Homoclinic points describe the asymptotic behavior of group actions on spaces and play an important role in general theory of dynamical systems. In 1999, Doug Lind and Klaus Schmidt established relations between homoclinic points and entropy
properties for expansive algebraic actions of Z^d. Their proof depends heavily on the commutative factorial Noetherian ring structure of the integral group ring of Z^d.

In a joint work with Hanfeng Li, we extend their results to expansive algebraic actions of polycyclic-by-finite groups. We use three ingredients to do this: characterizations of expansive algebraic actions, local entropy theory for actions of countable amenable groups on compact groups, and comparison between entropies of dual algebraic actions.

Applying our results to the field of von Neumann algebras, we get a positive answer to a question of Deninger about the Fuglede-Kadison determinant to the case group is amenable. We also prove that for an amenable group, an element in the integral
group ring is a non-zero divisor if and only if the entropy of corresponding principal algebraic action is finite.