Hopf decomposition


In mathematics, the Hopf decomposition, named after Eberhard Hopf, gives a canonical decomposition of a measure space 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∞ induced by T, there is a unique τ-invariant projection p in A such that pA is conservative and A is dissipative.

Definitions

  • Wandering sets and dissipative actions. A measurable subset W of X is wandering if its characteristic function q = χW in A = L∞ satisfies qτn = 0 for all n; thus, up to null sets, the translates Tn are pairwise disjoint. An action is called dissipative if X = ∐ Tn a.e. for some wandering set W.
  • Conservative actions. If X has no wandering subsets of positive measure, the action is said to be conservative.
  • Incompressible actions. An action is said to be incompressible if whenever a measurable subset Z satisfies T ⊆ Z then has measure zero. Thus if q = χZ and τ ≤ q, then τ = q a.e.
  • Recurrent actions. An action T is said to be recurrent if q ≤ τ ∨ τ2 ∨ τ3 ∨... a.e. for any q = χY.
  • Infinitely recurrent actions. An action T is said to be infinitely recurrent if q ≤ τm ∨ τm + 1 ∨ τm+2 ∨... a.e. for any q = χY and any m ≥ 1.

    Recurrence theorem

Theorem. If T is an invertible transformation on a measure space preserving null sets, then the following conditions are equivalent on T :
  1. T is conservative;
  2. T is recurrent;
  3. T is infinitely recurrent;
  4. T is incompressible.
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 ∧ ∨ τ2 ∨ τ3⊥ = q ∧ τ ∧ τ2 ∧ τ3 ∧... is wandering so that if q < 1, necessarily r = 0. Hence q ≤ τ ∨ τ2 ∨ τ3 ∨ ⋅⋅⋅, so that T is recurrent.
If T is recurrent, then q ≤ τ ∨ τ2 ∨ τ3 ∨ ⋅⋅⋅ Now assume by induction that q ≤ τk ∨ τk+1 ∨ ⋅⋅⋅. Then τk ≤ τk+1 ∨ τk+2 ∨ ⋅⋅⋅ ≤. Hence q ≤ τk+1 ∨ τk+2 ∨ ⋅⋅⋅. 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, then τ ≤ q. In fact in this case τn is a decreasing sequence. But by recurrence, q ≤ τ ∨ τ2 ∨ τ3 ∨ ⋅⋅⋅, so q ≤ τ and hence q = τ.
Finally suppose that T is incompressible. If T is not conservative there is a p ≠ 0 in A with the τn disjoint. But then q = p ⊕ τ ⊕ τ2 ⊕ ⋅⋅⋅ satisfies τ < q with, contradicting incompressibility. So T is conservative.

Hopf decomposition

Theorem. If T is an invertible transformation on a measure space preserving null sets and inducing an automorphism τ of A = L∞, then there is a unique τ-invariant p = χC in A such that τ is conservative on pA = L∞ and dissipative on A = L∞ 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 Tn for n > 1.
Corollary. If an invertible transformation T acts ergodically but non-transitively on the measure space preserving null sets and B is a subset with μ > 0, then the complement of B ∪ TB ∪ T2B ∪ ⋅⋅⋅ has measure zero.

Hopf decomposition for a non-singular flow

Let be a measure space and St a non-sngular flow on X inducing a 1-parameter group of automorphisms σt of A = L∞. 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 A.
  • For s, t ≠ 0 the conservative and dissipative parts of Ss and St coincide if s/''t is rational.
  • If S''1 is dissipative on A = L∞, then there is an invariant measure λ on A and p in A such that
  • The conservative and dissipative parts of St for t ≠ 0 are independent of t.