Real extensions of distal minimal flows and continuous topological ergodic decompositions
aa r X i v : . [ m a t h . D S ] A p r REAL EXTENSIONS OF DISTAL MINIMAL FLOWS ANDCONTINUOUS TOPOLOGICAL ERGODIC DECOMPOSITIONS
GERNOT GRESCHONIG
Abstract.
We prove a structure theorem for topologically recurrent real skewproduct extensions of distal minimal compact metric flows with a compactlygenerated Abelian acting group (e.g. Z d -flows and R d -flows). The main resultstates that every such extension apart from a coboundary can be representedby a perturbation of a so-called Rokhlin skew product. We obtain as a corollarythat the topological ergodic decomposition of the skew product extension intoprolongations is continuous and compact with respect to the Fell topology onthe hyperspace. The right translation acts minimally on this decomposition,therefore providing a minimal compact metric analogue to the Mackey ac-tion. This topological Mackey action is a distal (possibly trivial) extension ofa weakly mixing factor (possibly trivial), and it is distal if and only if pertur-bation of the Rokhlin skew product is defined by a topological coboundary. Introduction and main results
The study of real-valued topological cocycles and real skew product extensionshas been initiated by Besicovitch, Gottschalk, and Hedlund. Besicovitch [Be] provedthe existence of point transitive real skew product extensions of an irrational ro-tation on the one-dimensional torus. Furthermore, he proved that none of them isminimal, i.e. there are always non-transitive points for a point transitive real skewproduct extension. The main result in Chapter 14 of [GoHe] can be rephrased tothe dichotomy that a topologically conservative real skew product extension of aminimal rotation on a torus (finite or infinite dimensional) is either point transi-tive or it is defined by a topological coboundary and almost periodic. This resultand a generalisation to skew product extensions of a Kronecker transformation (cf.[LemMe]) exploit the isometric behaviour of a minimal rotation. A correspondingresult apart from isometries is based on homotopy conditions for the class of dis-tal minimal homeomorphisms usually called Furstenberg transformations (cf. [Gr]).However, in general this dichotomy is not valid, and counterexamples can be pro-vided by the Rokhlin skew products of the so-called topological type
III . Thismotivates the study of topologically conservative real skew product extensions ofcompact flows apart from isometries and toral extensions, which is carried out inthis note for distal minimal flows with Abelian compactly generated acting groups.Throughout this note we shall denote by T a compactly generated Abelian Haus-dorff topological group acting continuously on a compact metric phase space (
X, d )so that (
X, T ) is a compact metric flow . In the monograph [GoHe] such an act-ing group T is called generative , and notions of recurrence are provided for such Mathematics Subject Classification.
Abelian acting groups apart from Z and R . For a Z -action on X we let T be theself-homeomorphism of X generating the action by ( n, x ) T n x , while in the caseof a real flow we shall use the notation { φ t : t ∈ R } for the acting group. We calla flow minimal if the whole phase space is the only non-empty invariant closedsubset, and then for every x ∈ X the orbit closure ¯ O T ( x ) = { τ x : τ ∈ T } is all of X . A flow ( X, T ) is topologically transitive if for arbitrary open neighbourhoods U , V ⊂ X there exists some τ ∈ T with τ U ∩ V 6 = ∅ , and it is weakly mixing if theflow ( X × X, T ) with the diagonal action is topologically transitive. For a topologi-cally transitive flow (
X, T ) with complete separable metric phase space there existsa dense G δ -set of transitive points x with ¯ O T ( x ) = X , and a flow with transitivepoints is point transitive . If ( X, T ) and (
Y, T ) are flows with the same acting group T and π : X −→ Y is a continuous onto mapping with π ( τ x ) = τ π ( x ) for every τ ∈ T and x ∈ X , then ( Y, T ) = π ( X, T ) is called a factor of (
X, T ) and (
X, T ) iscalled an extension of (
Y, T ). Such a mapping π is called a homomorphism of theflows ( X, T ) and (
Y, T ). The set of bicontinuous bijective homomorphisms of a flow(
X, T ) onto itself is the topological group Aut(
X, T ) of automorphisms of (
X, T )with the topology of uniform convergence. Two points x, y ∈ X are called distal ifinf τ ∈ T d ( τ x, τ y ) > , otherwise they are called proximal . For a general compact Hausdorff flow ( X, T )distality of two points x, y ∈ X is defined by the absence of any nets { τ n } n ∈ I ⊂ T with lim τ n x = lim τ n y . A flow is called distal if any two distinct points are distal,and an extension of flows is called distal if any two distinct points in the same fibreare distal. An important property of distal compact flows is the partitioning of thephase space into invariant closed minimal subsets, even if the flow is not minimal.Suppose that A is an Abelian locally compact second countable (Abelian l.c.s.)group with zero element A , and let A ∞ denote its one point compactification withthe convention that g + ∞ = ∞ + g = ∞ for every g ∈ A . A cocycle of a compactmetric flow ( X, T ) is a continuous mapping f : T × X −→ A with the identity f ( τ, τ ′ x ) + f ( τ ′ , x ) = f ( τ τ ′ , x )for all τ, τ ′ ∈ T and x ∈ X . Given a compact metric Z -flow ( X, T ) and a continuousfunction f : X −→ A , we can define a cocycle f : Z × X −→ A with f (1 , · ) ≡ f by f ( n, x ) = P n − k =0 f ( T k x ) if n ≥ , A if n = 0 , − f ( − n, T n x ) if n < . Moreover, there is a natural occurrence of cocycles of R -flows as solutions to ODE’s.Suppose that ( M, { φ t : t ∈ R } ) is a smooth flow on a compact manifold M and A : M −→ R is a smooth function. Then a continuous real valued cocycle f ( t, m )of the flow ( M, { φ t : t ∈ R } ) is given by the fundamental solution to the ODE d f ( t, m ) d t = A ( φ t ( m ))with the initial condition f (0 , m ) = 0. The skew product extension of the flow ( X, T )by a cocycle f : T × X −→ A is defined by the homeomorphisms e τ f ( x, a ) = ( τ x, f ( τ, x ) + a ) EAL EXTENSIONS OF DISTAL MINIMAL FLOWS 3 of X × A for all τ ∈ T , which provide a continuous action ( τ, x, a ) e τ f ( x, a ) of T on X × A by the cocycle identity. For a Z -flow ( X, T ) this action is generated by e T f ( x, a ) = ( T x, f ( x ) + a ) . The essential property of a skew product is that the T -action on X × A commuteswith the right translation action of the group A on X × A , which is defined by R b ( x, a ) = ( x, a − b )for every b ∈ A . The orbit closure of ( x, a ) ∈ X × A under e τ f will be denoted by¯ O T,f ( x, a ) = { e τ f ( x, a ) : τ ∈ T } .The prolongation D T ( x ) of x ∈ X under the group action of T is defined by D T ( x ) = \ { ¯ O T ( U ) : U is an open neighbourhood of x } , and we shall use the notation D T,f ( x, a ) for the prolongation of a point ( x, a ) ∈ X × A under the skew product action e τ f .While the inclusion of the orbit closure in the prolongation is obvious, the coin-cidence of these sets is generic by a result from the paper [Gl3]. This result, one ofour main tools, is usually referred to as “topological ergodic decomposition”. Fact 1.1.
For every compact metric flow ( X, T ) there exists a T -invariant dense G δ set F ⊂ X so that for every x ∈ F holds ¯ O T ( x ) = D T ( x ) . For a skew product extension e τ f of ( X, T ) by a cocycle f : T × X −→ A there existsa T -invariant dense G δ set F ⊂ X so that for every x ∈ F and every a ∈ A holds ¯ O T,f ( x, a ) = D T,f ( x, a ) . This assertion holds as well for the extension of e τ f to X × A ∞ which is definedby ( x, ∞ ) ( τ x, ∞ ) for every x ∈ X , and given an R -valued topological cocycle g = ( g , g ) : T × X −→ R for the extension of e τ g to X × ( R ∞ ) which is definedby ( x, s, ∞ ) ( τ x, s + g ( x ) , ∞ ) , ( x, ∞ , t ) ( τ x, ∞ , t + g ( x )) , and ( x, ∞ , ∞ ) ( τ x, ∞ , ∞ ) , for every x ∈ X and s, t ∈ R .Proof. The statement for a compact metric phase space and a general acting groupis according to Theorem 1 of [AkGl]. The other statements can be verified bymeans of the extension of e τ f onto the compactification of X × A . The coincidenceof ¯ O T,f ( x, a ) and D T,f ( x, a ) for some ( x, a ) ∈ X × A implies this coincidence for all( x, a ′ ) ∈ { x } × A ∞ , since the extension of e τ f to X × A ∞ commutes with the righttranslation on X × A ∞ . (cid:3) Remark . If y ∈ ¯ O T ( x ) and z ∈ ¯ O T ( y ), then z ∈ ¯ O T ( x ) follows by a diagonalisa-tion argument. A corresponding statement for prolongations is not valid, howeverfollows from x ∈ ¯ O T ( y ) and z ∈ D T ( y ) that z ∈ D T ( x ).We shall consider more general Abelian acting groups than Z and R , hence thedefinition of recurrence requires the notions of a replete semigroup and an extensivesubset of the Abelian compactly generated group T (cf. [GoHe]). We recall that asemigroup P ⊂ T is replete if for every compact subset K ⊂ T there exists a τ ∈ T with τ K ⊂ P , and a subset E ⊂ T is extensive if it intersects every repletesemigroup. Therefore, a subset E of T = Z or T = R is extensive if and only if E contains arbitrarily large positive and arbitrarily large negative elements. GERNOT GRESCHONIG
Definition 1.3.
We call a cocycle f ( τ, x ) of a minimal compact metric flow ( X, T ) topologically recurrent if for arbitrary neighbourhoods U ⊂ X and U ( A ) ⊂ A of A there exists an extensive set of elements τ ∈ T with U ∩ τ − ( U ) ∩ { x ∈ X : f ( τ, x ) ∈ U ( A ) } 6 = ∅ . Since e τ f and the right translation on X × A commute, this is equivalent to the regional recurrence of the skew product action e τ f on X × A , i.e. for every openneighbourhood U ⊂ X × A there exists an extensive set of elements τ ∈ T with e τ f ( U ) ∩ U = ∅ . A non-recurrent cocycle is called transient .A point ( x, a ) ∈ X × A is e τ f -recurrent if for every neighbourhood U ⊂ X × A of ( x, a ) the set of τ ∈ T with e τ f ( x, a ) ∈ U is extensive. Moreover, a point ( x, a ) ∈ X × A is regionally e τ f -recurrent if for every neighbourhood U of ( x, a ) the set of τ ∈ T with e τ f ( U ) ∩ U = ∅ is extensive. Remarks . If f ( τ, x ) is recurrent, then by Theorems 7.15 and 7.16 in [GoHe]there exists a dense G δ set of e τ f -recurrent points in X × R .Given a regionally e τ f -recurrent point ( x, a ) ∈ X × A , every point in { x } × A is regionally e τ f -recurrent. The minimality of ( X, T ) and Theorem 7.13 in [GoHe]imply that every point in X × A is regionally e τ f -recurrent, hence f ( τ, x ) is recurrent.A cocycle f ( n, x ) of a Z -flow is topologically recurrent if and only if e T f is topo-logically conservative, i.e. for every open neighbourhood U ⊂ X × A there exists aninteger n = 0 so that e T nf ( U ) ∩ U = ∅ .One of the most important concepts in the study of cocycles is the essentialrange, originally introduced in the measure theoretic category by Schmidt [Sch]. Definition 1.5.
Let f ( τ, x ) be a cocycle of a minimal compact metric flow ( X, T ).An element a ∈ A is in the set E ( f ) of topological essential values if for arbitraryneighbourhoods U ⊂ X and U ( a ) ⊂ A of a there exists an element τ ∈ T so that U ∩ τ − ( U ) ∩ { x ∈ X : f ( τ, x ) ∈ U ( a ) } is non-empty. The set E ( f ) is also called the topological essential range . The cocycleidentity implies that f ( T , x ) = A for all x ∈ X and hence A ∈ E ( f ). Moreover,the essential range is always a closed subgroup of A (cf. [LemMe], Proposition 3.1,which carries over from the case of a minimal Z -action to a general Abelian groupacting minimally). Fact 1.6. If f ( τ, x ) is a cocycle with full topological essential range E ( f ) = A , then D T,f ( x, a ) ⊂ { x } × A holds for every ( x, a ) ∈ X × A . By Fact 1.1 there exists a T -invariant dense G δ set F ⊂ X with { x } × A ⊂ ¯ O T,f ( x, a ) for every ( x, a ) ∈ F × A .For every τ ∈ T follows that { τ x } × A ⊂ ¯ O T,f ( x, g ) , and by the minimality of theflow ( X, T ) every ( x, a ) ∈ F × A is a transitive point for e τ f . Throughout this note we shall use a notion of “relative” triviality of cocycles.
Definition 1.7.
Let f ( τ, x ) and f ( τ, x ) be R -valued cocycles of a minimal com-pact metric flow ( X, T ). We shall call the cocycle f ( τ, x ) relatively trivial with re-spect to f ( τ, x ), if for every sequence { ( τ k , x k ) } k ≥ ⊂ T × X with d ( x k , τ k x k ) → f ( τ k , x k ) → f ( τ k , x k ) → k → ∞ . For a sequence { τ k } k ≥ ⊂ T and a point ¯ x ∈ X so that τ k ¯ x and f ( τ k , ¯ x ) are convergent, thisimplies that also f ( τ k , ¯ x ) is convergent.By the following lemma it suffices to verify an essential value condition “locally”. EAL EXTENSIONS OF DISTAL MINIMAL FLOWS 5
Lemma 1.8.
