In mathematics, the Hopf decomposition, named after Eberhard Hopf, gives a canonical decomposition of a measure space (X, μ) with respect to an invertible non-singular transformation T:X→X, i.e. a transformation which with its inverse is measurable and carries null sets onto null sets.
Thus, if τ is the automorphism of A = L∞(X) induced by T, there is a unique τ-invariant projection p in A such that pA is conservative and (I–p)A is dissipative.
If T is an invertible transformation on a measure space (X,μ) preserving null sets, then the following conditions are equivalent on T (or its inverse):[1] Since T is dissipative if and only if T−1 is dissipative, it follows that T is conservative if and only if T−1 is conservative.
If T is an invertible transformation on a measure space (X,μ) preserving null sets and inducing an automorphism τ of A = L∞(X), then there is a unique τ-invariant p = χC in A such that τ is conservative on pA = L∞(C) and dissipative on (1 − p)A = L∞(D) where D = X \ C.[2] Corollary.
If an invertible transformation T acts ergodically but non-transitively on the measure space (X,μ) preserving null sets and B is a subset with μ(B) > 0, then the complement of B ∪ TB ∪ T2B ∪ ⋅⋅⋅ has measure zero.
Let (X,μ) be a measure space and St a non-sngular flow on X inducing a 1-parameter group of automorphisms σt of A = L∞(X).