  The new notion of adjoint algebraic entropy of endomorphisms of Abelian
groups is introduced. Various examples and basic properties are provided. It is
proved that the adjoint algebraic entropy of an endomorphism equals the
algebraic entropy of the adjoint endomorphism of the Pontryagin dual. As
applications, we compute the adjoint algebraic entropy of the shift
endomorphisms of direct sums, and we prove an Addition Theorem for the adjoint
algebraic entropy of bounded Abelian groups. A dichotomy is established,
stating that the adjoint algebraic entropy of any endomorphism can take only
values zero or infinity. As a consequence, we obtain the following surprising
discontinuity criterion for endomorphisms: every endomorphism of a compact
abelian group, having finite positive algebraic entropy, is discontinuous.
