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.



Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms
The answer is "yes". It follows as part of F. Bonahon's determination of the bordism group of surface diffeomorphisms. See
*Bonahon, Francis*, [**Cobordism of automorphisms of surfaces**](http://dx.doi.org/10.24033/asens.1448), Ann. Sci. Éc. Norm. Supér. (4) 16, 237-270 (1983). [ZBL0535.57016](https://zbmath.org/?q=an:0535.57016).
Bonahon justified the relevant step by appeal to some 3-dimensional topology, including geometrization. Independently about the same time J. Ewing and I proved the same result using 2-dimensional techniques and some number theory related to the G-signature theorem. 
*Edmonds, A. L.; Ewing, J. H.*, [**Remarks on the cobordism group of surface diffeomorphisms**](http://dx.doi.org/10.1007/BF01466055), Math. Ann. 259, 497-504 (1982). [ZBL0468.57023](https://zbmath.org/?q=an:0468.57023).
We wrote a conference proceedings note that has exactly the result you ask for as its title! See *Ewing, John; Edmonds, Allan*, Periodic surface diffeomorphisms which bound, bound periodically, Publ., Secc. Mat., Univ. Auton. Barc. 26, No. 3, 37-42 (1982). [ZBL0545.57009](https://zbmath.org/?q=an:0545.57009).


