Topological entropy of a Lie group automorphism

Let {\phi} be an automorphism on a connected Lie group G. Through several G-subgroups associated to the dynamics of {\phi} we analyze their topological entropy. Assume that G belongs to the class of finite semisimple center Lie groups which admits a {\phi} invariant Levi subgroup. Then we prove that the topological entropy information of {\phi} is contained in the toral component of the unstable subgroup of {\phi} in the radical of G. We specialize the main result in a couple of interesting situations.