KNOWPIA
WELCOME TO KNOWPIA

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.

**Wandering sets and dissipative actions.**A measurable subset*W*of*X*is*wandering*if its characteristic function*q*= χ_{W}in*A*= L^{∞}(*X*) satisfies*q*τ^{n}(*q*) = 0 for all*n*; thus, up to null sets, the translates*T*^{n}(*W*) are pairwise disjoint. An action is called*dissipative*if*X*= ∐*T*^{n}(*W*) 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*) ⊆*Z*then*Z*\*TZ*has measure zero. Thus if*q*= χ_{Z}and τ(*q*) ≤*q*, then τ(*q*) =*q*a.e.**Recurrent actions.**An action*T*is said to be*recurrent*if*q*≤ τ(*q*) ∨ τ^{2}(*q*) ∨ τ^{3}(*q*) ∨ ... a.e. for any*q*= χ_{Y}.**Infinitely recurrent actions.**An action*T*is said to be*infinitely recurrent*if*q*≤ τ^{m}(*q*) ∨ τ^{m + 1}(*q*) ∨ τ^{m+2}(*q*) ∨ ... a.e. for any*q*= χ_{Y}and any*m*≥ 1.

**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]}

*T*is conservative;*T*is recurrent;*T*is infinitely recurrent;*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* ∧ (τ(*q*) ∨ τ^{2}(*q*) ∨ τ^{3}(*q*) ∨ ⋅⋅⋅)^{⊥} = *q* ∧ τ(1 - *q*) ∧ τ^{2}(1 -*q*) ∧ τ^{3}(*q*) ∧ ... is wandering so that if *q* < 1, necessarily *r* = 0. Hence *q* ≤ τ(*q*) ∨ τ^{2}(*q*) ∨ τ^{3}(*q*) ∨ ⋅⋅⋅, so that *T* is recurrent.

If *T* is recurrent, then *q* ≤ τ(*q*) ∨ τ^{2}(*q*) ∨ τ^{3}(*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*) ∨ τ^{2}(*q*) ∨ τ^{3}(*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*) ⊕ τ^{2}(*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* = *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*.^{[2]}

- Without loss of generality it can be assumed that μ is a probability measure. If
*T*is conservative there is nothing to prove, since in that case*C*=*X*. Otherwise there is a wandering set*W*for*T*. Let*r*=*χ*_{W}and*q*= ⊕*τ*^{n}(*r*). Thus*q*is*τ*-invariant and dissipative. Moreover*μ*(*q*) > 0. Clearly an orthogonal direct sum of such*τ*-invariant dissipative*q*′s is also*τ*-invariant and dissipative; and if*q*is*τ*-invariant and dissipative and*r*<*q*is*τ*-invariant, then*r*is dissipative. Hence if*q*_{1}and*q*_{2}are*τ*-invariant and dissipative, then*q*_{1}∨*q*_{2}is*τ*-invariant and dissipative, since*q*_{1}∨*q*_{2}=*q*_{1}⊕*q*_{2}(1 −*q*_{1}). Now let*M*be the supremum of all*μ*(*q*) wirh*q**τ*-invariant and dissipative. Take*q*_{n}*τ*-invariant and dissipative such that*μ*(*q*_{n}) increases to*M*. Replacing*q*_{n}by*q*_{1}∨ ⋅⋅⋅ ∨*q*_{n},*t*can be assumed that*q*_{n}is increasing to*q*say. By continuity*q*is*τ*-invariant and*μ*(*q*) =*M*. By maximality*p*=*I*−*q*is conservative. Uniqueness is clear since no*τ*-invariant*r*<*p*is dissipative and every*τ*-invariant*r*<*q*is dissipative.

**Corollary.** The Hopf decomposition for *T* coincides with the Hopf decomposition for *T*^{−1}.

- Since a transformation is dissipative on a measure space if and only if its inverse is dissipative, the dissipative parts of
*T*and*T*^{−1}coincide. Hence so do the conservative parts.

**Corollary.** The Hopf decomposition for *T* coincides with the Hopf decomposition for *T*^{n} for *n* > 1.

- If
*W*is a wandering set for*T*then it is a wandering set for*T*^{n}. So the dissipative part of*T*is contained in the dissipative part of*T*^{n}. Let σ = τ^{n}. To prove the converse, it suffices to show that if σ is dissipative, then τ is dissipative. If not, using the Hopf decomposition, it can be assumed that σ is dissipative and τ conservative. Suppose that*p*is a non-zero wandering projection for σ. Then τ^{a}(*p*) and τ^{b}(*p*) are orthogonal for different*a*and*b*in the same congruence class modulo*n*. Take a set of τ^{a}(*p*) with non-zero product and maximal size. Thus |*S*| ≤*n*. By maximality,*r*is wandering for τ, a contradiction.

**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.

- Note that ergodicity and non-transitivity imply that the action of
*T*is conservative and hence infinitely recurrent. But then*B*≤*T*^{m}(*B*) ∨*T*^{m + 1}(*B*) ∨*T*^{m+2}(*B*) ∨ ... for any*m*≥ 1. Applying*T*^{−m}, it follows that*T*^{−m}(*B*) lies in*Y*=*B*∪*TB*∪*T*^{2}*B*∪ ⋅⋅⋅ for every*m*> 0. By ergodicity*μ*(*X*\*Y*) = 0.

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*.

