Hopf decomposition

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", i.e. a transformation which with its inverse is measurable and carries null sets onto null sets. Up to null sets, "X" can be written as a disjoint union "C" ∐ "D" of "T"-invariant sets where the actions of "T" on "C" and "D" are conservative and dissipative. 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.
Recurrence theorem.
Theorem. 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):
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 conservative, then "r" = "q" ∧ (τ("q") ∨ τ2("q") ∨ τ3("q") ∨ ⋅⋅⋅)⊥ = "q" ∧ τ(1 - "q") ∧ τ2(1 -"q") ∧ τ3("q") ∧ ... is wandering so that if "q" < 1, necessarily "r" = 0. Hence "q" ≤ τ("q") ∨ τ2("q") ∨ τ3("q") ∨ ⋅⋅⋅, so that "T" is recurrent.
If "T" is recurrent, then "q" ≤ τ("q") ∨ τ2("q") ∨ τ3("q") ∨ ⋅⋅⋅ Now assume by induction that "q" ≤ τ"k"("q") ∨ τ"k"+1("q") ∨ ⋅⋅⋅. Then τ"k"("q") ≤ τ"k"+1("q") ∨ τ"k"+2("q") ∨ ⋅⋅⋅ ≤ . Hence "q" ≤ τ"k"+1("q") ∨ τ"k"+2("q") ∨ ⋅⋅⋅. So the result holds for "k"+1 and thus "T" is infinitely recurrent. Conversely by definition an infinitely recurrent transformation is recurrent.
Now suppose that "T" is recurrent. To show that "T" is incompressible it must be shown that, if τ("q") ≤ "q", then τ("q") ≤ "q". In fact in this case τ"n"("q") is a decreasing sequence. But by recurrence, "q" ≤ τ("q") ∨ τ2("q") ∨ τ3("q") ∨ ⋅⋅⋅ , so "q" ≤ τ("q") and hence "q" = τ("q").
Finally suppose that "T" is incompressible. If "T" is not conservative there is a "p" ≠ 0 in "A" with the τ"n"("p") disjoint (orthogonal). But then "q" = "p" ⊕ τ("p") ⊕ τ2("p") ⊕ ⋅⋅⋅ satisfies τ("q") < "q" with , contradicting incompressibility. So "T" is conservative.
Hopf decomposition.
Theorem. 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" = "χ""C" in "A" such that "τ" is conservative on "pA" = "L"∞("C") and dissipative on (1 − "p")"A" = "L"∞("D") where "D" = "X" \ "C".
Corollary. The Hopf decomposition for "T" coincides with the Hopf decomposition for "T"−1.
Corollary. The Hopf decomposition for "T" coincides with the Hopf decomposition for "T""n" for "n" > 1.
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" ∪ "T"2"B" ∪ ⋅⋅⋅ has measure zero.
Hopf decomposition for a non-singular flow.
Let ("X",μ) be a measure space and "S""t" a non-sngular flow on "X" inducing a 1-parameter group of automorphisms σ"t" of "A" = L∞("X"). It will be assumed that the action is faithful, so that σ"t" is the identity only for "t" = 0. For each "S""t" or equivalently σ"t" with "t" ≠ 0 there is a Hopf decomposition, so a "p""t" fixed by σ"t" such that the action is conservative on "p""t""A" and dissipative on (1−"p""t")"A".