ICDEA 2012: Difference Equations, Discrete Dynamical Systems and Applications pp 171-178 | Cite as
Periods of Homeomorphisms on Closed Surfaces
Abstract
The goal of this paper is to show what information on the set of periodic points of a homeomorphism on a closed surface can be obtained using the action of this homeomorphism on the homological groups of the closed surface.
Keywords
Periodic point Period Homeomorphism Closed surfaceNotes
Acknowledgments
The first author of this work was partially supported by MICINN/FEDER grant number MTM2011–22587, Junta de Comunidades de Castilla-La Mancha, grant number PEII09-0220-0222. The second author was partially supported by a MINECO grant MTM2013-40998-P, an AGAUR grant number 2014SGR-568, and the grants FP7-PEOPLE-2012-IRSES 318999 and 316338. The third author was partially supported by Fundación Séneca de la Región de Murcia grant number 12001/PI/09.
References
- 1.Brown, R.F.: The Lefschetz Fixed Point Theorem. Scott, Foresman and Company, Glenview, IL (1971)Google Scholar
- 2.Franks, J.: Homology and Dynamical Systems, CBMS Regional Conference Series, vol. 49, American Mathematical Society, Providence (1982)Google Scholar
- 3.Franks, J., Llibre, J.: Periods of surface homeomorphisms. Contemp. Math. 117, 63–77 (1991)MathSciNetCrossRefMATHGoogle Scholar
- 4.Fuller, F.B.: The existence of periodic points. Ann. Math. 57, 229–230 (1953)MathSciNetCrossRefMATHGoogle Scholar
- 5.Halpern, B.: Fixed point for iterates. Pacific J. Math. 25, 255–275 (1968)MathSciNetCrossRefMATHGoogle Scholar
- 6.Munkres, J.R.: Elements of Algebraic Topology. Addison–Wesley, Boston (1984)Google Scholar
- 7.Vicks, J.W.: Homology Theory. An Introduction to Algebraic Topology. Springer, New York (1994). (Academic Press, New York, 1973)Google Scholar