Most research in this area is about proving *nonexistence* of such rational maps for a sufficiently general hypersurface, other than the obvious rational maps: constant maps (which some disallow since they are not dominant) and the identity map.  One of the people who studies this is Amerik.  This is also related to birational superrigidity of general hypersurfaces, the evolution of the Iskovskikh-Manin method.

Of course *every* smooth cubic hypersurface has many such nontrivial birational automorphisms.  For every point $p$ in $X$, for a general point $q$ in $X\setminus\{p\}$, the line $L$ spanned by $p$ and $q$ intersects $X$ in a third point $r$.  There is a rational involution that sends $q$ to $r$.