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The objective of the present paper is to develop a comprehensive theory of entropy in the realm of a  gs -space(  L  ,  p  ){(L,p)}    involving the notion of a generalized  s -map, and to study dynamics of a  p -preserving system(  L  ,  p  ,  ϕ  ){(L,p,\phi)}   . It is obtained that the entropy of  p -preserving systems is isomorphism invariant, and also entropy of an invertible  p -preserving system vanishes provided the corresponding Rokhlin space admits a one-sided generator.

Kolmogorov--Sinai entropy for p -preserving systems