}</math>

defines a homotopy from the [[identity function]] to it. The identity function has degree one at every point. In particular, the identity function has degree one at the origin, so <math>g</math> also has degree one at the origin. As a consequence, the preimage <math>g^{-1}(0)</math> is not empty. The elements of <math>g^{-1}(0)</math> are precisely the fixed points of the original function ''f''.

This requires some work to make fully general. The definition of degree must be extended to singular values of ''f'', and then to continuous functions. The more modern advent of [[homology theory]] simplifies the construction of the degree, and so has become a standard proof in the literature.

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:<math>{\mathbf X}({\mathbf x},t)=(-t\,{\mathbf w}({\mathbf x}), {\mathbf x}\cdot {\mathbf w}({\mathbf x})).</math>

By construction {{mvar|'''X'''}} is a continuous vector field on the unit sphere of {{mvar|''W''}}, satisfying the tangency condition {{mvar|'''y'''}} ⋅ {{mvar|'''X'''}}({{mvar|'''y'''}})&nbsp;=&nbsp;0. Moreover, {{mvar|'''X'''}}({{mvar|'''y'''}}) is nowhere vanishing (because, if {{var|'''x'''}} has norm 1, then {{mvar|'''x'''}} ⋅ {{mvar|'''w'''}}({{mvar|''x''}}) is non-zero; while if {{mvar|'''x'''}} has norm strictly less than 1, then {{mvar|''t''}} and {{mvar|'''w'''}}({{mvar|'''x'''}}) are both non-zero). This contradiction proves the fixed point theorem when {{mvar|''n''}} is even. For {{mvar|''n''}} odd, one can apply the fixed point theorem to the closed unit ball {{mvar|''B''}} in {{mvar|''n''}} + 1}} dimensions and the mapping {{mvar|'''F'''}}({{mvar|'''x'''}},{{mvar|''y''}}) = ({{mvar|'''f'''}}({{mvar|'''x'''}}),0).

The advantage of this proof is that it uses only elementary techniques; more general results like the [[Borsuk-Ulam theorem]] require tools from [[algebraic topology]].<ref name="Milnor78">{{harvnb|Milnor|1978}}</ref>

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===A proof by Hirsch===

There is also a quick proof, by [[Morris Hirsch]], based on the impossibility of a differentiable retraction. TheLet [[indirect''f'' proof]]denote startsa bycontinuous notingmap from the unit ball Dⁿ in n-dimensional Euclidean space to itself and assume that ''f'' fixes no point. By continuity and the fact that Dⁿ is compact, it follows that for some ε > 0, ∥x - ''f''(x)∥ > ε for all x in Dⁿ. Then the map ''f'' can be approximated by a smooth map retaining the property of not fixing a point; this can be done by using the [[Weierstrass approximation theorem]] or by [[convolution|convolving]] with smooth [[bump function]]s. One then defines a retraction as above whichby sending each x to the point of ∂Dⁿ where the unique ray from x through ''f''(x) intersects ∂Dⁿ, and this must now be a differentiable mapping. Such a retraction must have a non-singular value p ∈ ∂Dⁿ, by [[Sard's theorem]], which is also non-singular for the restriction to the boundary (which is just the identity). Thus the inverse image ''f''^-1(p) would be a compact 1-manifold with boundary. TheSuch a boundary would have to contain at least two end pointsendpoints, bothand of whichthese would have to lie on the boundary of the original ball—whichball. isThis impossiblewould inmean that the inverse image of one point on ∂Dⁿ contains a different point on ∂Dⁿ, contradicting the definition of a retraction Dⁿ → ∂Dⁿ.<ref>{{harvnb|Hirsch|1988}}</ref>

R. Bruce Kellogg, Tien-Yien Li, and [[James A. Yorke]] turned Hirsch's proof into a [[Computability|computable]] proof by observing that the retract is in fact defined everywhere except at the fixed points.{{sfn|Kellogg|Li|Yorke|1976}} For almost any point, ''q'', on the boundary, (— assuming it is not a fixed point) the— onethe 1-manifold with boundary mentioned above does exist and the only possibility is that it leads from ''q'' to a fixed point. It is an easy numerical task to follow such a path from ''q'' to the fixed point so the method is essentially computable.{{sfn|Chow|Mallet-Paret|Yorke|1978}} gave a conceptually similar path-following version of the homotopy proof which extends to a wide variety of related problems.

===A proof using oriented area===

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==External links==

* [http://www.cut-the-knot.org/do_you_know/poincare.shtml#brouwertheorem Brouwer's Fixed Point Theorem for Triangles] at [[cut-the-knot]]

* {{PlanetMath|BrouwerFixedPointTheorem}}

* [http://planetmath.org/encyclopedia/BrouwerFixedPointTheorem.html Brouwer theorem] {{Webarchive|url=https://web.archive.org/web/20070319191655/http://planetmath.org/encyclopedia/BrouwerFixedPointTheorem.html |date=2007-03-19 }}, from [[PlanetMath]] with attached proof.

* [http://www.mathpages.com/home/kmath262/kmath262.htm Reconstructing Brouwer] at MathPages

* [https://web.archive.org/web/20110314100800/http://mathforum.org/mathimages/index.php/Brouwer_Fixed_Point_Theorem Brouwer Fixed Point Theorem] at Math Images.

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