# American Institute of Mathematical Sciences

February  2014, 34(2): 599-611. doi: 10.3934/dcds.2014.34.599

## Dynamics of random selfmaps of surfaces with boundary

 1 Dept. of Mathematics, Kyungsung University, Busan 608-736, South Korea 2 Dept. of Mathematics and Computer Science, Fairfield University, Fairfield CT, 06824, United States

Received  July 2011 Revised  April 2013 Published  August 2013

We use Wagner's algorithm to estimate the number of periodic points of certain selfmaps on compact surfaces with boundary. When counting according to homotopy classes, we can use the asymptotic density to measure the size of sets of selfmaps. In this sense, we show that almost all'' such selfmaps have periodic points of every period, and that in fact the number of periodic points of period $n$ grows exponentially in $n$. We further discuss this exponential growth rate and the topological and fundamental-group entropies of these maps.
Since our approach is via the Nielsen number, which is homotopy and homotopy-type invariant, our results hold for selfmaps of any space which has the homotopy type of a compact surface with boundary.
Citation: Seung Won Kim, P. Christopher Staecker. Dynamics of random selfmaps of surfaces with boundary. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 599-611. doi: 10.3934/dcds.2014.34.599
##### References:
 [1] G. Arzhantseva and A. Ol'shanskiĭ, Generality of the class of groups in which subgroups with a lesser number of generators are free, Mathematical Notes, 59 (1996), 350-355. doi: 10.1007/BF02308683. [2] E. Fadell and S. Husseini, The Nielsen number on surfaces, in "Topological Methods in Nonlinear Functional Analysis" (Toronto, Ont., 1982), Contemporary Mathematics, 21, Amer. Math. Soc., Providence, RI, (1983), 59-98. doi: 10.1090/conm/021/729505. [3] E. Hart, P. Heath and E. Keppelmann, Algorithms for Nielsen type periodic numbers of maps with remnant on surfaces with boundary and on bouquets of circles. I, Fundamenta Mathematicae, 200 (2008), 101-132. doi: 10.4064/fm200-2-1. [4] E. Hart, P. Heath and E. Keppelmann, An algorithm for Nielsen type periodic numbers of maps with remnant on surfaces with boundary and on bouquets of circles. II, in preparation. [5] B. Jiang, "Lectures on Nielsen Fixed Point Theory," Contemporary Mathematics, 14, American Mathematical Society, Providence, RI, 1983. [6] B. Jiang, Estimation of the number of periodic orbits, Pacific Journal of Mathematics, 172 (1996), 151-185. [7] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. [8] R. Lyndon and P. Schupp, "Combinatorial Group Theory," Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. [9] P. C. Staecker, Typical elements in free groups are in different doubly-twisted conjugacy classes, Topology and its Applications, 157 (2010), 1736-1741. doi: 10.1016/j.topol.2010.02.017. [10] J. Wagner, An algorithm for calculating the Nielsen number on surfaces with boundary, Transactions of the American Mathematical Society, 351 (1999), 41-62. doi: 10.1090/S0002-9947-99-01827-9.

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##### References:
 [1] G. Arzhantseva and A. Ol'shanskiĭ, Generality of the class of groups in which subgroups with a lesser number of generators are free, Mathematical Notes, 59 (1996), 350-355. doi: 10.1007/BF02308683. [2] E. Fadell and S. Husseini, The Nielsen number on surfaces, in "Topological Methods in Nonlinear Functional Analysis" (Toronto, Ont., 1982), Contemporary Mathematics, 21, Amer. Math. Soc., Providence, RI, (1983), 59-98. doi: 10.1090/conm/021/729505. [3] E. Hart, P. Heath and E. Keppelmann, Algorithms for Nielsen type periodic numbers of maps with remnant on surfaces with boundary and on bouquets of circles. I, Fundamenta Mathematicae, 200 (2008), 101-132. doi: 10.4064/fm200-2-1. [4] E. Hart, P. Heath and E. Keppelmann, An algorithm for Nielsen type periodic numbers of maps with remnant on surfaces with boundary and on bouquets of circles. II, in preparation. [5] B. Jiang, "Lectures on Nielsen Fixed Point Theory," Contemporary Mathematics, 14, American Mathematical Society, Providence, RI, 1983. [6] B. Jiang, Estimation of the number of periodic orbits, Pacific Journal of Mathematics, 172 (1996), 151-185. [7] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. [8] R. Lyndon and P. Schupp, "Combinatorial Group Theory," Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. [9] P. C. Staecker, Typical elements in free groups are in different doubly-twisted conjugacy classes, Topology and its Applications, 157 (2010), 1736-1741. doi: 10.1016/j.topol.2010.02.017. [10] J. Wagner, An algorithm for calculating the Nielsen number on surfaces with boundary, Transactions of the American Mathematical Society, 351 (1999), 41-62. doi: 10.1090/S0002-9947-99-01827-9.
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