Generalized dilations and homogeneity

The goal of this article is to show that the class of homogeneous systems can be made very general if one considers generalized dilations (which are a class of group actions) and defines homogeneity with respect to them. It turns out that uniqueness of solutions (in both directions of time) is indeed a sufficient condition for a system to be homogeneous with respect to some generalized dilation. The relation between homogeneity and monotonicity is also studied and it is shown that if a system is monotone with respect to some V (a positive function) then there exists a generalized dilation with respect to which both the system and V are homogeneous. Another result presented in the paper is the equivalence of local monotonicity and global monotonicity under homogeneity.