  In this paper we address the problem of classification of simple weight
modules over weak generalized Weyl algebras of rank one. The principal
difference between weak generalized Weyl algebras and generalized weight
algebras is that weak generalized Weyl algebras are defined using an
endomorphism rather than an automorphism of a commutative ring $R$. We reduce
classification of simple weight modules over weak generalized Weyl algebras to
description of the dynamics of the action of the above mentioned endomorphism
on the set of maximal ideals. We also describe applications of our results to
the study of generalized Heisenberg algebras.