- For
*s*,*t*≠ 0 the conservative and dissipative parts of*S*_{s}and*S*_{t}coincide if*s*/*t*is rational.^{[3]}

- This follows from the fact that for any non-singular invertible transformation the conservative and dissipative parts of
*T*and*T*^{n}coincide for*n*≠ 0.

- If
*S*_{1}is dissipative on*A*= L^{∞}(*X*), then there is an invariant measure λ on*A*and*p*in*A*such that

*p*> σ_{t}(*p*) for all*t*> 0- λ(
*p*– σ_{t}(*p*)) =*t*for all*t*> 0 - σ
_{t}(*p*) 1 as*t*tends to −∞ and σ_{t}(*p*) 0 as*t*tends to +∞.

- Let
*T*=*S*_{1}. Take*q*a wandering set for*T*so that ⊕ τ^{n}(*q*) = 1. Changing μ to an equivalent measure, it can be assumed that μ(*q*) = 1, so that μ restricts to a probability measure on*qA*. Transporting this measure to τ^{n}(*q*)*A*, it can further be assumed that μ is τ-invariant on*A*. But then λ = ∫^{1}_{0}μ ∘ σ_{t}*dt*is an equivalent σ-invariant measure on*A*which can be rescaled if necessary so that λ(*q*) = 1. The*r*in*A*that are wandering for*Τ*(or τ) with ⊕ τ^{n}(*r*) = 1 are easily described: they are given by*r*= ⊕ τ^{n}(*q*_{n}) where*q*= ⊕*q*_{n}is a decomposition of*q*. In particular λ(*r*) =1. Moreover if*p*satisfies*p*> τ(*p*) and τ^{–n}(*p*) 1, then λ(*p*– τ(*p*)) = 1, applying the result to*r*=*p*– τ(*p*). The same arguments show that conversely, if*r*is wandering for τ and λ(*r*) = 1, then ⊕ τ^{n}(*r*) = 1.

- Let
*Q*=*q*⊕ τ(*q*) ⊕ τ^{2}(*q*) ⊕ ⋅⋅⋅ so that τ^{k}(*Q*) <*Q*for*k*≥ 1. Then*a*= ∫^{∞}_{0}σ_{t}(*q*)*dt*= Σ_{k≥0}∫^{1}_{0}σ_{k+t}(*q*)*dt*= ∫^{1}_{0}σ_{t}(*Q*)*dt*so that 0 ≤ a ≤ 1 in*A*. By definition σ_{s}(*a*) ≤*a*for*s*≥ 0, since*a*− σ_{s}(*a*) = ∫^{∞}_{s}σ_{t}(*q*)*dt*. The same formulas show that σ_{s}(*a*) tends 0 or 1 as*s*tends to +∞ or −∞. Set*p*= χ_{[ε,1]}(a) for 0 < ε < 1. Then σ_{s}(*p*) = χ_{[ε,1]}(σ_{s}(*a*)). It follows immediately that σ_{s}(*p*) ≤*p*for*s*≥ 0. Moreover σ_{s}(*p*) 0 as*s*tends to +∞ and σ_{s}(*p*) 1 as*s*tends to − ∞. The first limit formula follows because 0 ≤ ε ⋅ σ_{s}(*p*) ≤ σ_{s}(*a*). Now the same reasoning can be applied to τ^{−1}, σ_{−t}, τ^{−1}(*q*) and 1 – ε in place of τ, σ_{t},*q*and ε. Then it is easily checked that the quantities corresponding to*a*and*p*are 1 −*a*and 1 −*p*. Consequently σ_{−t}(1−*p*) 0 as*t*tends to ∞. Hence σ_{s}(*p*) 1 as*s*tends to − ∞. In particular*p*≠ 0 , 1.

- So
*r*=*p*− τ(*p*) is wandering for τ and ⊕ τ^{k}(*r*) = 1. Hence λ(*r*) = 1. It follows that λ(*p*−σ_{s}(*p*) ) =*s*for*s*= 1/*n*and therefore for all rational*s*> 0. Since the family σ_{s}(*p*) is continuous and decreasing, by continuity the same formula also holds for all real*s*> 0. Hence*p*satisfies all the asserted conditions.

- The conservative and dissipative parts of
*S*_{t}for*t*≠ 0 are independent of*t*.^{[4]}

- The previous result shows that if
*S*_{t}is dissipative on*X*for*t*≠ 0 then so is every*S*_{s}for*s*≠ 0. By uniqueness,*S*_{t}and*S*_{s}preserve the dissipative parts of the other. Hence each is dissipative on the dissipative part of the other, so the dissipative parts agree. Hence the conservative parts agree.

**^**Krengel 1985, pp. 16–17**^**Krengel 1985, pp. 17–18**^**Krengel 1985, p. 18**^**Krengel 1968, p. 183

- Aaronson, Jon (1997),
*An introduction to infinite ergodic theory*, Mathematical Surveys and Monographs, vol. 50, American Mathematical Society, ISBN 0-8218-0494-4 - Hopf, Eberhard (1937),
*Ergodentheorie*(in German), Springer - Krengel, Ulrich (1968), "Darstellungssätze für Strömungen und Halbströmungen I",
*Math. Annalen*(in German),**176**: 181−190 - Krengel, Ulrich (1985),
*Ergodic theorems*, De Gruyter Studies in Mathematics, vol. 6, de Gruyter, ISBN 3-11-008478-3