Celebratio Mathematica

[1]M. Brown: “A countable connected Hausdorff space,” Bull. Amer. Math. Soc. 59 (1953), pp. 367. Abstract only (inside The April meeting in New York); unpublished. article

[2]M. Brown: Continuous collections of higher dimensional hereditarily indecomposable continua. Ph.D. thesis, The University of Wisconsin, Madison, 1958. Advised by R. H. Bing. MR 2612711

[3]M. Brown: “Weak \( n \)-homogeneity implies weak \( (n-1) \)-homogeneity,” Proc. Amer. Math. Soc. 10 (1959), pp. 644–647. MR 0107857 Zbl 0093.36604

[4]M. Brown: “Some applications of an approximation theorem for inverse limits,” Proc. Amer. Math. Soc. 11 (1960), pp. 478–483. MR 0115157 Zbl 0113.37705

[5]M. Brown: “On the inverse limit of Euclidean \( N \)-spheres,” Trans. Amer. Math. Soc. 96 (1960), pp. 129–134. MR 0119182 Zbl 0136.19805

[6]M. Brown: “A proof of the generalized Schoenflies theorem,” Bull. Amer. Math. Soc. 66 (1960), pp. 74–76. MR 0117695 Zbl 0132.20002

[7]M. Brown: “The monotone union of open \( n \)-cells is an open \( n \)-cell,” Proc. Amer. Math. Soc. 12 (1961), pp. 812–814. MR 0126835 Zbl 0103.39305

[8]M. Brown: “Locally flat embeddings of topological manifolds,” pp. 83–91 in Topology of 3-manifolds and related topics (Univ. of Georgia Institute, 1961). Prentice-Hall (Englewood Cliffs, NJ), 1962. MR 0158373 Zbl 1246.57059

[9]M. Brown: “A mapping theorem for untriangulated manifolds,” pp. 92–94 in Topology of 3-manifolds and related topics (Univ. of Georgia Institute, 1961). Prentice-Hall (Englewood Cliffs, NJ), 1962. MR 0158374 Zbl 1246.57052

[10]M. Brown: “Locally flat imbeddings of topological manifolds,” Ann. of Math. (2) 75 (1962), pp. 331–341. MR 0133812 Zbl 0201.56202

[11]M. Brown: “On a theorem of Fisher concerning the homeomorphism group of a manifold,” Michigan Math. J. 9 (1962), pp. 403–405. MR 0150747 Zbl 0204.23901

[12]M. Brown and H. Gluck: “Stable structures on manifolds,” Bull. Amer. Math. Soc. 69 (1963), pp. 51–58. MR 0145497 Zbl 0118.39104

[13]M. Brown and H. Gluck: “Stable structures on manifolds, III: Applications,” Ann. of Math. (2) 79 (1964), pp. 45–58. MR 0158385

[14]M. Brown and H. Gluck: “Stable structures on manifolds, I: Homeomorphisms of \( S^{n} \),” Ann. of Math. (2) 79 (1964), pp. 1–17. MR 0158383

[15]M. Brown and H. Gluck: “Stable structures on manifolds, II: Stable manifolds,” Ann. of Math. (2) 79 (1964), pp. 18–44. MR 0158384

[16]M. Brown: “Wild cells and spheres in higher dimensions,” Michigan Math. J. 14 (1967), pp. 219–224. MR 0221481 Zbl 0147.23902

[17]M. Brown: “A note on Kister’s isotopy,” Michigan Math. J. 14 (1967), pp. 95–96. MR 0214077 Zbl 0185.51503

[18]M. Brown: “Pushing graphs around,” pp. 19–22 in Conference on the topology of manifolds (Michigan State Univ., E. Lansing, MI, 1967). Prindle, Weber & Schmidt (Boston), 1968. MR 0234468 Zbl 0185.50601

[19]M. Brown: “A note on Cartesian products,” Amer. J. Math. 91 (1969), pp. 32–36. MR 0239555 Zbl 0179.51405

[20]M. Brown: “Sets of constant distance from a planar set,” Michigan Math. J. 19 (1972), pp. 321–323. MR 0315714 Zbl 0244.54019

[21]M. Brown: “An application of homology theory to 4-coloring problems,” Nederl. Akad. Wetensch. Proc. Ser. A 75 (1972), pp. 353–354. Also published in Indag. Math. 34. MR 0317312

[22]M. Brown and R. Connelly: “On graphs with a constant link,” pp. 19–51 in New directions in the theory of graphs (Univ. Michigan, Ann Arbor, MI, 1971). Academic Press (New York), 1973. MR 0347685 Zbl 0258.05104

[23]M. Brown and R. Connelly: “On graphs with a constant link, II,” Discrete Math. 11 (1975), pp. 199–232. MR 0364016 Zbl 0304.05102

[24] M. Brown and W. D. Neumann: “Proof of the Poincaré–Birkhoff fixed point theorem,” Mich. Math. J. 24 : 1 (1977), pp. 21–31. MR 448339 Zbl 0402.55001 article

The Poincaré–Birkhoff fixed point theorem (also called Poincaré’s last geometric theorem) asserts the existence of at least two fixed points for a so-called area-preserving twist homeomorphism of the annulus. It was formulated as a conjecture and proved in special cases by Poincaré [1912], shortly before his death. In [1913] George Birkhoff published a proof which, though correct for one fixed point, overlooked the passibility that this fixed point might have index 0 in deducing the existence of a second fixed point. This error was corrected in his paper [1926], in which a generalization of the theorem in question is proven, with “area-preserving” replaced by a purely topological condition and “homeomorphism” replaced by a more general situation. However, some mathematicians have claimed that this proof too is incorrect, and the last few years have seen some extensive efforts to try to find a correct proof for the second fixed point.

