Geometry of autonomous discrete Painlev\'e equations related to the Weyl group $W(E_8^{(1)})$

Discrete Painlev\'e equations are integrable two-dimensional birational maps associated to a family of generalized Halphen surfaces. The latter can be seen either as $\mathbb P^2$ blown up at nine points or as $\mathbb P^1\times\mathbb P^1$ blown up at eight points. These maps become autonomous if the blow-up points are in a special position (support a pencil of cubic curves in $\mathbb P^2$, respectively a pencil of biquadratic curves in $\mathbb P^1\times\mathbb P^1$), so that the generalized Halphen surfaces become rational elliptic surfaces. In the generic case, the symmetry of a discrete Painlev\'e equation is the Weyl group $W(E_8^{(1)})$. One has a system of commuting maps which correspond to translational elements of $W(E_8^{(1)})$ associated to the roots of the lattice $E_8^{(1)}$. In the present note, we give a geometric construction of these commuting maps. For this, we use some novel birational involutions based on the above mentioned pencils of curves.