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. Up to null sets, X can be written as a disjoint union C ∐ D of T-invariant sets where the action of T on C is conservative and the action of T on D is 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.
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): [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 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 ≤ τk(q) ∨ τk+1(q) ∨ ⋅⋅⋅. Then τk(q) ≤ τk+1(q) ∨ τk+2(q) ∨ ⋅⋅⋅ ≤ . Hence q ≤ τk+1(q) ∨ τk+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 τn(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 τn(p) disjoint (orthogonal). But then q = p ⊕ τ(p) ⊕ τ2(p) ⊕ ⋅⋅⋅ satisfies τ(q) < q with q − τ(q) = p ≠ 0, contradicting incompressibility. So T is conservative.
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 = 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. 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 Tn 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 ∪ 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). It will be assumed that the action is faithful, so that σt is the identity only for t = 0. For each St or equivalently σt with t ≠ 0 there is a Hopf decomposition, so a pt fixed by σt such that the action is conservative on ptA and dissipative on (1−pt)A.
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. Up to null sets, X can be written as a disjoint union C ∐ D of T-invariant sets where the action of T on C is conservative and the action of T on D is 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.
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): [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 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 ≤ τk(q) ∨ τk+1(q) ∨ ⋅⋅⋅. Then τk(q) ≤ τk+1(q) ∨ τk+2(q) ∨ ⋅⋅⋅ ≤ . Hence q ≤ τk+1(q) ∨ τk+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 τn(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 τn(p) disjoint (orthogonal). But then q = p ⊕ τ(p) ⊕ τ2(p) ⊕ ⋅⋅⋅ satisfies τ(q) < q with q − τ(q) = p ≠ 0, contradicting incompressibility. So T is conservative.
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 = 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. 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 Tn 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 ∪ 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). It will be assumed that the action is faithful, so that σt is the identity only for t = 0. For each St or equivalently σt with t ≠ 0 there is a Hopf decomposition, so a pt fixed by σt such that the action is conservative on ptA and dissipative on (1−pt)A.