Let ( X, T ) be a minimal compact metric flow with an Abelian group T acting, and let f ( τ, x ) be a cocycle of ( X, T ) with values in an Abelian l.c.s.group A . If there exists a sequence { ( τ k , x k ) } k ≥ ⊂ T × X with d ( x k , τ k x k ) → and f ( τ k , x k ) → a ∈ A ∞ ( R ∞ × R ∞ for A = R , respectively) as k → ∞ , then for every x ∈ X it holds that ( x, a ) ∈ D T,f ( x, A ) . Hence if a ∈ A is finite, then a ∈ E ( f ) .Now let g = ( g , g ) : T × X −→ R be a cocycle of the flow ( X, T ) with a sequence { ( τ k , x k ) } k ≥ ⊂ T × X so that d ( x k , τ k x k ) → , g ( τ k , x k ) → , and g ( τ k , x k ) as k → ∞ . Then there exist a point ¯ x ∈ X and a sequence { ¯ τ k } k ≥ ⊂ T sothat d (¯ x, ¯ τ k ¯ x ) → and g (¯ τ k , ¯ x ) → (0 , ∞ ) as k → ∞ . Moreover, for an extension ( Y, T ) of ( X, T ) = π ( Y, T ) there exists a sequence { (˜ τ k , y k ) } k ≥ ⊂ T × Y with d Y ( y k , ˜ τ k y k ) → and ( g ◦ π )(˜ τ k , y k ) → (0 , ∞ ) as k → ∞ .Proof. We let { ( τ k , x k ) } k ≥ ⊂ T × X be a sequence with the properties above, andwe may assume that x k → x ′ ∈ X as k → ∞ . For arbitrary neighbourhoods U ⊂ X and U ( a ) of a ∈ A ∞ we can fix an element τ ∈ T with τ x ′ ∈ U , and since the group T is Abelian it holds that τ x k → τ x ′ and τ k τ x k = τ τ k x k → τ x ′ as k → ∞ . Fromthe cocycle identity and the continuity of f ( τ, · ) follows f ( τ k , τ x k ) = f ( τ, τ k x k ) + f ( τ k , x k ) + f ( τ − , τ x k )= f ( τ, τ k x k ) + f ( τ k , x k ) − f ( τ, x k ) → a as k → ∞ , and for all k large enough it holds that τ x k , τ k τ x k ∈ U and f ( τ k , τ x k ) ∈ U ( a ). Since the neighbourhoods U and U ( a ) were arbitrary, we have ( x, a ) ∈D T,f ( x, A ) for every x ∈ X and a ∈ E ( f ) if a = ∞ .If g ( τ k , x k ) (0 , ∞ ) as k → ∞ , then E ( g ) has an element (0 , c ) with c ∈ R \ { } . Since E ( g ) is a closed subspace of R , we can start over with a sequence { ( τ k , x k ) } k ≥ ⊂ T × X so that d ( x k , τ k x k ) → g ( τ k , x k ) →
0, and | g ( τ k , x k ) | →∞ as k → ∞ . The statement above implies that ( x, , ∞ ) ∈ D T,g ( x, ,
0) for every x ∈ X , and by Fact 1.1 we can select ¯ x ∈ X and a sequence { ¯ τ k } k ≥ ⊂ T so that¯ τ k ¯ x → ¯ x and g (¯ τ k , ¯ x ) → (0 , ∞ ). For an arbitrary point ¯ y ∈ π − (¯ x ) we can selectan increasing sequence of positive integers { k l } l ≥ with d Y (¯ τ k l +1 ¯ y, ¯ τ k l ¯ y ) → g ◦ σ )(¯ τ k l +1 (¯ τ k l ) − , ¯ τ k l ¯ y ) → (0 , ∞ ) and put { (˜ τ l , y l ) = (¯ τ k l +1 (¯ τ k l ) − , ¯ τ k l ¯ y ) } l ≥ . (cid:3) Definition 1.9.
Let f ( τ, x ) be a cocycle of a minimal compact metric flow ( X, T )with values in an Abelian l.c.s. group A , and let b : X −→ A be a continuousfunction. Another cocycle of the flow ( X, T ) can be defined by the A -valued function g ( τ, x ) = f ( τ, x ) + b ( τ x ) − b ( x ) . The cocycle g ( τ, x ) is called topologically cohomologous to the cocycle f ( τ, x ) withthe transfer function b ( x ). A cocycle g ( τ, x ) = b ( τ x ) − b ( x ) topologically cohomol-ogous to zero is bounded on T × X and called a topological coboundary .The Gottschalk-Hedlund theorem ([GoHe], Theorem 14.11) characterises topo-logical coboundaries of a minimal Z -action as cocycles bounded on at least onesemi-orbit. The generalisation to an Abelian group T acting minimally is natural. Fact 1.10.
A real valued topological cocycle f ( τ, x ) of a minimal compact metricflow ( X, T ) with an Abelian acting group T is a coboundary if and only if thereexists a point ¯ x ∈ X so that the function τ f ( τ, ¯ x ) is bounded on T . For thegroups T = Z and T = R acting, the boundedness on a semi-orbit is sufficient.A real valued cocycle f ( τ, x ) is also a topological coboundary if for every sequence { ( τ k , x k ) } k ≥ ⊂ T × X with d ( x k , τ k x k ) → the set { f ( τ k , x k ) } k ≥ ⊂ R is bounded. GERNOT GRESCHONIG
Proof.
Suppose that τ f ( τ, ¯ x ) is bounded on T . By the cocycle identity holds f ( τ, τ ′ ¯ x ) = f ( τ τ ′ , ¯ x ) − f ( τ ′ , ¯ x )for all τ, τ ′ ∈ T , and by the density of the T -orbit of ¯ x follows the boundednessof f ( τ, x ) on T × X and thus the triviality of the subgroup E ( f ) = { } . By thedensity of the T -orbit of ¯ x and the boundedness of τ f ( τ, ¯ x ), the intersection { x }× R ∩ ¯ O T,f (¯ x,
0) is non-empty for every x ∈ X . For every x ∈ X this intersectionis a singleton, since otherwise Lemma 1.8 proves a non-zero element in E ( f ). Hencethe compact set ¯ O T,f (¯ x,
0) is the graph of a continuous function b : X −→ R with f ( τ, ¯ x ) = b ( τ ¯ x ), and thus f ( τ, x ) = b ( τ x ) − b ( x ) holds for every ( τ, x ) ∈ T × X .For T = Z and T = R the set of limit points of a semi-orbit is a T -invariant closedsubset of X , which is non-empty by compactness and equal to X by minimality.We can conclude the proof as above, but using the semi-orbit.Now suppose that f ( τ, x ) is not a topological coboundary and let ¯ x ∈ X bearbitrary. Then there exists a sequence { τ ′ l } l ≥ ⊂ T with | f ( τ ′ l , ¯ x ) | → ∞ , and wemay assume that τ ′ l ¯ x → x ′ as l → ∞ . Since ( X, T ) is minimal, there exists sequence { τ ′′ k } k ≥ ⊂ T with τ ′′ k x ′ → ¯ x as k → ∞ . A diagonalisation with a sufficientlyincreasing sequence of positive integers { l k } l ≥ yields for τ k = τ ′′ k τ ′ l k that τ k ¯ x → ¯ x and | f ( τ k , ¯ x ) | = | f ( τ ′′ k , τ ′ l k ¯ x ) + f ( τ ′ l k , ¯ x ) | → ∞ as k → ∞ . (cid:3) The following lemma appeared originally in the paper [At] in a setting for R d -valued cocycles of a minimal rotation on a torus. Lemma 1.11.
Let f ( τ, x ) be a real valued topological cocycle of a minimal compactmetric flow ( X, T ) with an Abelian acting group T . If the skew product action e τ f is not point transitive on X × R , then for every neighbourhood U ⊂ R of there exista compact symmetric neighbourhood K ⊂ U of and an ε > so that for every τ ∈ T holds { x ∈ X : d ( x, τ x ) < ε and f ( τ, x ) ∈ K \ K } = ∅ . (1) Proof.
Suppose that f ( τ, x ) is real valued and e τ f is not point transitive. By Fact1.6 the essential range E ( f ) is a proper closed subgroup of R , and thus there existsa compact symmetric neighbourhood K ⊂ U of 0 with (2 K \ K ) ∩ E ( f ) = ∅ .If the assertion is false for the neighbourhood K , then there exists a sequence { ( τ k , x k ) } k ≥ ⊂ T × X with d ( x k , τ k x k ) → f ( τ k , x k ) → t ∈ K \ K . NowLemma 1.8 implies t ∈ E ( f ) ∩ K \ K , in contradiction to the choice of K . (cid:3) We shall commence the study of cocycles of distal minimal flows by the general-isation of the results for minimal rotations in [GoHe] and [LemMe].
Proposition 1.12.
Let ( X, T ) be a minimal compact isometric flow with a com-pactly generated Abelian acting group T , and let f ( τ, x ) be a topologically recurrentreal valued cocycle of ( X, T ) . Then the cocycle f ( τ, x ) is either a coboundary or itsskew product extension e τ f is point transitive on X × R .Proof. Suppose that the cocycle f ( τ, x ) is not a coboundary and e τ f is not pointtransitive. Then by Lemma 1.11 there exist a compact symmetric neighbourhood K of 0 and an ε > τ ∈ T . Furthermore, if L ⊂ T is a compact generative subset, then ε > τ ′ ∈ L and x, x ′ ∈ X with d ( x, x ′ ) < ε it holds that f ( τ ′ , x ) − f ( τ ′ , x ′ ) ∈ K . EAL EXTENSIONS OF DISTAL MINIMAL FLOWS 7
By Fact 1.10 we can fix a pair (¯ τ , ¯ x ) ∈ T × X with d (¯ x, ¯ τ ¯ x ) < ε and f (¯ τ , ¯ x ) / ∈ K ,since f ( τ, x ) is not a coboundary. The Abelian group T acts on X isometrically,and thus d (¯ x, ¯ τ ¯ x ) < ε implies that d ( τ ′ ¯ x, ¯ τ τ ′ ¯ x ) = d ( τ ′ ¯ x, τ ′ ¯ τ ¯ x ) < ε . Together withequality (1) we can conclude for every τ ′ ∈ L that f (¯ τ , τ ′ ¯ x ) = f (¯ τ , ¯ x ) − f ( τ ′ , ¯ x ) + f ( τ ′ , ¯ τ ¯ x ) / ∈ K, and hence both of the real numbers f (¯ τ , ¯ x ) and f (¯ τ , τ ′ ¯ x ) are elements of the oneand the same of the disjoint sets R + \ K and R − \ K . Since the set L is generativein the Abelian group T acting minimally on X , it follows by induction that f (¯ τ , x )is in the closure of one of the sets R + \ K and R − \ K for every x ∈ X . Thuswe have a constant c > | f (¯ τ k , x ) | > | k | c for every integer k , and we define asubset P ⊂ T by P = ∪ k ≥ ¯ τ k · { τ ∈ T : f ( τ, · ) < | k | c/ } . Given two integers k, k ′ ≥ τ k τ , ¯ τ k ′ τ ′ ∈ P with f ( τ, · ) < | k | c/ f ( τ ′ , · ) < | k ′ | c/
2, we can conclude that ¯ τ k τ ¯ τ k ′ τ ′ = ¯ τ k + k ′ ( τ τ ′ ) with f ( τ τ ′ , · ) < | k + k ′ | c/ P is a semigroup. Moreover, the semigroup P contains a translate of everycompact set L ⊂ T , since for large enough k ≥ f ( τ, x ) < | k | c/ τ ∈ L and every x ∈ X . Therefore P is a replete semigroup in T sothat | f ( τ, x ) | > c/ τ, x ) ∈ P × X , which contradicts the existenceof a dense G δ set of e τ f -recurrent points (cf. Remarks 1.4). (cid:3) The
Rokhlin extensions and the
Rokhlin skew products have been studied in themeasure theoretic setting in [LemLes] and [LemPa]. We shall introduce the notionof a perturbed Rokhlin skew product , which will be inevitable in our main result.
Definition 1.13.
Suppose that (
X, T ) is a distal minimal compact metric flow and( M, { φ t : t ∈ R } ) is a distal minimal compact metric R -flow. Let f : T × X −→ R be a cocycle of ( X, T ) with a point transitive skew product e τ f on X × R . We definethe Rokhlin extension τ φ,f on X × M by τ φ,f ( x, m ) = ( τ x, φ f ( τ,x ) ( m )) , which is an action of the group T on X × M due to the cocycle identity for f ( τ, x ).If (¯ x,
0) is a transitive point for e τ f , then by the minimality of ( M, { φ t : t ∈ R } )every point (¯ x, m ) with m ∈ M is a transitive point for ( X × M, T ). Since the flow( X × M, T ) is distal by the distality of its components, it is even minimal.The skew product extension of ( X × M, T ) by the cocycle ( τ, x, m ) f ( τ, x ) isthe Rokhlin skew product e τ φ,f on X × M × R with e τ φ,f ( x, m, t ) = ( τ x, φ f ( τ,x ) ( m ) , t + f ( τ, x )) . Let g : R × M −→ R be a cocycle of the flow ( M, { φ t : t ∈ R } ). The R -valued map( τ, x, m ) f ( τ, x ) + g ( f ( τ, x ) , m )defined on T × X × M turns out to be a cocycle of the flow ( X × M, T ) due to thecocycle identity for g ( t, m ). The skew product extension of this cocycle with e τ φ,f,g ( x, m, t ) = ( τ x, φ f ( τ,x ) ( m ) , t + f ( τ, x ) + g ( f ( τ, x ) , m )) . is called a perturbed Rokhlin skew product e τ φ,f,g on X × M × R .We shall present at first the basic example of a topological Rokhlin skew productof topological type III , i.e. recurrent with a trivial topological essential range butnot a topological coboundary. GERNOT GRESCHONIG
Example . Let f : T −→ R be a continuous function with a point transitive skewproduct extension e T f of the irrational rotation T by α on the torus, and let β ∈ (0 , R -flow ( T , { φ t : t ∈ R } ) defined by φ t ( y, z ) = ( y + t, z + βt )is minimal and distal. The minimal and distal Rokhlin extension T φ,f on T is T φ,f ( x, y, z ) = ( x + α, y + f ( x ) , z + βf ( x )) , and putting h ( x, y, z ) = f ( x ) for all ( x, y, z ) ∈ T gives a topological type III cocycle h ( n, ( x, y, z )) of the homeomorphism T φ,f with the skew product extension e T φ,f . Indeed, since e T f is point transitive, the cocycle h ( n, ( x, y, z )) is recurrent,but it is not bounded and therefore no topological coboundary. Furthermore, asequence { t n } n ≥ ⊂ R with t n mod 1 → βt n ) mod 1 → E ( h ) = { } . For a point ¯ x ∈ T sothat (¯ x, ∈ T × R is transitive under e T f and arbitrary y, z ∈ T the orbit closureof (¯ x, y, z,
0) under the skew product extension of T φ,f by h is of the form¯ O e T φ,f ((¯ x, y, z ) ,
0) = ¯ O T φ,f ,h ((¯ x, y, z ) ,
0) = T × { ( φ t ( y, z ) , t ) ∈ T × R : t ∈ R } . The collection of these sets is a partition of T × R into e T φ,f -orbit closures.The next example makes clear that the perturbation of a Rokhlin skew productby a cocycle is an essential component, which in general cannot be eliminated bycontinuous cohomology. Example . Let T , f , h , and { φ t : t ∈ R } be defined as in Example 1.14, and sup-pose that g ( t, ( y, z )) is a point transitive R -valued cocycle of the flow { φ t : t ∈ R } .From the unique ergodicity of the flow { φ t : t ∈ R } follows R T g ( t, ( y, z )) dλ ( y, z ) = 0for every t ∈ R , and after rescaling g we can assume that | g ( t, ( y, z )) | < | t | / t ∈ R and ( y, z ) ∈ T . We define a function¯ h ( x, y, z ) = f ( x ) + g ( f ( x ) , ( y, z ))so that the cocycle of T φ,f is ¯ h ( n, ( x, y, z )) = f ( n, x ) + g ( f ( n, x ) , ( y, z )) for every n .Since the perturbation g ( f ( n, x ) , ( y, z )) is unbounded, there cannot be a continuoustransfer function defined on T so that ¯ h and h are cohomologous. However, due tothe condition | g ( t, ( y, z )) | < | t | / T × { ( φ t ( y, z ) , t + g ( t, ( y, z ))) ∈ T × R : t ∈ R } is closed, and it coincides with ¯ O e T φ,f,g ((¯ x, y, z ) ,
0) if the point (¯ x, ∈ T × R istransitive under e T f . Thus the structure of the orbit closures is preserved as well asthese sets provide a partition of T × R . Remark . The structure of Example 1.14 can be revealed from the toral exten-sions of T φ,f by the function ( γh ) mod 1 for all γ ∈ R . This distal homeomorphismof T is transitive and hence minimal for rationally independent 1, β , and γ . How-ever, for γ = 1 and γ = β the orbit closures collapse to graphs representing thedependence of h and the action on the coordinates of the torus. The same approachwill not be successful with respect to Example 1.15. It can be verified that for every γ ∈ R the toral extension of T φ,f by the function ( γ ¯ h ) mod 1 is minimal on T .The main result of this note puts these examples into a structure theorem. Structure theorem.
