Construction of geodesics on Teichmüller spaces of Riemann surfaces with $\mathbb{Z}$ action

Teichmüller space $\mathrm{T}\mathrm{e}\mathrm{i}\mathrm{c}\mathrm{h}(R)$ of a Riemann surface $R$ is a deformation space of $R$. In this paper, we prove a sufficient condition for extremality of the Beltrami coefficients when $R$ has the $\mathbb{Z}$ action. As an application, we discuss the construction of geodesics. Earle-Kra-Krushkaĺ proved that the necessary and sufficient conditions for the geodesics connecting $[0]$ and $[\mu ]$ to be unique are $\mathrm{\parallel }{\mu }_0{\mathrm{\parallel }}_{\mathrm{\infty }}=\mathrm{\mid }{\mu }_0\mathrm{\mid }(z)$ (a.e.$z$) and unique extremality''. As a byproduct of our results, we show that we cannot exclude unique extremality’'.To show the above claim, we construct a point $[{\mu }_0]$ in $\mathrm{T}\mathrm{e}\mathrm{i}\mathrm{c}\mathrm{h}(\mathbb{C}\setminus \mathbb{Z})$, satisfying $\mathrm{\parallel }{\mu }_0{\mathrm{\parallel }}_{\mathrm{\infty }}=\mathrm{\mid }{\mu }_0\mathrm{\mid }(z)$ (a.e.$z$) and there exists a family of geodesics $\{{\gamma }_{\lambda }{\}}_{\lambda \in D}$ connecting $[0]$ and $[{\mu }_0]$ with complex analytic parameter, where $D$ is an open set in $l^{\mathrm{\infty }}$.