As far as I see, Souriau is using unconventional notions in his book "Structure of Dynamical Systems". He explains these notions in §2. of Chapter I, but it is a puzzle for me. Mainly because he regularly talks about mappings without mentioning the domain or the co-domain of these mappings. Could somebody tell me in the language of the conventional differential geometry what the exact meaning (and perhaps the benefit) of Souriau's $\delta$ symbol is?
Edit 1:
The relevant pages of the book are shown here.
Edit 2:
Here it is the complete book (French original).