Suppose that ( X, T ) is a distal minimal compact metric flowwith a compactly generated Abelian acting group T and that f : T × X −→ R is a EAL EXTENSIONS OF DISTAL MINIMAL FLOWS 9 topologically recurrent cocycle which is not a coboundary. Then there exist a factor ( X α , T ) = π α ( X, T ) , a topological cocycle f α : T × X α −→ R of ( X α , T ) , and a distalminimal compact metric R -flow ( M, { φ t : t ∈ R } ) , so that the Rokhlin extension ( X α × M, T ) with the action τ φ,f α is a factor ( Y, T ) = π Y ( X, T ) of ( X, T ) . Thecocycle f ( τ, x ) is topologically cohomologous to ( f Y ◦ π Y )( τ, x ) = f ( τ, x ) + b ( τ x ) − b ( x ) with a topological cocycle f Y : T × Y −→ R of the flow ( Y, T ) so that D T,f Y ◦ π Y ( x, ∩ ( π − α ( π α ( x )) × { } ) = π − Y ( π Y ( x )) × { } (2) holds for all x ∈ X . Moreover, there exists a cocycle g : R × M −→ R of the R -flow ( M, { φ t : t ∈ R } ) so that the cocycle ( + g )( t, m ) = t + g ( t, m ) is topologically transient and f Y ( τ, ( x, m )) = f α ( τ, x ) + g ( f α ( τ, x ) , m ) = ( + g )( f α ( τ, x ) , m ) (3) holds for every τ ∈ T and ( x, m ) ∈ Y = X α × M . Thus the skew product e τ f Y on Y × R is the perturbed Rokhlin skew product e τ φ,f α ,g . We shall conclude the proof of this theorem in the next section of this note.The application of the structure theorem for a topological ergodic decompositionrequires a suitable topology on the hyperspace of the non-compact space X × R .We shall use the Fell topology on the hyperspace of non-empty closed subsets of alocally compact separable metric space. Given finitely many open neighbourhoods U , . . . , U k and a compact set K , an element of the Fell topology base consistsof all non-empty closed subsets which intersect each of the open neighbourhoods U , . . . , U k while being disjoint from K . This topology is separable, metrisable, and σ -compact (cf. [HLP]). The Fell topology was introduced in [Fe] as a compacttopology on the hyperspace of all closed subsets, with the empty set as infinity. Decomposition theorem.
Suppose that f : T × X −→ R is a topologically re-current cocycle of a distal minimal compact metric flow ( X, T ) with a compactlygenerated Abelian acting group T . The prolongations D T,f ( x, s ) ⊂ X × R of the skewproduct action e τ f with ( x, s ) ∈ X × R define a partition of X × R . The mapping ( x, s )
7→ D
T,f ( x, s ) is continuous with respect to the Fell topology on the hyperspaceof non-empty closed subsets of X × R , and the right translation on X × R is aminimal continuous R -action on the set of prolongations. If the cocycle f ( τ, x ) isnot a topological coboundary, then the set of all prolongations in the skew productis Fell compact. Topological Mackey action.
A recurrent cocycle f ( τ, x ) apart from a coboundaryhas a minimal compact metric flow as a topological version of the Mackey action . Itsphase space is the set of prolongations in the skew product with the Fell topology,with the right translation of R acting on the prolongations. This flow is a distalextension (possibly the trivial extension) of a weakly mixing compact metric flow(possibly the trivial flow). The Mackey action is distal if and only if the perturbationcocycle g ( t, m ) in the structure theorem is a topological coboundary. While most of the properties of the topological Mackey action are part of thedecomposition theorem, its structure as a distal extension of a weakly mixing flowwill be verified in the next section of this note. The proof of the decompositiontheorem depends on the following general lemma on transient cocycles of minimal R -flows, which might be of independent interest. Lemma 1.17.
Let ( M, { φ t : t ∈ R } ) be a minimal compact metric R -flow andlet h ( t, m ) : R × M −→ R be a transient cocycle of ( M, { φ t : t ∈ R } ) . Then forevery point ( m, s ) ∈ M × R the orbit O φ,h ( m, s ) , the orbit closure ¯ O φ,h ( m, s ) , andthe prolongation D φ,h ( m, s ) under the skew product extension e φ th coincide. Themapping from points to their orbits in M × R is continuous with a compact rangewith respect to the Fell topology, and the right translation on M × R provides aminimal continuous R -action on the set of orbits. Moreover, for every m ∈ M themapping t h ( t, m ) maps R onto R .Proof. Since prolongations are closed sets, it suffices to verify that for every ( m, s ) ∈ M × R the orbit and the prolongation coincide. Otherwise, there exist two points( m, s ) , ( m ′ , s ′ ) ∈ M × R so that ( m ′ , s ′ ) is not in the e φ th -orbit of ( m, s ), howeverthere exists a sequence { ( t k , m k ) } k ≥ ⊂ R × M so that ( t k , m k ) → (+ ∞ , m ) and f φ t k h ( m k , s ) = ( φ t k ( m k ) , s + h ( t k , m k )) → ( m ′ , s ′ ) . If there exists a compact set L ⊂ R with h ([0 , t k ] , m k ) ⊂ L for all k ≥
1, then h ([0 , ∞ ) , m ) ⊂ L since m k → m , and by Fact 1.10 the cocycle h ( t, m ) is a cobound-ary in contradiction to its transience. Therefore we have an increasing sequence ofintegers { k l } l ≥ , a sequence { t ′ l } l ≥ ⊂ R with t ′ l ∈ [0 , t k l ], and S ∈ { +1 , − } so that S · h ( t ′ l , m k l ) = max t ∈ [0 ,t kl ] S · h ( t, m k l ) → + ∞ as l → ∞ . For every limit point ¯ m of the sequence { φ t ′ l ( m k l ) } l ≥ it holds that S · h ( t, ¯ m ) ≤ t ∈ R , and the mapping t h ( t, ¯ m ) maps each of the sets R + and R − onto S · R − . Hence for every t ∈ R + there exists a t ′ ∈ R − with h ( t, ¯ m ) = h ( t ′ , ¯ m ), and by the density of the semi-orbit { φ t ( ¯ m ) : t ∈ R + } (cf. theproof of Fact 1.10) and the cocycle identity the open set M k = { m ∈ M : | h ( t, m ) | < − k for some t < − k } is dense for every integer k ≥
1. For a point m k in the dense G δ set T t ∈ Q φ t ( M k ),we can find rational numbers t , . . . , t k < − k so that φ t + ··· + t l ( m k ) ∈ M k and | h ( t + · · · + t l , m k ) | < k − k for all 1 ≤ l ≤ k . Since M is compact, there existsa point ˜ m ∈ M each neighbourhood of which contains at least two distinct pointsout of the finite sequence φ t ( m k ) , . . . , φ t + ··· + t k ( m k ) ∈ M k for an infinite set ofintegers k ≥
1. Thus ( ˜ m,
0) is a regionally recurrent point for the skew product e φ th ,in contradiction to the Remarks 1.4 and the transience of the cocycle h ( t, m ).Suppose for some m ∈ M the mapping t h ( t, m ) is not onto R , then eitherthere exists a compact set L ⊂ R so that one of the inclusions h ([0 , ∞ ) , m ) ⊂ L , h (( −∞ , , m ) ⊂ L holds, or there exists a point ¯ m ∈ M as above. In each of thesecases, we can obtain a contradiction to the transience of h ( t, m ) as above.Let ( m, s ) ∈ M × R be arbitrary, and let a Fell neighbourhood of its closed e φ th -orbit be defined by the open neighbourhoods U , . . . , U k and a compact set K . Obviously the e φ th -orbit of every point in a suitable neighbourhood of ( m, s )intersects each of the neighbourhoods U , . . . , U k . Moreover, the e φ th -orbit is dis-joint from K for every point in a suitable neighbourhood of ( m, s ), since otherwisethe e φ th -prolongation of ( m, s ) intersects the compact set K . This contradicts thecoincidence of orbits and prolongations of e φ th , and hence the mapping of a point( m, s ) ∈ M × R to its e φ th -orbit is Fell continuous. By the surjectivity of the mapping EAL EXTENSIONS OF DISTAL MINIMAL FLOWS 11 t h ( t, m ), every e φ th -orbit intersects the set { ( m,
0) : m ∈ M } . The Fell compact-ness of the set of e φ th -orbits in M × R follows, and the right translation on M × R isa Fell continuous R -action by the definition of the Fell neighbourhoods. Since thepoints ( φ t ( m ) ,
0) and ( m, − h ( m, t )) are within the same e φ th -orbit, the minimalityof ( M, { φ t : t ∈ R } ) implies the minimality of the right translation action. (cid:3) Proof of the decomposition theorem.
For a topological coboundary f ( τ, x ) the as-sertions are trivial, since the prolongations are just the right translates of the graphof the transfer function. If the cocycle f ( τ, x ) is not a coboundary, then we applythe structure theorem. For a point ¯ x α ∈ X α with ¯ O T,f α (¯ x α ,
0) = X α × R and every m ∈ M follows from equality (3) that X α × O φ, + g ( m, ⊂ ¯ O T,f Y (¯ x α , m, ⊂ X α × ¯ O φ, + g ( m, O φ, + g ( m,
0) = ¯ O φ, + g ( m,
0) = D φ, + g ( m, x α ∈ X α with ¯ O T,f α (¯ x α ,
0) = X α × R is a dense G δ set, and for arbitrary ( x α , m ) ∈ X α × M and ( x ′ α , m ′ , t ′ ) ∈ D T,f Y ( x α , m,
0) we canfind a sequence { ( τ k , ¯ x k , m k ) } ⊂ T × X α × M with (¯ x k , m k ) → ( x α , m ), τ k (¯ x k , m k ) → ( x ′ α , m ′ ), f Y ( τ k , (¯ x k , m k )) → t ′ , and X α ×O φ, + g ( m k ,
0) = ¯ O T,f Y (¯ x k , m k , m ′ , t ′ ) ∈ D φ, + g ( m,
0) so that D T,f Y ( x α , m, ⊂ X α × D φ, + g ( m, x α , m, s ) ∈ X α × M × R that D T,f Y ( x α , m, s ) = X α × { ( φ t ( m ) , s + t + g ( t, m )) : t ∈ R } , and these sets define a partition of X α × M × R . The Fell continuity of the mapping( x α , m, s )
7→ D
T,f Y ( x α , m, s ), the compactness of its range, and the minimality ofthe right translation follow directly from Lemma 1.17.Suppose that ( x, s ) ∈ X × R and ( x ′ α , m ′ , s ′ ) ∈ D T,f Y ( π Y ( x ) , s ), and let { τ k } k ≥ ⊂ T and { y k } k ≥ ⊂ X α × M be sequences with y k → π Y ( x ), τ k y k → ( x ′ α , m ′ ), and f Y ( τ k , y k ) → s ′ − s . Since π Y is a distal homomorphism, it is an open onto map-ping, and the mapping y π − Y ( y ) is continuous with respect to the Hausdorffmetric d H (cf. [Ku], p. 68, Theorem 1, and p. 47). Therefore we can define a se-quence { x k ∈ π − Y ( y k ) } k ≥ ⊂ X so that x k → x , τ k x k → x ′ ∈ π − Y ( x ′ α , m ′ ), and( f Y ◦ π Y )( τ k , x k ) → s ′ − s . By equality (2), for every x ′′ ∈ π − Y ( π Y ( x ′ )) there existsa sequence { (¯ τ k , ¯ x k ) } k ≥ with ¯ x k sufficiently close to x k so that ¯ x k → x , τ k ¯ x k → x ′ ,( f Y ◦ π Y )( τ k , ¯ x k ) → s ′ − s , d (¯ τ k τ k ¯ x k , x ′′ ) < d H ( π − Y ( π Y ( τ k x k )) , π − Y ( π Y ( x ′ ))) + 2 − k ,and ( f Y ◦ π Y )(¯ τ k , τ k ¯ x k ) < − k . From d H ( π − Y ( π Y ( τ k x k )) , π − Y ( π Y ( x ′ ))) → τ k τ k ¯ x k → x ′′ and ( f Y ◦ π Y )(¯ τ k τ k , ¯ x k ) → s ′ − s , hence ( x ′′ , s ′ ) ∈ D T,f Y ◦ π Y ( x, s ).Thus the e τ f Y ◦ π Y -prolongations in X × R are exactly the pre-images of the e τ f Y -prolongations under the mapping π Y × id R . If a Fell neighbourhood of D T,f Y ◦ π Y ( x, s )is defined by the open neighbourhoods U , . . . , U k and a compact set K in X × R ,then by the openness of π Y the π Y × id R -images of these sets define a Fell neigh-bourhood of D T,f Y ( π Y ( x ) , s ). The Fell continuity of ( x, s )
7→ D
T,f Y ◦ π Y ( x, s ), thecompactness of the range, and the minimality of the right translation follow.Let b : X −→ R be the transfer function in the structure theorem. The homeo-morphism ( x, s ) ( x, s + b ( x )) on X × R defines a homeomorphism of the hyper-space with the Fell topology commuting with the right translation. Since it maps D T,f ( x, s ) onto D T,f Y ◦ π Y ( x, s + b ( x )), it is a topological isomorphism of the righttranslation actions and all properties carry over. (cid:3) Remarks . Though the compact metric flow (
X, T ) is not necessarily itself aRokhlin extension, the existence of a real-valued recurrent non-coboundary cocycle f ( τ, x ) with a non-transitive skew product extension e τ f implies the existence of a Rokhlin extension factor (
Y, T ) = π Y ( X, T ) with a non-trivial flow { φ t : t ∈ R } and a cocycle f Y ( τ, y ) of the flow ( Y, T ) with f Y ◦ π Y cohomologous to f .The surjectivity of the mapping t ( + g )( t, m ) for every m ∈ M implies that { f ( τ, x ) : τ ∈ T } is dense in R for all x in a dense G δ subset of X .For other well-known topologies on the hyperspace like the Vietoris topology andthe Hausdorff topology with respect to the product metric on X × R , the continuityof the mapping ( x, s )
7→ D
T,f ( x, s ) can be disproved by Example 1.15.2. Proof of the structure theorem
Furstenberg’s structure theorem for distal minimal flows shall be our main toolfor studying the structure of cocycles.
