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'':''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.

Definitions

*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 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'') ∨ τ²(''q'') ∨ τ³(''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.

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):

# ''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'') ∨ 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''.

: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.

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''.

*For ''s'', ''t'' ≠ 0 the conservative and dissipative parts of ''S''_''s'' and ''S''_''t'' coincide if ''s''/''t'' is rational.

: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'') $\uparrow$ 1 as ''t'' tends to −∞ and σ_''t''(''p'') $\downarrow$ 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 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'') $\uparrow$ 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 .

:Let ''Q'' = ''q'' ⊕ τ(''q'') ⊕ τ² (''q'') ⊕ ⋅⋅⋅ so that τ^''k'' (''Q'') < ''Q'' for ''k'' ≥ 1. Then so that 0 ≤ a ≤ 1 in ''A''. By definition σ_''s''(''a'') ≤ ''a'' for ''s'' ≥ 0, since . The same formulas show that σ_''s''(''a'') tends 0 or 1 as ''s'' tends to +∞ or −∞. Set ''p'' = χ _{�,1/sub>(a) for 0 < ε < 1. Then σ''s''(''p'') = χ �,1/sub>(σ''s''(''a'')). It follows immediately that σ''s''(''p'') ≤ ''p'' for ''s'' ≥ 0. Moreover σ''s''(''p'') $\downarrow$ 0 as ''s'' tends to +∞ and σ''s''(''p'') $\uparrow$ 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'') $\downarrow$ 0 as ''t'' tends to ∞. Hence σ''s''(''p'') $\uparrow$ 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''.

: 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.

See also

*Ergodic flow

Notes

References

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*{{citation, last=Krengel, first=Ulrich, title=Ergodic theorems, series=De Gruyter Studies in Mathematics, volume=6, publisher=de Gruyter, year= 1985, isbn=3-11-008478-3

Ergodic theory
Measure theory