  In this paper, we determine the topological entropy $h(\phi)$ of a continuous
endomorphism $\phi$ of a Lie group $G$. This computation is a classical topic
in ergodic theory which seemed to have long been solved. But, when $G$ is
noncompact, the well known Bowen's formula for the entropy $h_{d}(\phi)$
associated to a left invariant distance $d$ just provides an upper bound to
$h(\phi)$, which is characterized by the so called variational principle. We
prove that \[ h\left(\phi\right) = h\left(\phi|_{T(G_\phi)}\right) \] where
$G_\phi$ is the maximal connected subgroup of $G$ such that $\phi(G_\phi) =
G_\phi$, and $T(G_\phi)$ is the maximal torus in the center of $G_\phi$. This
result shows that the computation of the topological entropy of a continuous
endomorphism of a Lie group reduces to the classical formula for the
topological entropy of a continuous endomorphism of a torus. Our approach
explores the relation between null topological entropy and the nonexistence of
Li-Yorke pairs and also relies strongly on the structure theory of Lie groups.