We present here an elementary proof for two fixed points which is a simple modification of Birkhoff’s well known original proof for one fixed point. Our modification to get the second fixed point is essentially the same modification that Birkhoff sketches in the 1926 proof of his topological version to get from one fixed point to two.

This paper is therefore in a sense an expository paper, and to make the proof as transparent as possible we shall restrict to the simplest situation — a twist homeomorphism of the annulus which is just a rotation by a fixed angle on each boundary circle. As we point out in a final section, the proof goes through almost word for word without this restriction. It also extends to more general measures than the standard Lebesgue measure on the annulus.

Since our proof is so chase to Birkhoff’s proof, which has met with some skepticism, we have felt it advisable to give somewhat more detail than would otherwise be necessary. This is also in keeping with the view of this paper as an expository one.

[25]M. Brown: “A short short proof of the Cartwright–Littlewood theorem,” Proc. Amer. Math. Soc. 65 : 2 (1977), pp. 372. MR 0461491 Zbl 0369.57001

[26]M. Brown and A. G. Wasserman: “Arithmetic invariants of simplicial complexes,” Canad. J. Math. 32 : 6 (1980), pp. 1306–1310. MR 604685 Zbl 0413.57013

[27]B. Brechner and M. Brown: “Mapping cylinder neighborhoods in the plane,” Proc. Amer. Math. Soc. 84 : 3 (1982), pp. 433–436. MR 640248 Zbl 0478.54030

[28]M. Brown: “A new proof of Brouwer’s lemma on translation arcs,” Houston J. Math. 10 : 1 (1984), pp. 35–41. MR 736573 Zbl 0551.57005

[29]M. Brown and J. M. Kister: “Invariance of complementary domains of a fixed point set,” Proc. Amer. Math. Soc. 91 : 3 (1984), pp. 503–504. MR 744656 Zbl 0547.57010

[30]M. Brown: “Homeomorphisms of two-dimensional manifolds,” Houston J. Math. 11 : 4 (1985), pp. 455–469. MR 837985 Zbl 0605.57005

[31]M. Brown: “The mathematical work of R. H. Bing,” pp. 1–25 in Proceedings of the 1987 topology conference (Birmingham, AL, 1987), published as Topology Proc. 12 : 1. Issue edited by G. Gruenhage, D. Bennett, and L. Mohler. Auburn University (Birmingham, AL), 1987. MR 951703 Zbl 0661.01017 incollection

Donna Bennett	Related
Gary Gruenhage	Related
Lee K. Mohler	Related
R H Bing	Related	Volume

[32]M. Brown, E. E. Slaminka, and W. Transue: “An orientation preserving fixed point free homeomorphism of the plane which admits no closed invariant line,” Topology Appl. 29 : 3 (1988), pp. 213–217. MR 953953 Zbl 0668.54024

Edward E. Slaminka	Related
William Raoul Reagle Transue	Related

[33]M. Brown: “Fixed points for orientation preserving homeomorphisms of the plane which interchange two points,” Pacific J. Math. 143 : 1 (1990), pp. 37–41. MR 1047399 Zbl 0728.55001

[34]M. Brown: “On the fixed point index of iterates of planar homeomorphisms,” Proc. Amer. Math. Soc. 108 : 4 (1990), pp. 1109–1114. MR 994772 Zbl 0686.58028

[35]M. Barge and M. Brown: “Problems in dynamics on continua,” pp. 177–182 in Continuum theory and dynamical systems (Arcata, CA, 1989). Contemp. Math. 117. Amer. Math. Soc. (Providence, RI), 1991. MR 1112814 Zbl 0726.54026

[36]M. Brown: “Fundamental regions of planar homeomorphisms,” pp. 49–56 in Continuum theory and dynamical systems (Arcata, CA, 1989). Contemp. Math. 117. Amer. Math. Soc. (Providence, RI), 1991. MR 1112802 Zbl 0732.54029

[37]M. Brown: “A periodic homeomorphism of the plane,” pp. 83–87 in Continuum theory and dynamical systems. Lecture Notes in Pure and Appl. Math. 149. Dekker (New York), 1993. MR 1235347 Zbl 0791.58074

[38]K. Boucher, M. Brown, and E. E. Slaminka: “A Nielsen-type theorem for area-preserving homeomorphisms of the two disc,” pp. 43–50 in Continuum theory and dynamical systems. Lecture Notes in Pure and Appl. Math. 149. Dekker (New York), 1993. MR 1235344 Zbl 0807.58027

Kenneth M. Boucher	Related
Edward E. Slaminka	Related

Morton Brown

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