Definition 2.1.
Let X and Y be compact metric spaces, let π be a continuousmapping from of X onto Y , and let M be a compact homogeneous metric space.Suppose that ρ ( x , x ) is a continuous real valued function defined on the set R π = { ( x, x ′ ) ∈ X : π ( x ) = π ( x ′ ) } so that for every y ∈ Y the function ρ is a metric on the fibre π − ( y ) with anisometric mapping between π − ( y ) and M . Then X is called an M - bundle over Y .Now let ( X, T ) and (
Y, T ) = π ( X, T ) be compact metric flows with X an M -bundle over Y . If the function ρ satisfies ρ ( x, x ′ ) = ρ ( τ x, τ x ′ ) for all ( x, x ′ ) ∈ R π and τ ∈ T , then ( X, T ) is called an isometric extension of (
Y, T ). Fact 2.2 (Furstenberg’s structure theorem [Fu]) . Let ( X, T ) be a distal minimalcompact metric flow. Then there exist a countable ordinal η and factors ( X ξ , T ) = π ξ ( X, T ) for each ordinal ≤ ξ ≤ η with the following properties: (i) ( X η , T ) = ( X, T ) and ( X , T ) is the trivial flow. (ii) ( X ξ , T ) = π ζξ ( X ζ , T ) is a factor of ( X ζ , T ) for all ordinals ≤ ξ < ζ ≤ η . (iii) For every ordinal ≤ ξ < η the flow ( X ξ +1 , T ) is an isometric extensionof ( X ξ , T ξ ) . (iv) For a limit ordinal < ξ ≤ η the flow ( X ξ , T ) is the inverse limit of theflows { ( X ζ , T ) : 0 ≤ ζ < ξ } .A system { ( X ξ , T ) : 0 ≤ ξ ≤ η } with the properties above is called a quasi-isometricsystem or I -system. Definition 2.3. An I -system { ( X ξ , T ) : 0 ≤ ξ ≤ η } is called normal if for eachordinal 0 ≤ ξ < η the flow ( X ξ +1 , T ) is the maximal isometric extension of ( X ξ , T )in ( X η , T ). This I -system gives the minimal ordinal η to represent the compactmetric flow ( X, T ) = ( X η , T ) (cf. [Fu], Proposition 13.1, Definitions 13.2 and 13.3).It will be essential that the fibres of all the isometric extensions are connected,with a possible exception of extension from the trivial flow to the minimal isometricflow ( X , T ). For a normal I -system this property will be ensured by results of thepaper [MaWu]. These results require the acting group to be generated by every openneighbourhood of a compact subset, and the group T is even compactly generated. Proposition 2.4.
Let { ( X ξ , T ) : 0 ≤ ξ ≤ η } be a normal I -system with a compactlygenerated group T acting minimally and distally. Then for all ordinals ≤ ξ < ζ ≤ η the extension from ( X ξ , T ) to ( X ζ , T ) has connected fibres. EAL EXTENSIONS OF DISTAL MINIMAL FLOWS 13
Proof.
In the following arguments we shall refer to terminology and results providedin the paper [MaWu]. At first suppose that 1 ≤ ξ < η is not a limit ordinal.Let S ( π ξ − ) ⊂ X denote the relativised equicontinuous structure relation of thehomomorphism π ξ − : ( X, T ) −→ ( X ξ − , T ), hence the flow ( X, T ) /S ( π ξ − ) is themaximal isometric extension ( X ξ , T ) of the flow ( X ξ − , T ) in ( X, T ). By Theorem3.7 of [MaWu] the homomorphism π ξ : ( X, T ) −→ ( X ξ , T ) has connected fibres.For a limit ordinal 1 < ξ < η and an ordinal 0 ≤ ζ < ξ , the same argument showsthe connectedness of the fibres of π ζ +1 : ( X, T ) −→ ( X ζ +1 , T ). Since( π ξ ) − ( x ξ ) = \ ≤ ζ<ξ ( π ζ +1 ) − ( π ξζ +1 ( x ξ ))holds for every x ξ ∈ X ξ , the fibre ( π ξ ) − ( x ξ ) is connected as the limit of a sequenceof connected sets in a compact metric space (cf. [Ku], p. 170, Theorem 14). The hy-pothesis follows, since for all ordinals 1 ≤ ξ < ζ ≤ η the fibres of the homomorphism π ζξ are the images under π ζ of the connected fibres of π ξ . (cid:3) We shall henceforth assume that { ( X ξ , T ) : 0 ≤ ξ ≤ η } is a normal I -system with( X η , T ) = ( X, T ). For every ordinal 1 ≤ ξ < η we shall define a projection of thecocycles of ( X, T ) to the cocycles of ( X ξ , T ) by families of probability measures,using the fact that every distal extension of compact metric flows is a so-called RIM -extension (relatively invariant measure, cf. [Gl1]). For an isometric extensionthis
RIM is unique (cf. [Gl1]), and within the I -system the RIM’s obey to anintegral decomposition formula.
Fact 2.5.
For every ordinal ≤ ξ ≤ η there exists a family of probability measures { µ ξ,x ξ : x ξ ∈ X ξ } on X so that for every x ξ ∈ X ξ and τ ∈ T holds µ ξ,x ξ ( π − ξ ( x ξ )) = 1 and µ ξ,x ξ ◦ τ − = µ ξ,τx ξ . The mapping x ξ µ ξ,x ξ is continuous with respect to the weak-* topology on C ( X ) ∗ . For a continuous function ϕ ∈ C ( X ) and ordinals ≤ ξ < ζ ≤ η wehave for all x ξ ∈ X ξ the equality that µ ξ,x ξ ( ϕ ) = Z X ζ µ ζ,y ζ ( ϕ ) d ( µ ξ,x ξ ◦ π − ζ )( y ζ ) . (4) Proof.
The proof follows the inductive construction of an invariant measure for(
X, T ) out of the unique
RIM’s of the extensions ( X ξ , T ) = π ξ +1 ξ ( X ξ +1 , T ) with0 ≤ ξ < η (cf. Chapter 12 of [Fu]). The generalisation to a relatively invariantmeasure is provided in [dVr], p. 494, where the induction process is initiated withthe family of point measures on X ξ instead of the point measure on the trivial space X . Since the I -system and the RIM’s for the isometric extensions remain fixed,the decomposition formula follows from the inductive construction. (cid:3)
Given a cocycle f ( τ, x ) of the flow ( X, T ) and an ordinal 1 ≤ ξ < η , the RIM { µ ξ,x ξ : x ξ ∈ X ξ } defines a continuous function f ξ : T × X ξ −→ R by f ξ ( τ, x ξ ) = µ ξ,x ξ ( f ( τ, · )) = Z X f ( τ, x ) dµ ξ,x ξ ( x ) . The properties of the
RIM imply for all τ, τ ′ ∈ T and x ξ ∈ X ξ that f ξ ( τ, τ ′ x ξ ) + f ξ ( τ ′ , x ξ ) = µ ξ,τ ′ x ξ ( f ( τ, · )) + µ ξ,x ξ ( f ( τ ′ , · )) == µ ξ,x ξ ( f ( τ, · ) ◦ τ ′ ) + µ ξ,x ξ ( f ( τ ′ , · )) = f ξ ( τ τ ′ , x ξ ) , therefore f ξ is a cocycle of the flow ( X ξ , T ). Furthermore, for ordinals ξ , ζ with1 ≤ ξ < ζ ≤ η and every x ξ ∈ X ξ , the integral of the cocycle ( f ζ − f ξ ◦ π ζξ )( τ, x ζ )by the measure µ ξ,x ξ ◦ π − ζ on X ζ vanishes for every τ ∈ T : Z X ξ ( f ζ − f ξ ◦ π ζξ )( τ, x ζ ) d ( µ ξ,x ξ ◦ π − ζ )( x ζ ) == Z X ξ (cid:0) µ ζ,x ζ ( f ( τ, · )) (cid:1) d ( µ ξ,x ξ ◦ π − ζ )( x ζ ) − µ ξ,π ζξ ( x ζ ) ( f ( τ, · )) = 0Since the measure µ ξ,x ξ ◦ π − ζ is supported by the connected fibre ( π ζξ ) − ( x ξ ) in X ζ , for every τ ∈ T and every x ξ ∈ X ξ the function x ζ ( f ζ − f ξ ◦ π ζξ )( τ, x ζ )assumes zero on the fibre ( π ζξ ) − ( x ξ ). This property will be essential, as well as therepresentation of extensions of distal flows by so-called regular extensions. Fact 2.6.
Let ( X, T ) be a distal minimal compact metric flow with a factor ( Y, T ) = σ ( X, T ) . Then there exist a distal minimal compact Hausdorff flow ( ˜ X, T ) with ( X, T ) = π ( ˜ X, T ) as a factor and a Hausdorff topological group G ⊂ Aut( ˜
X, T ) acting freely on ˜ X (i.e. g (˜ x ) = ˜ x for some ˜ x ∈ ˜ X implies g = G ). The group G actsstrictly transitive on the fibres ˜ σ − (˜ σ (˜ x )) = { g (˜ x ) : g ∈ G } of the homomorphism ˜ σ = σ ◦ π for every ˜ x ∈ ˜ X , and ( X, T ) is the orbit space of a subgroup H of G in ˜ X so that π is the mapping of a point in ˜ X to its H -orbit (cf. [El] [MaWu] , Proposition 1.1).For an isometric extension ( X, T ) of ( Y, T ) , the flow ( ˜ X, T ) is metric and an iso-metric extension of ( Y, T ) , with a compact metric group G and a compact subgroup H , hence called a compact metric group extension.Remark . The construction above is also called the regulariser of an extension.In [Gl2] it is verified that a compact Hausdorff flow ( ˜
X, T ) with these properties ismetrisable if and only if the extension from (
Y, T ) to (
X, T ) is isometric.Studying the skew product extensions e τ f ξ : X ξ × R −→ X ξ × R for the ordinals0 ≤ ξ ≤ η will require the following technical lemma. Lemma 2.8.
Let the minimal compact metric flow ( Z, T ) be an extension of theflow ( Y, T ) = σ ( Z, T ) and let g ( τ, y ) be a real-valued cocycle of ( Y, T ) . Let h ( τ, z ) be a real-valued cocycle of ( Z, T ) so that for every τ ∈ T and z ′ ∈ Z the image of σ − ( σ ( z ′ )) under the function z h ( τ, z ) is connected and includes zero. Supposethat there exist a compact symmetric neighbourhood K ⊂ R of and ε > so thatfor all z ∈ Z and τ ∈ T with d Z ( z, τ z ) < ε holds ( g ◦ σ + h )( τ, z ) / ∈ K \ K .Suppose that there exists a δ > so that for all τ ∈ T and z ∈ Z with d Z ( z, τ z ) <δ holds d Z ( z ′ , τ z ′ ) < ε for every z ′ ∈ σ − ( σ ( z )) . Given ¯ z ∈ Z and a sequence { ¯ τ k } ⊂ T so that ¯ τ k ¯ z converges and ( g ◦ σ )(¯ τ k , ¯ z ) → as k → ∞ , the sequence { ( g ◦ σ + h )(¯ τ k , ¯ z ) } k ≥ is bounded. Similarly, for a sequence { ¯ τ k } ⊂ T so that ¯ τ k ¯ z converges and ( g ◦ σ + h )(¯ τ k , ¯ z ) → , the sequence { ( g ◦ σ )(¯ τ k , ¯ z ) } k ≥ is bounded.Proof. There exists a k ≥ k, k ′ ≥ k holds d Z (¯ τ k ¯ z, ¯ τ k ′ ¯ z ) < δ and( g ◦ σ )(¯ τ k ′ , ¯ z ) − ( g ◦ σ )(¯ τ k , ¯ z ) = ( g ◦ σ )(¯ τ k ′ ¯ τ − k , ¯ τ k ¯ z ) ∈ K . By the choice of K , ε , and δ follows that ( g ◦ σ + h )(¯ τ k ′ ¯ τ − k , z ) / ∈ K \ K for all z ∈ σ − ( σ (¯ τ k ¯ z )). Since the range of ( g ◦ σ + h )(¯ τ k ′ ¯ τ − k , z ) on the fibre σ − ( σ (¯ τ k ¯ z )) EAL EXTENSIONS OF DISTAL MINIMAL FLOWS 15 is connected and intersects K , we can conclude that ( g ◦ σ + h )(¯ τ k ′ ¯ τ − k , ¯ τ k ¯ z ) ∈ K for all k, k ′ ≥ k . Therefore the sequence { ( g ◦ σ + h )(¯ τ k , ¯ z ) } k ≥ is bounded.Provided a sequence { ¯ τ k } ⊂ T with convergent ¯ τ k ¯ z and ( g ◦ σ + h )(¯ τ k , ¯ z ) → k ≥ k, k ′ ≥ k holds d Z (¯ τ k ¯ z, ¯ τ k ′ ¯ z ) < δ and( g ◦ σ + h )(¯ τ k ′ ¯ τ − k , ¯ τ k ¯ z ) ∈ K . We conclude as above that ( g ◦ σ + h )(¯ τ k ′ ¯ τ − k , z ) ∈ K for all k, k ′ ≥ k and z ∈ σ − ( σ (¯ τ k ¯ z )). Since h (¯ τ k ′ ¯ τ − k , z ) = 0 for some z ∈ σ − ( σ (¯ τ k ¯ z )), the sequence { ( g ◦ σ )(¯ τ k , ¯ z ) } k ≥ is bounded. (cid:3) At first the step from an ordinal to its successor by an isometric extension shallbe considered. The “local” behaviour within the fibres of a compact group extensionis similar to a skew product extension by a compact metric group, even if the globalstructure might be different since it does not necessarily split into a product.
