# Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms

Let $\Sigma$ be an oriented compact surface with non-empty boundary that is not a disk or a cylinder (i.e. negative Euler characteristic). Let $\phi, \psi: \Sigma \to \Sigma$ be two orientation preserving periodic homeomorphisms, that is, $\phi^n = \psi^m = id$ with $n,m$ their periods. Suppose that $\phi$ is isotopic to $\psi$ (so in particular $n=m$).

Does there exist an isotopy $H(x,t)$ between these two homeomorphisms such that $H(\cdot, t)$ is a periodic homeomorphism for all $t$?

Edit: Just in case this helps to solve the question. Notice that since $\phi$ and $\psi$ are isotopic and periodic, they have the same fixed-point data. This is enough to say that they are conjugate in $Homeo^+(\Sigma)$ by the classical theory of Nielsen. So $\phi = \gamma \circ \psi \circ \gamma^{-1}$. To prove the original question it would be enough to show that $\gamma$ is isotopic to the identity.