Kulkarni showed that, if $g$ is greater than $3$, a periodic map on an oriented surface $\Sigma_g$ of genus $g$ with order not smaller than $4g$ is uniquely determined by its order, up to conjugation and power. In this paper, we show that, if $g$ is greater than $30$, the same phenomenon happens for periodic maps on the surfaces with orders more than $8g/3$, and, for any integer $N$, there is $g > N$ such that there are periodic maps of $\Sigma_g$ of order $8g/3$ which are not conjugate up to power each other. Moreover, as a byproduct of our argument, we provide a short proof of Wiman's classical theorem: the maximal order of periodic maps of $\Sigma_g$ is $4g+2$.
"A uniqueness of periodic maps on surfaces." J. Math. Soc. Japan 68 (4) 1777 - 1787, October, 2016. https://doi.org/10.2969/jmsj/06841777