Lemma 2.9.
Let γ be an ordinal with ≤ γ < η . If there exists a sequence { ( τ k , x k ) } k ≥ ⊂ T × X γ +1 with d γ +1 ( x k , τ k x k ) → so that ( f γ ◦ π γ +1 γ )( τ k , x k ) → and f γ +1 ( τ k , x k ) as k → ∞ (or equivalently ( f γ +1 ◦ π γ +1 γ )( τ k , x k ) and f γ ( τ k , x k ) → ), then the skew product e τ f γ +1 is necessarily point transitive.Therefore, if f γ ( τ, x γ ) is transient, then f γ +1 ( τ, x γ +1 ) is either transient or theskew product e τ f γ +1 is point transitive.Proof. Suppose that e τ f γ +1 is not point transitive and let G ⊂ Aut(
Z, T ) define acompact metric group extension of ( X γ , T ) with ( X γ +1 , T ) = π ( Z, T ). Then theskew product extension e τ f γ +1 ◦ π of the flow ( Z, T ) is also not point transitive, andLemma 1.11 provides K ⊂ R and ε >
0. Since G acts uniformly equicontinuous,there exists a δ > z ∈ Z and τ ∈ T with d Z ( z, τ z ) < δ follows d Z ( k ( z ) , k ( τ z )) = d Z ( k ( z ) , τ k ( z )) < ε for all k ∈ K . For every z ∈ Z the G -orbit of z is all of the fibre ( π γ +1 γ ◦ π ) − (( π γ +1 γ ◦ π )( z )). Since the π γ +1 γ -fibres are connected,for every τ ∈ T and z ′ ∈ Z the range of ( f γ +1 − f γ ◦ π γ +1 γ )( τ, π ( z )) on the fibre( π γ +1 γ ◦ π ) − (( π γ +1 γ ◦ π )( z ′ )) is connected and contains zero. Hence Lemma 2.8 applieswith ( Y, T ) = ( X γ , T ), σ = π γ +1 γ ◦ π , g = f γ , and h ( τ, z ) = ( f γ +1 − f γ ◦ π γ +1 γ )( τ, π ( z )).However, given the sequence { ( τ k , x k ) } k ≥ ⊂ T × X γ +1 in the hypothesis, Lemma1.8 provides a point ¯ x ∈ X γ +1 and a sequence { ¯ τ k } ⊂ T so that ( f γ ◦ π γ +1 γ )(¯ τ k , ¯ x ) → f γ +1 (¯ τ k , ¯ x ) → ∞ (or ( f γ ◦ π γ +1 γ )(¯ τ k , ¯ x ) → ∞ and f γ +1 (¯ τ k , ¯ x ) → z ∈ Z with π (¯ z ) = ¯ x and changing to a subsequence of { ¯ τ k } ⊂ T with ¯ τ k ¯ z convergent, this contradicts to Lemma 2.8.Now suppose that f γ ( τ, x γ ) is transient and f γ +1 ( τ, x γ +1 ) is recurrent. Let x ′ ∈ X γ +1 be so that ( x ′ ,
0) is e τ f γ +1 -recurrent (cf. Remarks 1.4). Since ( x ′ ,
0) is cannotbe e τ f γ ◦ π γ +1 γ -recurrent, there exist a neighbourhood V ⊂ X γ +1 × R of ( x ′ ,
0) anda replete semigroup P ⊂ T so that e τ f γ ◦ π γ +1 γ ( x ′ , / ∈ V for every τ ∈ P . Givenan arbitrary compact set C ⊂ T , by Theorem 6.32 in [GoHe] there exists a repletesemigroup Q ⊂ P \ C . Since ( x ′ ,
0) is e τ f γ +1 -recurrent, we can inductively construct asequence { τ k } k ≥ ⊂ P with τ k x ′ → x ′ , f γ +1 ( τ k , x ′ ) →
0, and ( f γ ◦ π γ +1 γ )( τ k , x ′ ) e τ f γ +1 follows by the preceding statement. (cid:3) Furthermore, we shall study the case of transfinite induction to a limit ordinal.The arguments are quite similar, however with an approximation of a limit ordinalinstead of an isometric group extension.
Lemma 2.10.
Suppose that γ is a limit ordinal with < γ ≤ η . (i) If for every ordinal < α < γ there exists an ordinal α ≤ ξ < γ so that f ξ ( τ, x ξ ) has a point transitive skew product extension, then f γ ( τ, x γ ) hasa point transitive skew product extension. (ii) If there exist an ordinal ≤ α < γ and a sequence { ( τ k , x k ) } k ≥ ⊂ T × X γ with d γ ( x k , τ k x k ) → so that ( f ξ ◦ π γξ )( τ k , x k ) → for every α ≤ ξ < γ and f γ ( τ k , x k ) as k → ∞ (or equivalently ( f ξ ◦ π γξ )( τ k , x k ) for every α ≤ ξ < γ and f γ ( τ k , x k ) → ), then e τ f γ is necessarily point transitive. (iii) If there exists an ordinal ≤ α < γ so that for all α ≤ ξ < γ the cocycle f ξ ( τ, x ξ ) is transient, then f γ ( τ, x γ ) is either transient or its skew productextension is point transitive.Proof. Suppose that the skew product of τ f γ on X γ × R is not point transitive, andlet K ⊂ R and ε > γ is a limit ordinal and( X γ , T ) is the inverse limit of the flows { ( X ξ , T ) : 0 ≤ ξ < γ } , we can choose anordinal ζ < γ so that for all x, x ′ ∈ X γ with π γζ ( x ) = π γζ ( x ′ ) holds d γ ( x, x ′ ) < ε/ δ = ε/
3, then d γ ( x ′ , τ x ′ ) < δ for x ′ ∈ X γ and τ ∈ T implies d γ ( x, τ x ) < ε for all x ∈ ( π γζ ) − ( π γζ ( x ′ )). These conditions remain valid even if the ordinal ζ willbe increased later. Since the π γζ -fibres are connected, for every τ ∈ T and x ′ γ ∈ X γ the range of ( f γ − f ζ ◦ π γζ )( τ, x γ ) on the π γζ -fibre of x ′ γ is connected and contains 0.Under the hypothesis (i), we can choose ( τ, x ζ ) ∈ T × X ζ so that f ζ ( τ, x ζ ) ∈ K \ K and d γ ( x ′ γ , τ x ′ γ ) < δ for some x ′ γ ∈ ( π γζ ) − ( x ζ ). Thus d γ ( x γ , τ x γ ) < ε holdsfor all x γ ∈ ( π γζ ) − ( x ζ ), and for x γ with ( f γ − f ζ ◦ π γζ )( τ, x γ ) = 0 this contradicts f γ ( τ, x γ ) / ∈ K \ K . Thus assertion (i) is verified.We apply then Lemma 2.8 with ( Z, T ) = ( X γ , T ), ( Y, T ) = ( X ζ , T ), σ = π γζ , h = ( f γ − f ζ ◦ π γζ ), and g = f ζ . However, given the sequence { ( τ k , x k ) } k ≥ ⊂ T × X γ in hypothesis (ii), Lemma 1.8 provides a point ¯ x ∈ X γ and a sequence { ¯ τ k } ⊂ T so that ( f ζ ◦ π γζ , f γ − f ζ ◦ π γζ )(¯ τ k , ¯ x ) = ( g ◦ σ, h )(¯ τ k , ¯ x ) → (0 , ∞ ) (or ( ∞ , τ k ¯ x → ¯ z as k → ∞ . This is a contradiction to Lemma 2.8 and verifies (ii).Now suppose that f ζ ( τ, x ζ ) is transient and f γ ( τ, x γ ) is recurrent, and choose x ′ ∈ X γ so that ( x ′ ,
0) is e τ f γ -recurrent. Since ( x ′ ,
0) is cannot be e τ f ζ ◦ π γζ -recurrent,there exist a neighbourhood V ⊂ X γ × R of ( x ′ ,
0) and a replete semigroup P ⊂ T with e τ f ζ ◦ π γζ ( x ′ , / ∈ V for every τ ∈ P . By induction exists a sequence { τ k } k ≥ ⊂ P with g ( τ k ) f γ ( x ′ , → ( x ′ ,
0) as k → ∞ , and by Lemma 1.8 there exist a point ¯ x ∈ X γ and sequence { ¯ τ k } ⊂ T so that ( f γ , f ζ ◦ π γζ )(¯ τ k , ¯ x ) = ( g ◦ σ + h, g ◦ σ )(¯ τ k , ¯ x ) → (0 , ∞ )and ¯ τ k ¯ x → ¯ x . This contradiction to Lemma 2.8 verifies the statement (iii). (cid:3) Proposition 2.11.
If the real-valued cocycle f ( τ, x ) is topologically recurrent apartfrom a coboundary, then there exists a maximal ordinal ≤ α ≤ η so that the skewproduct extension e τ f α is point transitive on X α × R . The cocycle ( f − f α ◦ π α )( τ, x ) is relatively trivial with respect to ( f α ◦ π α )( τ, x ) .Proof. Let us first suppose that the cocycle f ξ ( τ, x ξ ) is recurrent for every ordinal1 ≤ ξ < η , and let O = { ≤ ξ ≤ η : f ξ ( τ, x ξ ) is not a coboundary } . This set is non-empty since f η ( τ, x ) is not a coboundary, and let β be its minimal element. If β = 1,then by Proposition 1.12 the recurrent skew product e τ f of the isometric flow ( X , T )is point transitive. If β >
1, then Fact 1.10 provides a sequence { ( τ k , x k ) } k ≥ ⊂ T × X β with d β ( x k , τ k x k ) → f β ( τ k , x k ) → ∞ . For all 1 ≤ ζ < β holds( f β ◦ π βζ )( τ k , x k ) →
0, and by the Lemmas 2.9 and 2.10 (ii) e τ f β is point transitive. EAL EXTENSIONS OF DISTAL MINIMAL FLOWS 17 If f ξ ( τ, x ξ ) is transient for an ordinal 1 ≤ ξ < η , then let β be the minimalelement of the set O = { ξ < ζ ≤ η : f ζ ( τ, x ζ ) is topologically recurrent } . Thisset is non-empty since f η ( τ, x η ) is topologically recurrent, and it follows from theLemmas 2.9 and 2.10 (iii) that e τ f β is even point transitive.Now let O = { ≤ ξ ≤ η : e τ f ζ is not point transitive for all ξ ≤ ζ ≤ η } . If O isempty, then e τ f η is point transitive and α = η . Otherwise, the set O has a minimalelement γ > e τ f β is point transitive for some 1 ≤ β ≤ η . Since γ cannotbe a limit ordinal by Lemma 2.10 (i), there exists a maximal ordinal α ≥ e τ f α . If { ( τ k , x k ) } k ≥ ⊂ T × X is a sequence with d ( x k , τ k x k ) → f α ◦ π α )( τ k , x k ) →
0, then transfinite induction using the maximality of α andLemmas 2.9, 2.10 (ii) verifies that ( f ξ ◦ π ξ )( τ k , x k ) → α ≤ ξ ≤ η . (cid:3) After the flow ( X α , T ) with a point transitive skew product extension e τ f α hasbeen identified, we shall study the extension from ( X α , T ) to ( X, T ). There mightbe infinitely many isometric extensions in between, and therefore this extensionis in general a distal extension. Since our construction will use the regulariser ofthis extension, it is necessary to leave the category of compact metric flows for thecategory of compact Hausdorff flows during the following construction (cf. Remark2.7). However, the flow which will be constructed by means of the regulariser willbe metric as a factor of the compact metric flow (
X, T ). Proposition 2.12.
