Functiones et Approximatio Commentarii Mathematici

Homotopy minimal periods of holomorphic maps on surfaces

Jaume Llibre and Wacław Marzantowicz

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In this paper we study the minimal periods on a holomorphic map which are preserved by any of its deformation considering separately the case of continuous and holomorphic homotopy. A complete description of the set of such minimal periods for holomorphic self-map of a compact Riemann surface is given. It shows that a nature of answer depends on the geometry of the surface distinguishing the parabolic case of the Riemann sphere, elliptic case of tori and the hyperbolic case of a surface of genus $\geq 2$.

Article information

Funct. Approx. Comment. Math., Volume 40, Number 2 (2009), 309-326.

First available in Project Euclid: 1 July 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M20: Fixed points and coincidences [See also 54H25]
Secondary: 57N05: Topology of $E^2$ , 2-manifolds 57N10: Topology of general 3-manifolds [See also 57Mxx]

Set of periods periodic points holomorphic maps homotopy Riemann surfaces


Llibre, Jaume; Marzantowicz, Wacław. Homotopy minimal periods of holomorphic maps on surfaces. Funct. Approx. Comment. Math. 40 (2009), no. 2, 309--326. doi:10.7169/facm/1246454033.

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  • Ll. Alsedà, S. Baldwin, J. Llibre, R. Swanson, W. Szlenk, Torus maps and Nielsen numbers, Contemporary Math. 152 (1993), 1--7.
  • A. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics 132, Springer--Verlag, New York, 1991.
  • P. Boyland, Isotopy stability of dynamics on surfaces, Geometry and topology in dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999), 17--45, Contemp. Math., 246, Amer. Math. Soc., Providence, RI, 1999.
  • R. F. Brown, The Lefschetz fixed point theorem, Scott, Foresman and Company, Glenview, IL, 1971.
  • C. Corrales, J. M. Gamboa, G. Gromadzki, Automorphisms of Klein surfaces with fixed points, Glagow Math. J. 41 (1999), 183--189.
  • A. Dold, Fixed point indices of iterated maps, Invent. math. 74 (1983), 419--435.
  • D. B. Epstein, Curves on $2$-manifolds and isotopies, Acta Math. 115, (1966), 83--107.
  • N. Fagella, J. Llibre, Periodic points of holomorphic maps via Lefschetz numbers, Trans. Amer. Math. Soc. 352 (2000), 4711--4730.
  • H. M. Farkas, K. Kra, Riemann Surfaces, Graduate Texts in Mathematics 71, Springer--Verlag, New York, 1980.
  • E. Hart, E. Keppelmann, Nielsen periodic point theory for periodic maps on orientable surfaces, Topology Appl. 153, (2006), 1399--1420.
  • B. von Kerejarto, Vorlesungen uber Topologie. I Flachentopologie, Springer Verlag, 1923.
  • J. Jezierski, W. Marzantowicz, Homotopy minimal periods for maps of three dimensional nilmanifolds, Pacific J. Math. 209 (2003), 85--101.
  • J. Jezierski, W. Marzantowicz, Homotopy methods in topological fixed and periodic points theory, Topological Fixed Point Theory and Its Applications 3, Springer, Dordrecht, 2006.
  • B. Jiang, J.H. Guo, Fixed points of surface diffeomorphisms, Pacific J. Math. 160 (1993), 67--89.
  • B. Jiang, J. Llibre Minimal sets of periods for torus maps, Discrete and Continuous Dynamical Systems 4 (1998), 301--320.
  • J. Llibre, Lefschetz numbers for periodic points, Contemporary Math. 152 (1993), 215--227.
  • J. W. Milnor, Topology from the differentiable viewpoint, based on notes by David W. Weaver. Revised reprint of the 1965 original. Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997.
  • I. Niven, H.S. Zuckerman, An introduction to the theory of numbers, fourth edition, John Wiley & Sons, New York, 1980.
  • M. Sierakowski, Sets of periods for automorphisms of compact Riemann surfaces, J. Pure Appl. Algebra 208 (2007), 561--574.
  • G. Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), 39--72.
  • W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988), 417--431.