In this paper we study the minimal periods on a holomorphic map which are preserved by any of its deformation considering separately the case of continuous and holomorphic homotopy. A complete description of the set of such minimal periods for holomorphic self-map of a compact Riemann surface is given. It shows that a nature of answer depends on the geometry of the surface distinguishing the parabolic case of the Riemann sphere, elliptic case of tori and the hyperbolic case of a surface of genus $\geq 2$.
"Homotopy minimal periods of holomorphic maps on surfaces." Funct. Approx. Comment. Math. 40 (2) 309 - 326, June 2009. https://doi.org/10.7169/facm/1246454033