There exists a factor ( Y, T ) = ( X α × M, T ) = π Y ( X, T ) whichis a Rokhlin extension of ( X α , T ) = ρ α ( Y, T ) by a distal minimal compact metric R -flow ( M, { φ t : t ∈ R } ) and the cocycle f α ( τ, x α ) so that for every x ∈ X holds π − Y ( π Y ( x )) × { } = D T,f α ◦ π α ( x, ∩ ( π − α ( π α ( x )) × { } ) . (5) The R -flow { ψ t : t ∈ R } ⊂ Aut(
Y, T ) defined by ψ t ( x α , m ) = ( x α , φ t ( m )) for ( x α , m ) ∈ Y = X α × M fulfils for every y ∈ Y and every t ∈ R that ¯ O T,f α ◦ ρ α ( y, ∩ ( ρ − α ( ρ α ( y )) × { t } ) ⊂ { ( ψ t ( y ) , t ) } , (6) with coincidence of these sets if ( ρ α ( y ) , ∈ X α × R is transitive for e τ f α .Proof. We shall construct a factor (
Y, T ) of (
X, T ) and a flow { ϕ t : t ∈ R } ⊂ Aut(
Y, T ), and then we shall represent (
Y, T ) as a Rokhlin extension of ( X α , T ).Let ( ˜ X, T ) be a distal minimal compact Hausdorff flow with (
X, T ) = π ( ˜ X, T ) anda Hausdorff topological group G ⊂ Aut( ˜
X, T ) acting freely on the fibres of π α ◦ π so that ( X, T ) is the H -orbit space of a subgroup H ⊂ G (cf. Fact 2.6). For anarbitrary point ˜ z ∈ ˜ X and t ∈ R we define a closed subset of G by G ˜ z,t = { g ∈ G : ( π ( g (˜ z )) , t ) ∈ D T,f α ◦ π α ( π (˜ z ) , } . (7)The mapping π is open as a homomorphism of distal minimal compact flows, andhence for every g ∈ G ˜ z,t there exist nets { ˜ z i } i ∈ I ⊂ ˜ X and { τ i } i ∈ I ⊂ T with ˜ z i → ˜ z , τ i π (˜ z i ) → π ( g (˜ z )), and f α ( τ i , π α ◦ π (˜ z i )) → t . Since the cocycle ( f α ◦ π α )( τ, x α ) isconstant on the fibres of π α and T is Abelian, it follows for every fixed τ ∈ T that τ i π ( τ ˜ z i ) = τ i τ π (˜ z i ) = τ τ i π (˜ z i ) → τ π ( g (˜ z )) = π ( τ g (˜ z )) = π ( g ( τ ˜ z ))and by the cocycle identity f α ( τ i , π α ◦ π ( τ ˜ z i )) = f α ( τ i , π α ◦ π (˜ z i )) − f α ( τ, π α ◦ π (˜ z i ))+ f α ( τ, π α ◦ π ( τ i ˜ z i )) → t. By the density of the T -orbit of ˜ z and a diagonalisation of nets there exist for every˜ x ∈ ˜ X nets { ˜ x i } i ∈ I ⊂ ˜ X and { τ ′ i } i ∈ I ⊂ T with ˜ x i → ˜ x , τ ′ i π (˜ x i ) → π ( g (˜ x )), and f α ( τ i , π α ◦ π (˜ x i )) → t . Therefore( π ( g (˜ x )) , t ) ∈ D T,f α ◦ π α ( π (˜ x ) , g ∈ G ˜ x,t = G ˜ z,t = G t . By symmetry follows now that G − t = ( G t ) − .Then we fix a point x ′ ∈ X with ¯ O T,f α ( π α ( x ′ ) ,
0) = X α × R and D T,f α ◦ π α ( x ′ ,
0) =¯ O T,f α ◦ π α ( x ′ ,
0) (cf. Fact 1.1). The set G t is non-empty for every t ∈ R , since¯ O T,f α ( π α ( x ′ ) ,
0) = X α × R and the compactness of X ensure that¯ O T,f α ◦ π α ( x ′ , ∩ π − α ( π α ( x ′ )) × { t } 6 = ∅ . For arbitrary t, t ′ ∈ R and g ∈ G t , g ′ ∈ G t ′ , we select ˜ x, ˜ z ∈ ˜ X so that π (˜ x ) = x ′ and ˜ x = g ′ (˜ z ). Then we have( x ′ , t ′ ) = ( π ( g ′ (˜ z )) , t ′ ) ∈ D T,f α ◦ π α ( π (˜ z ) , , and for ˜ y = g (˜ x ) = gg ′ (˜ z ) it holds that ( π (˜ y ) , t ) ∈ ¯ O T,f α ◦ π α ( x ′ ,
0) = D T,f α ◦ π α ( x ′ , π (˜ y ) , t + t ′ ) ∈ D T,f α ◦ π α ( π (˜ z ) ,
0) so that gg ′ ∈ G t + t ′ . Hence G t G t ′ ⊂ G t + t ′ holds for all t, t ′ ∈ R , and from G − t = ( G t ) − follows ( G t ) − G t + t ′ = G − t G t + t ′ ⊂ G t ′ so that G t G t ′ = G t + t ′ . Thus the Hausdorff topological group˜ G = ∪ t ∈ R G t has the closed set G ⊃ H as a normal subgroup so that G t is a G -coset in ˜ G for every t ∈ R . Moreover, the mapping t G t is a group homomorphism from R into ˜ G/G . The group G is not necessarily compact, however its orbit space on ˜ X defines a partition into sets invariant under H ⊂ G . Hence this is also a partitionof X , and the equivalence relation R Y of this partition of X is T -invariant since G ⊂ Aut( ˜
X, T ). Moreover, R Y is closed in X , since definition (7) implies that( x, x ′ ) ∈ R Y if and only if ( x ′ , ∈ D T,f α ◦ π α ( x, ∩ ( π − α ( π α ( x )) × { } ). Thefactor ( Y, T ) = π Y ( X, T ) defined by the T -invariant closed equivalence relation R Y is an extension of ( X α , T ) = ρ α ( Y, T ), and equality (5) follows. The R -action { ϕ t : t ∈ R } ⊂ Aut(
Y, T ) is well defined for every y ∈ Y and t ∈ R by ϕ t ( y ) = G t (( π Y ◦ π ) − ( y )) = G t ( { ˜ x ∈ ˜ X : G (˜ x ) = y } ) . Let { ( t k , y k ) } k ≥ ⊂ R × Y be a sequence with ( t k , y k ) → ( t, y ), then ϕ t k ( y k ) = G g k (˜ x k ) for a sequence { ˜ x k } k ≥ ⊂ ˜ X with π Y ◦ π (˜ x k ) = y k and g k ∈ G t k . Wecan assume that ˜ x k → ˜ x and g k (˜ x k ) → ˜ z so that ( π (˜ z ) , t ) ∈ D T,f α ◦ π α ( π (˜ x ) ,
0) and˜ z = g t (˜ x ) for some g t ∈ G t . From π Y ◦ π (˜ x ) = y and ϕ t k ( y k ) = π Y ◦ π ( g k (˜ x k )) → π Y ◦ π (˜ z ) = ϕ t ( y ) follows the continuity of the action { ϕ t : t ∈ R } on Y .We turn to the inclusion (6). Suppose that ( y i , t ) ∈ ¯ O T,f α ◦ ρ α ( y, ∩ ρ − α ( x α ) ×{ t } for some x α ∈ X α and i ∈ { , } , and select x ∈ π − Y ( y ). By the compactness of X there exist points x i ∈ π − Y ( y i ) ⊂ π − α ( x α ) so that ( x i , t ) ∈ ¯ O T,f α ◦ π α ( x, x , ∈ D T,f α ◦ π α ( x , y = π Y ( x ) = π Y ( x ) = y , and thus for every y ∈ Y and t ∈ R holdscard { ¯ O T,f α ◦ ρ α ( y, ∩ ρ − α ( ρ α ( y )) × { t }} ≤ . (8)Moreover, for x α = ρ α ( y ) follows x = π ( g t (˜ x )) with g t ∈ G t and ˜ x ∈ π − ( x ) ⊂ ˜ X .Hence y = π Y ( x ) = ϕ t ( y ) and inclusion (6) is verified. For e τ f α -transitive ( ρ α ( y ) , y ∈ Y and t ∈ R , and for y ′ ∈ ρ − α ( ρ α ( y ))and x α ∈ X α holds ¯ O T,f α ◦ ρ α ( y ′ , ∩ ρ − α ( x α ) ×{ } = { ( y , } . We fix a point ¯ x ∈ X α EAL EXTENSIONS OF DISTAL MINIMAL FLOWS 19 with e τ f α -transitive (¯ x, y , ∈ D T,f α ◦ ρ α ( y ′ , ∩ ρ − α ( x α ) × { } , then Remark1.2 implies that ( y , ∈ D T,f α ◦ ρ α ( y , y = y . Hence¯ O T,f α ◦ ρ α ( y ′ , ∩ ρ − α ( x α ) × { } = D T,f α ◦ ρ α ( y ′ , ∩ ρ − α ( x α ) × { } (9)holds for every y ′ ∈ ρ − α (¯ x ) and x α ∈ X α . For distinct y ′ , y ′′ ∈ ρ − α (¯ x ) follows¯ O T,f α ◦ ρ α ( y ′ , ∩ ¯ O T,f α ◦ ρ α ( y ′′ , ∩ ρ − α ( x α ) × { } = ∅ . Indeed, given a point ¯ y in this intersection, for every sequence { τ k } k ≥ ⊂ T with τ k ¯ x → ρ α (¯ y ) and f α ( τ k , ¯ x ) → d Y ( τ k y ′ , τ k y ′′ ) →
0, incontradiction to the distality of (
Y, T ). Hence the mapping ι : X α × ρ − α (¯ x ) −→ Y ( x α , y ′ ) ρ − α ( x α ) ∩ { y ∈ Y : ( y, ∈ ¯ O T,f α ◦ ρ α ( y ′ , } is well-defined, one-to-one, and by equality (9) also continuous. For a dense setof points ¯ y ∈ Y holds the e τ f α -transitivity of ( ρ α (¯ y ) , ρ α is open. Wecan conclude for every y ∈ Y that D T,f α ◦ ρ α ( y, ∩ ρ − α (¯ x ) × { } 6 = ∅ , and thus D T,f α ◦ ρ α ( y ′ , ∩ { y } × { } = ¯ O T,f α ◦ ρ α ( y ′ , ∩ { y } × { } 6 = ∅ for some y ′ ∈ ρ − α (¯ x ).Hence ι is onto and by compactness Y and X α × ρ − α (¯ x ) are homeomorphic.Let { φ t : t ∈ R } be the restriction of { ϕ t : t ∈ R } to the { ϕ t : t ∈ R } -invariant compact metric space M = ρ − α (¯ x ). For every y ′ ∈ M and τ ∈ T holds ¯ O T,f α ◦ ρ α ( y ′ , ∩ ρ − α (¯ x ) × {− f α ( τ, ¯ x ) } = { ( φ − f α ( τ, ¯ x ) ( y ′ ) , − f α ( τ, ¯ x )) } and e τ f α ◦ π Y ( φ − f α ( τ, ¯ x ) ( y ′ ) , − f α ( τ, ¯ x )) ∈ ρ − α ( τ ¯ x ) ∩ { y ∈ Y : ( y, ∈ ¯ O T,f α ◦ ρ α ( y ′ , } .Therefore τ φ − f α ( τ, ¯ x ) ( y ′ ) = ι ( τ ¯ x, y ′ ) and τ y = ι ( τ φ,f α (¯ x, y )) for every y ∈ M and τ ∈ T . The minimality of ( Y, T ) implies that ( X α × M, T ) and (
Y, T ) are topologi-cally isomorphic via ι . Moreover, for the mapping ψ t ( x α , m ) = ( x α , φ t ( m )) with ψ ∈ Aut( X α × M, T ) and every m ∈ M = ρ − α (¯ x ) and t ∈ R holds ϕ t ( ι ( τ φ,f α (¯ x, m ))) = ϕ t ( τ m ) = τ ϕ t ( m ) = ι ( τ φ,f α (¯ x, φ t ( m ))) = ι ( ψ t ( τ φ,f α (¯ x, m ))). By the minimality of( X α × M, T ) follows ψ t = ι − ◦ ϕ t ◦ ι for every t ∈ R . The flow ( M, { φ t : t ∈ R } )is minimal and distal, since a non-transitive point m ′ ∈ M and a proximal pair( m ′ , m ′′ ) ∈ M , respectively, give rise a non-transitive point ( x α , m ′ ) ∈ Y and aproximal pair (( x α , m ′ ) , ( x α , m ′′ )) ∈ Y , respectively. (cid:3) It should be mentioned that an ordinal ξ ≤ η with ( Y, T ) = ( X ξ , T ) does notnecessarily exist. Therefore we shall define a cocycle f Y : T × Y −→ R independentlyof the cocycles f ξ ( τ, x ξ ), and it will turn out that ( f Y ◦ π Y )( τ, x ) can be chosentopologically cohomologous to f . Proposition 2.13.
There exists a topological cocycle f Y ( τ, y ) of the flow ( Y, T ) sothat ( f Y ◦ π Y )( τ, x ) is topologically cohomologous to f ( τ, x ) and f Y ( τ, y ) is relativelytrivial with respect to ( f α ◦ ρ α )( τ, y ) . We shall prove another technical lemma first.
Lemma 2.14.
Let ( Z, T ) be a distal minimal compact metric flow which extends ( X α , T ) = σ α ( Z, T ) , and let G ⊂ Aut(
Z, T ) be a Hausdorff topological group pre-serving the fibres of σ α . Suppose that there exists a continuous group homomorphism ϕ : G −→ R so that for every g ∈ G and every z ∈ Z it holds that ( g ( z ) , ϕ ( g )) ∈ D T,f α ◦ σ α ( z, . Furthermore, suppose that h ( τ, z ) is a real-valued cocycle of ( Z, T ) which is rela-tively trivial with respect to ( f α ◦ σ α )( τ, z ) . Then there exists a continuous cocycle ¯ h (( τ, g ) , z ) of the flow ( Z, T × G ) with the action { g ◦ τ : ( τ, g ) ∈ T × G } so that h ( τ, z ) = ¯ h (( τ, G ) , z ) holds for every ( τ, z ) ∈ T × Z and the mapping h ¯ h islinear. For z ∈ Z , g ∈ G , and a sequence { ( τ k , z k ) } k ≥ ⊂ T × Z with z k → z , τ k z k → g ( z ) , and ( f α ◦ ρ α )( τ k , z k ) → ϕ ( g ) holds h ( τ k , z k ) → ¯ h (( T , g ) , z ) as k → ∞ . (10) Proof.
