# 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 & Continuous Dynamical Systems - A, 2014, 34 (2) : 599-611. doi: 10.3934/dcds.2014.34.599
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