Fixed points and periodic points of orientationreversing planar homeomorphisms
Author:
J. P. Boronski
Journal:
Proc. Amer. Math. Soc. 138 (2010), 37173722
MSC (2010):
Primary 55M20; Secondary 54F15, 54H25, 58C30
Published electronically:
April 13, 2010
MathSciNet review:
2661570
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Two results concerning orientationreversing homeomorphisms of the plane are proved. Let be an orientationreversing planar homeomorphism with a continuum invariant (i.e. ). First, suppose there are at least bounded components of that are invariant under . Then there are at least components of the fixed point set of in . This provides an affirmative answer to a question posed by K. Kuperberg. Second, suppose there is a periodic orbit in with . Then there is a 2periodic orbit in , or there is a 2periodic component of . The second result is based on a recent result of M. Bonino concerning linked periodic orbits of orientationreversing homeomorphisms of the 2sphere . These results generalize to orientationreversing homeomorphisms of .
 1.
Harold
Bell, A fixed point theorem for plane homeomorphisms, Fund.
Math. 100 (1978), no. 2, 119–128. MR 0500879
(58 #18386)
 2.
Marc
Bonino, A Brouwerlike theorem for orientation reversing
homeomorphisms of the sphere, Fund. Math. 182 (2004),
no. 1, 1–40. MR 2100713
(2005m:37096), http://dx.doi.org/10.4064/fm18211
 3.
Marc
Bonino, Nielsen theory and linked periodic orbits of homeomorphisms
of 𝕊², Math. Proc. Cambridge Philos. Soc.
140 (2006), no. 3, 425–430. MR 2225641
(2007k:37054), http://dx.doi.org/10.1017/S030500410500914X
 4.
L. E. BROUWER, Beweis des Ebenen Translationssatzes, Math. Ann. 72 (1912), 3654.
 5.
Morton
Brown, A short short proof of the
CartwrightLittlewood theorem, Proc. Amer.
Math. Soc. 65 (1977), no. 2, 372. MR 0461491
(57 #1476), http://dx.doi.org/10.1090/S00029939197704614910
 6.
M.
Brown and J.
M. Kister, Invariance of complementary domains of
a fixed point set, Proc. Amer. Math. Soc.
91 (1984), no. 3,
503–504. MR
744656 (86c:57014), http://dx.doi.org/10.1090/S00029939198407446563
 7.
M.
L. Cartwright and J.
E. Littlewood, Some fixed point theorems, Ann. of Math. (2)
54 (1951), 1–37. With appendix by H. D. Ursell. MR 0042690
(13,148f)
 8.
O.
H. Hamilton, A short proof of the CartwrightLittlewood fixed point
theorem, Canadian J. Math. 6 (1954), 522–524.
MR
0064394 (16,276a)
 9.
Krystyna
Kuperberg, Fixed points of orientation reversing
homeomorphisms of the plane, Proc. Amer. Math.
Soc. 112 (1991), no. 1, 223–229. MR 1064906
(91h:54049), http://dx.doi.org/10.1090/S0002993919911064906X
 10.
Krystyna
Kuperberg, A lower bound for the number of fixed points of
orientation reversing homeomorphisms, The geometry of Hamiltonian
systems (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 22,
Springer, New York, 1991, pp. 367–371. MR 1123283
(92j:54044), http://dx.doi.org/10.1007/9781461397250_12
 11.
O.
M. Šarkovs′kiĭ, Coexistence of cycles of a
continuous mapping of the line into itself, Ukrain. Mat. Z.
16 (1964), 61–71 (Russian, with English summary). MR 0159905
(28 #3121)
 1.
 H. BELL, A fixed point theorem for plane homeomorphisms. Fund. Math. 100 (1978), 119128. MR 0500879 (58:18386)
 2.
 M. BONINO, A Brouwerlike theorem for orientation reversing homeomorphisms of the sphere. Fund. Math. 182 (2004), no. 1, 140. MR 2100713 (2005m:37096)
 3.
 M. BONINO, Nielsen theory and linked periodic orbits of homeomorphisms of . Math. Proc. Cambridge Philos. Soc. 140 (2006), no. 3, 425430. MR 2225641 (2007k:37054)
 4.
 L. E. BROUWER, Beweis des Ebenen Translationssatzes, Math. Ann. 72 (1912), 3654.
 5.
 M. BROWN, A short short proof of the CartwrightLittlewood theorem. Proc. Amer. Math. Soc., 65, no. 2 (1977), 372. MR 0461491 (57:1476)
 6.
 M. BROWN, J. M. KISTER, Invariance of complementary domains of a fixed point set. Proc. Amer. Math. Soc. 91 (1984), no. 3, 503504. MR 744656 (86c:57014)
 7.
 M. L. CARTWRIGHT, J. E. LITTLEWOOD, Some fixed point theorems. With appendix by H. D. Ursell, Ann. of Math. (2) 54 (1951), 137. MR 0042690 (13:148f)
 8.
 O. H. HAMILTON, A short proof of the CartwrightLittlewood fixed point theorem. Canadian J. Math. 6 (1954), 522524. MR 0064394 (16:276a)
 9.
 K. KUPERBERG, Fixed points of orientation reversing homeomorphisms of the plane. Proc. Amer. Math. Soc. 112 (1991), no. 1, 223229. MR 1064906 (91h:54049)
 10.
 K. KUPERBERG, A lower bound for the number of fixed points of orientation reversing homeomorphisms. The geometry of Hamiltonian systems (Berkeley, CA, 1989), 367371, Math. Sci. Res. Inst. Publ., 22, Springer, New York, 1991. MR 1123283 (92j:54044)
 11.
 O. M. SARKOVSKII, Coexistence of cycles of a continuous mapping of the line into itself. Ukrain. Mat. Z. 16 (1964), 6171. MR 0159905 (28:3121)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
55M20,
54F15,
54H25,
58C30
Retrieve articles in all journals
with MSC (2010):
55M20,
54F15,
54H25,
58C30
Additional Information
J. P. Boronski
Affiliation:
Department of Mathematics and Statistics, Auburn University, Auburn, Alabama 36849
Email:
boronjp@auburn.edu
DOI:
http://dx.doi.org/10.1090/S0002993910103608
PII:
S 00029939(10)103608
Keywords:
Fixed point,
periodic point,
planar homeomorphism,
continuum
Received by editor(s):
August 8, 2009
Received by editor(s) in revised form:
December 31, 2009
Published electronically:
April 13, 2010
Additional Notes:
The author was supported in part by NSF Grant #DMS0634724
Dedicated:
Dedicated to the memory of Professor Andrzej Lasota (1932–2006)
Communicated by:
Bryna Kra
Article copyright:
© Copyright 2010 American Mathematical Society