We put F = ( f α ◦ σ α , h ) : T × Z −→ R and fix a point ¯ z ∈ Z so that¯ O T,F (¯ z, ,
0) and D T,F (¯ z, ,
0) coincide in Z × ( R ∞ ) (cf. Fact 1.1). For every g ∈ G we fix a sequence { τ gk } k ≥ ⊂ T with g ( τ gk ) f α ◦ σ α (¯ z, → ( g (¯ z ) , ϕ ( g )) as k → ∞ , with { τ G k = T } k ≥ . Since g ∈ Aut(
Z, T ) and f α ◦ σ α ◦ g = f α ◦ σ α , we can conclude forevery t ∈ T that τ gk t ¯ z = tτ gk ¯ z → tg (¯ z ) = g ( t ¯ z ) as k → ∞ as well as( f α ◦ σ α )( τ gk , t ¯ z ) = ( f α ◦ σ α )( t, τ gk ¯ z ) + ( f α ◦ σ α )( τ gk , ¯ z ) − ( f α ◦ σ α )( t, ¯ z ) → ϕ ( g ) . (11)By the relative triviality of h ( τ, z ) with respect to ( f α ◦ σ α )( τ, z ), the sequence { h ( τ τ gk , t ¯ z ) } k ≥ converges for all τ, t ∈ T . Thus we can put¯ h (( τ, g ) , t ¯ z ) = lim k →∞ h ( τ τ gk , t ¯ z ) = h ( τ, g ( t ¯ z )) + lim k →∞ h ( τ gk , t ¯ z ) (12)for every ( τ, g, t ¯ z ) ∈ T × G × Z . Suppose that there exist sequences { ( τ ik , z ik ) } k ≥ ⊂ T × Z for i = 1 , z ik → z , τ ik z ik → g ( z ) = lim k →∞ g ( z ik ), and( f α ◦ σ α )( τ ik , z ik ) → ϕ ( g ) as k → ∞ , while for i = 1 , h i = lim k →∞ h ( τ ik , z ik ) ∈ R ∞ are either distinctor both equal to ∞ . Then ( g ( z ) , ϕ ( g ) , ¯ h i ) ∈ D T,F ( z, ,
0) for i = 1 ,
2, and for every τ ′ ∈ T follows from g ∈ Aut(
Z, T ) and the cocycle identity that( g ( τ ′ z ) , ϕ ( g ) + ( f α ◦ σ α )( τ ′ , g ( z )) − ( f α ◦ σ α )( τ ′ , z ) , h ( τ ′ , g ( z )) + ¯ h i − h ( τ ′ , z )) =( g ( τ ′ z ) , ϕ ( g ) , h ( τ ′ , g ( z )) + ¯ h i − h ( τ ′ , z )) ∈ D T,F ( τ ′ z, , . Since ¯ O T ( z ) = Z , either there are distinct points a , a ∈ R ∞ with ( g (¯ z ) , ϕ ( g ) , a i ) ∈D T,F (¯ z, ,
0) or it holds that ( g (¯ z ) , ϕ ( g ) , ∞ ) ∈ D T,F (¯ z, , O T,F (¯ z, ,
0) = D T,F (¯ z, ,
0) in Z × ( R ∞ ) , this contradicts to the relative trivialityof h ( τ, z ) with respect to ( f α ◦ σ α )( τ, z ). Therefore equality (10) holds true, andthe definition (12) extends uniquely from the T -orbit of ¯ z to a continuous mapping¯ h : T × G × Z −→ R since the action of T × G on X and ϕ are continuous.For the cocycle identity let ( τ , g ) , ( τ , g ) ∈ T × G be arbitrary with sequences { τ g k } k ≥ , { τ g k } k ≥ ⊂ T . By equality (11) we select a sequence { k l } l ≥ ⊂ N with τ g k l τ g l ¯ z = τ g l τ g k l ¯ z → g ( g (¯ z )) and ( f α ◦ σ α )( τ g l τ g k l , ¯ z ) → ϕ ( g )+ ϕ ( g ) = ϕ ( g g )as l → ∞ . Thus we can put { τ g g l } l ≥ = { τ g l τ g k l } l ≥ , and for every t ∈ T theequality (11) implies that τ g k l τ t ¯ z → g ( τ t ¯ z ), τ g l τ g k l τ t ¯ z → ( g g )( τ t ¯ z ), and( f α ◦ σ α )( τ τ g l , τ g k l τ t ¯ z ) =( f α ◦ σ α )( τ , τ g l τ g k l τ t ¯ z ) + ( f α ◦ σ α )( τ g l τ g k l , τ t ¯ z ) − ( f α ◦ σ α )( τ g k l , τ t ¯ z ) → ( f α ◦ σ α )( τ , g ( τ t ¯ z )) + ϕ ( g g ) − ϕ ( g ) as l → ∞ . The uniqueness according to equality (10) verifies that ¯ h (( τ , g g g − ) , g ( τ t ¯ z )) =lim l →∞ h ( τ τ g l , τ τ g k l t ¯ z ), and therefore¯ h (( τ , g g g − ) , g ( τ t ¯ z )) + ¯ h (( τ , g ) , t ¯ z ) == lim l →∞ h ( τ τ g l , τ τ g k l t ¯ z ) + lim l →∞ h ( τ τ g k l , t ¯ z ) == lim l →∞ h ( τ τ g l τ τ g k l , t ¯ z ) = ¯ h (( τ τ , g g ) , t ¯ z ) . EAL EXTENSIONS OF DISTAL MINIMAL FLOWS 21
We substitute g − g g for g and obtain from ¯ O T (¯ z ) = Z and the continuity of ¯ h that cocycle identity is valid. (cid:3) Proof of Proposition 2.13.
Let ( Y c , T ) = π c ( X, T ) be the flow defined by the con-nected components of the fibres of π Y (cf. [MaWu], Definition 2.2), and let ρ be thehomomorphism from ( Y c , T ) onto ( Y, T ) = ρ ( Y c , T ). With a RIM { µ c,y : y ∈ Y c } forthe distal extension ( Y c , T ) = π c ( X, T ) we define a cocycle f c ( τ, y ) = µ c,y ( f ( τ, · )) forevery ( τ, y ) ∈ T × Y c . We fix a point ¯ x ∈ X with D T,f α ◦ π α ( τ ¯ x,
0) = ¯ O T,f α ◦ π α ( τ ¯ x, τ ∈ T . By equality (5) and π − c ( π c ( τ ¯ x )) ⊂ π − Y ( π Y ( τ ¯ x )) holds for all τ ∈ T ¯ O T,f α ◦ π α (¯ x, ∩ ( π − c ( π c ( τ ¯ x )) × R ) = π − c ( π c ( τ ¯ x )) × { ( f α ◦ π α )( τ, ¯ x ) } . Let F ( τ, x ) be the R -valued cocycle ( f α ◦ π α , f ). We shall verify that¯ O T,F (¯ x, , ∩ ( π − c ( π c ( τ ¯ x )) × { ( f α ◦ π α )( τ, ¯ x ) } × R ) = { ( x, ( f α ◦ π α )( τ, ¯ x ) , b τ ( x )) : x ∈ π − c ( π c ( τ ¯ x )) } (13)for every τ ∈ T , in which b τ : π − c ( π c ( τ ¯ x )) −→ R is a continuous function. Indeed,for a sequence { τ k } k ≥ ⊂ T with τ k τ ¯ x → x ∈ π − c ( π c ( τ ¯ x )) and ( f α ◦ π α )( τ k , τ ¯ x ) → f − f α ◦ π α )( τ, x ) the existence and uniquenessof the limit b τ ( x ) of f ( τ k , τ ¯ x ). It also follows that for every ε > δ > τ ∈ T and x, x ′ ∈ π − c ( π c ( τ ¯ x )) with d ( x, x ′ ) < δ holds | b τ ( x ) − b τ ( x ′ ) | < ε . Since the fibres of π c are connected, a covering of X by δ -neighbourhoods provides a constant D > | b τ ( x ) − b τ ( x ′ ) | < D for all τ ∈ T and x, x ′ ∈ π − c ( π c ( τ ¯ x )). Equality (13) shows for x ∈ π − c ( π c (¯ x )) and τ ∈ T that b T ( x )+ f ( τ, x ) = b τ ( τ x ) and hence | f ( τ, ¯ x ) − f ( τ, x ) | ≤ D . Since ( f − f c ◦ π c )( τ, x )assumes zero on each π c -fibre, for all τ ∈ T holds | ( f − f c ◦ π c )( τ, ¯ x ) | < D so thatthis is a coboundary.Due to Theorem 3.7 in [MaWu] the extension from ( Y, T ) to ( Y c , T ) is an iso-metric extension, and by Fact 2.6 there exists a compact group extension ( ˜ Y , T ) of(
Y, T ) = ρ ( Y c , T ) by G ⊂ Aut( ˜
Y , T ) so that ( Y c , T ) = σ ( ˜ Y , T ) is the orbit spaceof a compact subgroup H ⊂ G . For every sequence { ( τ k , ˜ y k ) } k ≥ ⊂ T × ˜ Y with d ˜ Y (˜ y k , τ k ˜ y k ) → f α ◦ ρ α ◦ ρ ◦ σ )( τ k , ˜ y k ) → f c ◦ σ )( τ k , ˜ y k ) → { (˜ τ k , x k ) } k ≥ ⊂ T × X so that d ( x k , ˜ τ k x k ) →
0, ( f α ◦ π α )(˜ τ k , x k ) →
0, and ( f c ◦ π c )(˜ τ k , x k ) → ∞ , which contradictsto Proposition 2.11 and the boundedness of the transfer function between f and f c ◦ π c . We can apply Lemma 2.14 for the flow ( ˜ Y , T ), the cocycle h = f c ◦ σ , thegroup G ⊂ Aut( ˜
Y , T ), and the group homomorphism ϕ ≡
0, and we obtain a realvalued cocycle ¯ h (( τ, h ) , ˜ y ) with ¯ h (( τ, G ) , ˜ y ) = ( f c ◦ σ )( τ, ˜ y ) for every ( τ, ˜ y ) ∈ T × ˜ Y .We define a topological cocycle of ( Y, T ) by f Y ( τ, y ) = µ Y,y (( f c ◦ σ )( τ, · )), where { µ Y,y : y ∈ Y } is the RIM for the extension (
Y, T ) = ρ ◦ σ ( ˜ Y , T ). From the cocycleidentity ¯ h (( τ, g ) , ˜ y ) = ( f c ◦ σ )( τ, g (˜ y )) + ¯ h (( T , g ) , ˜ y ) = ¯ h (( T , g ) , τ ˜ y ) + ( f c ◦ σ )( τ, ˜ y )and the boundedness of ¯ h (( T , g ) , ˜ y ) for ( g, ˜ y ) ∈ G × ˜ Y we can conclude that( f c ◦ σ − f Y ◦ ρ ◦ σ )( τ, ˜ y ) is uniformly bounded and a coboundary. Hence also( f c − f Y ◦ ρ )( τ, y c ) is a coboundary, and the relative triviality of f Y ( τ, y ) withrespect to ( f α ◦ ρ α )( τ, y ) can be verified as above for the cocycle ( f c ◦ σ )( τ, ˜ y ). (cid:3) Proposition 2.15.
The cocycle ( f Y − f α ◦ ρ α )( τ, y ) of the flow ( Y, T ) can be extendedto a cocycle ¯ f (( τ, t ) , y ) of the T × R -flow ( Y, { ψ t ◦ τ : ( τ, t ) ∈ T × R } ) so that ( f Y − f α ◦ ρ α )( τ, y ) = ¯ f (( τ, , y ) for every ( τ, y ) ∈ T × Y . We put L τ ( x α , m ) = ( τ x α , m ) = ψ − f α ( τ,x α ) ( τ ( x α , m )) for ( x α , m ) ∈ X α × M = Y . Then for arbitrary ¯ x ∈ X α there exists a continuousfunction ¯ b : Y −→ R with ¯ b ( ρ − α (¯ x )) = { } so that for every ( τ, y ) ∈ T × Y holds ¯ f (( τ, − ( f α ◦ ρ α )( τ, y )) , y ) = ¯ b ( ψ − ( f α ◦ ρ α )( τ,y ) ( τ y )) − ¯ b ( y ) = ¯ b ( L τ y ) − ¯ b ( y ) , (14) and ( τ, y ) ¯ f (( τ, − ( f α ◦ ρ α )( τ, y )) , y ) is a topological coboundary of the distal flow ( Y, { L τ : τ ∈ T } ) with transfer function ¯ b . For every ( x α , m ) ∈ Y and t ∈ R holds ¯ f (( T , t ) , ( x α , m )) = ¯ f (( T , t ) , (¯ x, m )) + ¯ b ( x α , φ t ( m )) − ¯ b ( x α , m ) . (15) Proof.
Since Lemma 2.14 can be applied to the cocycle h = f Y − f α ◦ ρ α , the group G = { ψ t : t ∈ R } ⊂ Aut(
Y, T ), and the group homomorphism ϕ = id R , it providesthe cocycle ¯ f (( τ, t ) , y ). The mapping f ′ (( τ, t ) , y ) = ¯ f (( τ, t − ( f α ◦ ρ α )( τ, y )) , y ) fulfils f ′ (( τ, t ) , ψ t ′ ( L τ ′ y )) + f ′ (( τ ′ , t ′ ) , y ) == ¯ f (( τ, t − ( f α ◦ ρ α )( τ, ψ t ′ ( L τ ′ y ))) , ψ t ′ ( L τ ′ y )) + ¯ f (( τ ′ , t ′ − ( f α ◦ ρ α )( τ ′ , y )) , y ) == ¯ f (( τ τ ′ , t + t ′ − ( f α ◦ ρ α )( τ τ ′ , y )) , y ) = f ′ (( τ τ ′ , t + t ′ ) , y )and is thus a cocycle of the minimal flow ( Y, { ψ t ◦ L τ : ( τ, t ) ∈ T × R } ). Nowlet ( τ, y ) ∈ T × Y be arbitrary. By equality (6) and the density of e τ f α -transitivepoints in X α , there exists a sequence { ( τ k , y k ) } k ≥ ⊂ T × Y so that y k → τ y , τ k y k → ψ − ( f α ◦ ρ α )( τ,y ) ( τ y ), and ( f α ◦ ρ α )( τ k , y k ) → − ( f α ◦ ρ α )( τ, y ) as k → ∞ , andby equality (10) holds ( f Y − f α ◦ ρ α )( τ k , y k ) → ¯ f (( T , − ( f α ◦ ρ α )( τ, y )) , τ y ). Thus( f Y − f α ◦ ρ α )( τ k τ, τ − y k ) → ¯ f (( T , − ( f α ◦ ρ α )( τ, y )) , τ y ) + ( f Y − f α ◦ ρ α )( τ, y )as k → ∞ , and this limit coincides with ¯ f (( τ, − ( f α ◦ ρ α )( τ, y )) , y ) = f ′ (( τ, , y ) dueto the cocycle identity for ¯ f (( τ, t ) , y ). Since f Y ( τ, y ) is relatively trivial with respectto ( f α ◦ ρ α )( τ, y ), for every ε > δ > τ ′ , y ′ ) ∈ T × Y with d Y ( y ′ , τ ′ y ′ ) < δ and | ( f α ◦ ρ α )( τ ′ , y ′ ) | < δ holds | ( f Y − f α ◦ ρ α )( τ ′ , y ′ ) | < ε .From τ − y k → y , τ k y k → L τ y , and ( f α ◦ ρ α )( τ k τ, τ − y k ) → τ, y ) ∈ T × Y with d Y ( y, L τ ( y )) < δ that f ′ (( τ, , y ) < ε . Fact 1.10 implies thatthe cocycle ( τ, y ) f ′ (( τ, , y ) of the distal flow ( Y, { L τ : τ ∈ T } ) is a coboundaryon the { L τ : τ ∈ T } -orbit closure X α × { m } with transfer function b m : X α −→ R for every m ∈ M . Since δ > { L τ : τ ∈ T } -orbit closures, the transferfunctions { b m : m ∈ M } are uniformly equicontinuous. We fix a point ¯ x ∈ X α andobtain from the cocycle identity for all ( τ, t ) ∈ T × R and ( x α , m ) ∈ Y that f ¯ x (( τ, t ) , ( x α , m )) = f ′ (( τ, t ) , ( x α , m )) − f ′ (( T , t ) , (¯ x, m )) == b φ t ( m ) ( τ x α ) − b φ t ( m ) (¯ x ) − b m ( τ x α ) + b m (¯ x ) . The function f ¯ x (( τ, t ) , ( x α , m )) is also a cocycle of ( Y, { ψ t ◦ L τ : ( τ, t ) ∈ T × R } ) andbounded on T × R × Y , hence a coboundary with a transfer function ¯ b : Y −→ R so that ¯ b ( ρ − α (¯ x )) = { } . Now equality (14) follows, and equality (15) follows from f ′ (( T , t ) , ( x α , m )) = ¯ f (( T , t ) , ( x α , m )) for all t ∈ R and ( x α , m ) ∈ Y . (cid:3) With these prerequisites we can conclude the proof of our main result.
