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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.

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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, Ann. Sci. Éc. Norm. Supér. (4) 16, 237-270 (1983). ZBL0535.57016. 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, Math. Ann. 259, 497-504 (1982). ZBL0468.57023. 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.

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  • $\begingroup$ Wow! This is great. I'm also somehow happy to hear that this is not trivial. $\endgroup$ – Paul Oct 29 '18 at 20:49

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