An overview is provided of several recent results, constructions and publications relating dynamical systems, wavelets, transfer operators satisfying covariance commutation relations associated to non-invertible dynamics, defining generalizations of crossed product operator algebras to non-invertible dynamics or actions by semigroups, ideals in the corresponding crossed product type algebras and commutants of elements and subsets in the algebras and in their representations. Some open directions and open problems on this rich interplay motivated by these constructions and results are also indicated.