Proof of the structure theorem.
We let all elements of the theorem and the flow { ψ t : t ∈ R } ⊂ Aut( Y, { L τ : τ ∈ T } ) ∩ Aut(
Y, T ) be defined according to thePropositions 2.11, 2.12, 2.13, and 2.15. We fix a point ¯ x ∈ X α so that (¯ x,
0) is
EAL EXTENSIONS OF DISTAL MINIMAL FLOWS 23 transitive for e τ f α and ¯ b ( ρ − α (¯ x )) = { } . Then we define a cocycle g ( t, m ) of thedistal minimal flow ( M, { φ t : t ∈ R } ) by g ( t, m ) = ¯ f (( T , t ) , (¯ x, m )) for all ( t, m ) ∈ R × M. From equalities (14) and (15) follows for all τ ∈ T and ( x α , m ) ∈ Y that( f Y − f α ◦ ρ α )( τ, ( x α , m )) = ¯ f (( τ, , ( x α , m )) == ¯ f (( T , f α ( τ, x α )) , L τ ( x α , m )) + ¯ b ( L τ ( x α , m )) − ¯ b ( x α , m ) == g ( f α ( τ, x α ) , m ) + ¯ b ◦ τ ( x α , m ) − ¯ b ( x α , m ) . Hence equality (3) holds for the cocycle f Y ( τ, y ) − ¯ b ( τ y ) + ¯ b ( y ) cohomologous to f Y ( τ, y ), and this cocycle will be substituted for f Y ( τ, y ) henceforth. For everysequence { ( τ k , x k ) } k ≥ ⊂ T × X with ( f α ◦ π α )( τ k , x k ) → f Y ◦ π Y )( τ k , x k ) → k → ∞ , and thus identity (5) implies identity (2).For every ordinal ξ with α ≤ ξ ≤ η we can apply the Propositions 2.12, 2.13,and 2.15 to the distal minimal flow ( X ξ , T ) and the cocycle f ξ ( τ, x ξ ). We obtain afactor ( Y ξ , T ) = π Y ξ ( X ξ , T ) with ( X α , T ) = ρ ξα ( Y ξ , T ), an R -flow { ψ tξ : t ∈ R } ⊂ Aut( Y ξ , T ), a cocycle f Y ξ ( τ, y ξ ) of ( X ξ , T ), and a cocycle ¯ f ξ (( τ, t ) , y ξ ) of the flow( Y ξ , T × R ) extending the cocycle ( f Y ξ − f α ◦ ρ ξα )( τ, y ξ ). Striving for a contradiction tothe maximality of the ordinal α (cf. Proposition 2.11), we assume that the cocycle( + g )( t, m ) of the minimal flow ( M, { φ t : t ∈ R } ) is recurrent so that the cocycle¯ f η (( T , t ) , y η ) + t of the minimal flow (( ρ ηα ) − (¯ x ) , { ψ tη : t ∈ R } ) is also recurrent.We let β be the minimal element of the non-empty set of ordinals { α ≤ ξ ≤ η : ¯ f ξ (( T , t ) , y ξ ) + t is a recurrent cocycle of (( ρ ξα ) − (¯ x ) , { ψ tξ : t ∈ R } ) } , with β > α since f Y α = f α and ¯ f α (( T , t ) , y α ) ≡
0. We fix a point ¯ x β ∈ ( π βα ) − (¯ x )so that ( π Y β (¯ x β ) ,
0) is a recurrent point for the skew product extension of the flow(( ρ βα ) − (¯ x ) , { ψ tβ : t ∈ R } ) by the cocycle ¯ f β (( T , t ) , y β ) + t . Then there exists asequence { ¯ τ k } k ≥ ⊂ T with f α (¯ τ k , π βα (¯ x β )) → ∞ so that¯ f β (( T , f α (¯ τ k , π βα (¯ x β ))) , π Y β (¯ x β )) + f α (¯ τ k , π βα (¯ x β )) = f Y β (¯ τ k , π Y β (¯ x β )) → k → ∞ , and by the cohomology of the cocycles f β ( τ, x β ) and ( f Y β ◦ π Y β )( τ, x β )the sequence f β (¯ τ k , ¯ x β ) is bounded. Hence there exists a sequence { k l } l ≥ ⊂ N so that f α (¯ τ k l +1 , π βα (¯ τ k l ¯ x β )) → ∞ , d β (¯ τ k l +1 ¯ x β , ¯ τ k l ¯ x β ) →
0, and f β (¯ τ k l , ¯ x β ) is con-vergent. Then the sequence { ( τ l , x l ) = (¯ τ k l +1 (¯ τ k l ) − , ¯ τ k l ¯ x β ) } l ≥ ⊂ T × X β fulfils f α ( τ l , π βα ( x l )) → ∞ , d β ( x l , τ l x l ) →
0, and f β ( τ l , x l ) → l → ∞ . However, forevery α ≤ ξ < β holds¯ f ξ (( T , f α ( τ l , π βα ( x l ))) , π Y ξ ◦ π βξ ( x l )) + f α ( τ l , π βα ( x l )) = f Y ξ ( τ l , π Y ξ ◦ π βξ ( x l )) → ∞ for l → ∞ . Otherwise, since | ¯ f ξ (( T , t ) , ( x α , m ξ )) − ¯ f ξ (( T , t ) , (¯ x, m ξ )) | is uniformlybounded for all t ∈ R , x α ∈ X α , and m ξ ∈ M ξ (cf. identity (15)), there exists anon-trivial prolongation in the skew product of the minimal flow (( ρ ξα ) − (¯ x ) , { ψ tξ : t ∈ R } ) and its cocycle ¯ f ξ (( T , t ) , y ξ ) + t , which sufficient for its recurrence (cf.Lemma 1.17). Therefore also f ξ ( τ l , π βξ ( x l )) → ∞ as l → ∞ , and depending on thetype of the ordinal β follows either from Lemma 2.9 or Lemma 2.10 that e τ f β ispoint transitive, in contradiction to the maximality of α . (cid:3) Proof of the structure of the topological Mackey action.
In the proof of the decom-position theorem it is verified that the topological Mackey actions for the cocycle f ( τ, x ) of ( X, T ) and the transient cocycle ( + g )( t, m ) of ( M, { φ t : t ∈ R } ) are topologically isomorphic. Let { ( M ξ , { φ tξ : t ∈ R } ) : 0 ≤ ξ ≤ θ } be the normal I -system for the distal minimal compact metric flow ( M, { φ t : t ∈ R } ) with thehomomorphisms σ ξ : M −→ M ξ . For every ordinal 0 ≤ ξ ≤ θ a cocycle g ξ ( t, m ξ ) of( M ξ , { φ tξ : t ∈ R } ) is defined by a RIM . Let β be the minimal element of the set θ ∈ { ≤ ξ ≤ θ : ( g − g ξ ◦ σ ξ )( t, m ) is a coboundary of ( M, { φ t : t ∈ R } ) } . The cocycle ( + g β )( t, m β ) is transient, since the cocycle ( + g )( t, m ) cohomologousto ( + g β ◦ σ β )( t, m ) is transient. By Lemma 1.17 the right translation action { R b : b ∈ R } acts minimally on the Fell compact space D of orbits in M β × R .The mapping χ : M β −→ D defined by m β
7→ O φ β , ( + g β ) ( m β ,
0) is Fell continuous,and for every t ∈ R holds χ ◦ φ tβ ( m β ) = R ( + g β )( t,m β ) ◦ χ ( m β ). For β = 0 the flow( D, { R b : b ∈ R } ) is trivial and thus weakly mixing. If β ≥
1, then ( D, { R b : b ∈ R } )is a non-trivial minimal compact metric flow. If it is not weakly mixing, then thereexists a non-trivial equicontinuous factor ( D , { ϕ t : t ∈ R } ) = ν ( D, { R b : b ∈ R } )with homomorphism ν (cf. [KeRo]). We shall use a generalised and relativisedversion of Theorem 1 in [Eg] to obtain a contradiction to the minimality of β .Since ( D , { ϕ t : t ∈ R } ) is a minimal and non-trivial flow, for each small enough ε > ϕ ε ( d ) = d for all d ∈ D . We shall verify as a sub-lemma thatthere are no sequences { t k } k ≥ ⊂ R , { m k } k ≥ , { m ′ k } k ≥ ⊂ M β so that m k → ¯ m , m ′ k → ¯ m , φ t k β ( m k ) → ¯ m ′ , φ t k β ( m ′ k ) → ¯ m ′ , and g β ( t k , m ′ k ) − g β ( t k , m k ) → ε . Indeed, ν ◦ χ ◦ φ t k β ( m ′ k ) = ϕ ( + g β )( t k ,m ′ k ) ◦ ν ◦ χ ( m ′ k ) → ϕ ε (lim ϕ ( + g β )( t k ,m k ) ◦ ν ◦ χ ( m ′ k ))and lim ϕ ( + g β )( t k ,m k ) ◦ ν ◦ χ ( m ′ k ) = lim ϕ ( + g β )( t k ,m k ) ◦ ν ◦ χ ( m k ) = ν ◦ χ ( ¯ m ′ ), bythe equicontinuity of ( D , { ϕ t : t ∈ R } ), imply that ν ◦ χ ( ¯ m ′ ) = ϕ ε ◦ ν ◦ χ ( ¯ m ′ ),which contradicts to the choice of ε .If β = γ + 1 for some ordinal 0 ≤ γ < θ , then the sub-lemma implies the uniformequicontinuity of the mapping m β g β ( t, m β ) restricted on the σ βγ -fibres and forall t ∈ R . Indeed, otherwise we can find sequences as above with σ βγ ( m k ) = σ βγ ( m ′ k )for all k ≥
1, and the condition on sufficiently small ε can be fulfilled by theconnectedness of the σ βγ -fibres. Thus for all t ∈ R the cocycle ( g β − g γ ◦ σ βγ )( t, m β )is uniformly equicontinuous and assumes zero on every connected σ βγ -fibre. By Fact1.10 this is then a coboundary, in contradiction to the minimality of β .If β is a limit ordinal, then the sub-lemma applies to sequences so that for everyordinal 0 ≤ ξ < β there exists an integer k ξ ≥ σ βξ ( m k ) = σ βξ ( m ′ k ) for all k ≥ k ξ . Hence there exists an ordinal 0 ≤ ζ < β so that | g β ( t, m β ) − g β ( t, m ′ β ) | < ε for all m β , m ′ β ∈ M β with σ βζ ( m β ) = σ βζ ( m ′ β ) and for all t ∈ R . It follows that( g β − g ζ ◦ σ βζ )( t, m β ) is a coboundary, in contradiction to the minimality of β .The topological Mackey action of the transient cocycle ( + g )( t, m ) is topo-logically isomorphic to the topological Mackey action of the cohomologous cocycle( + g β ◦ σ β )( t, m ) (cf. the proof of the decomposition theorem). The weakly mixingflow ( D, { R b : b ∈ R } ) is a factor of the topological Mackey action of the cocycle( + g β ◦ σ β )( t, m ), since for every ( m, s ) ∈ M × R the mapping σ β × id R maps theorbit O φ, ( + g β ◦ σ β ) ( m, s ) in M × R to the orbit O φ β , ( + g β ) ( σ β ( m ) , s ) in M β × R con-tinuously with respect to the Fell topologies. Suppose that there exists two distinctorbits O , O ′ in M × R within the same σ β × id R -fibre and { t k } k ≥ ⊂ R is a sequencewith R t k O → O ′′ and R t k O ′ → O ′′ . Since the mapping t ( + g β )( t, m β ) is onto R for every m β ∈ M β (cf. Lemma 1.17), there exists a point ¯ m ∈ M β and distinct EAL EXTENSIONS OF DISTAL MINIMAL FLOWS 25 m, m ′ ∈ σ − β ( ¯ m ) so that ( m, ∈ O and ( m ′ , ∈ O ′ . Moreover, for every integer k ≥ t ′ k so that ( + g β )( t ′ k , ¯ m ) = t k , and therefore( φ t ′ k ( m ) , ∈ R t k O as well as ( φ t ′ k ( m ′ ) , ∈ R t k O ′ . By changing to a subsequencewe can suppose that φ t ′ k ( m ) → m and φ t ′ k ( m ′ ) → m as k → ∞ with m = m ,by the distality of the flow ( M, { φ t : t ∈ R } ). However, since ( m , , ( m , ∈ O ′′ ,the point ( ¯ m,
0) is a periodic point in ( D, { R b : b ∈ R } ) in contradiction to thetransience of the cocycle ( + g β )( t, m β ). We can conclude that σ β × id R is a distalhomomorphism of the topological Mackey action of the cocycle ( + g β ◦ σ β )( t, m )onto the weakly mixing flow ( D, { R b : b ∈ R } ). (cid:3) Acknowledgement : The author would like to thank Professor Jon Aaronsonand Professor Eli Glasner for useful discussions and encouragement.
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