Skew products and Livsic Theory

William Parry and Mark Pollicott

June 9, 2006

Dedicated to Anatoly Vershik on the occasion of his 70th birthday

1 Introduction

This is a short exposition of various aspects of skew-products from the pointof view of Livsic theory — an essay, as it were, focussed on compact groupextensions only and hence partial in at least two respects. We have taken theliberty of expanding recent results (our real topic) so that placed in context,they might interest a wider readership. Experts, of course, will pass overmuch of this material.

Skew-products date back (at least) to von Neumann who proposed theirstudy in connection with the classification theory of ergodic measure-preservingtransformations of a probability space. Von Neumann provided examples ofsuch transformations which were non-isomorphic, but which shared the samespectral characteristics. This represented a first step away from the well-understood theory of transformations with discrete spectrum (due to hiswork done with Halmos). Subsequently Anzai, Abramov and Furstenberglaid the foundations for later research in this area. At that time the empha-sis was on skew-products over translations of compact abelian groups and soinvolved algebraic structure at two levels — the base and fibre — giving riseto noticeable rigidity of behaviour.

We shall be interested in skew-products where the base is chaotic, ormore precisely hyperbolic, and where motion in the fibres is determined bycompact group translations. These systems therefore incorporate randomand deterministic features.

The purpose of this paper is to give an exposition of three aspects of thetheory:

1. ergodic decompositions of skew-products (heavily indebted to [6]);

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2. cohomology theory (Livsic theory proper [17]);

3. ergodic stability problems (motivated initially in [15] and recently re-vived in [1], [22], [14], [12], [10]).

The amount of detail provided varies with each section in accordance withour estimate of what is and what is not ‘well-known’.

Let (X, d) be a compact metric space. For 0 < α < 1, we say that acontinuous function f : X −→ Cd is Holder continuous (with exponent α) if

|f(x)− f(y)| ≤ Cd(x, y)α

for some constant C and for all x, y ∈ X. Here, as elsewhere, | · | denotesthe Euclidean norm. Likewise f : X → M(d), (d × d complex matrices) isHolder if |f(x)− f(y) ≤ Cd(x, y)α for some constant C and for all x, y ∈ X.(Here | · | represents the Euclidean operator norm.) The least such C aboveis denoted |f |α and we define Cα(X,Cd), Cα(X,M(d)) to be the spaces ofHolder continuous functions, which with respect to the norm ‖f‖ = |f |∞ +|f |α are Banach spaces.

Let φ : X −→ X be a continuous surjective map. Then two functionsf, g ∈ Cα(X,Cd) are said to be cohomologous (with respect to φ) if thereexists h ∈ Cα(X,Cd) such that

f − g = hφ− h .

If f is cohomologous to 0 then f is called a coboundary. Similarly f, g ∈Cα(X,G), where G ⊂ U(d) ⊂ M(d), are said to be cohomologous if thereexists h ∈ Cα(X,G) such that f = h−1ghφ and when f is cohomologous to1 (the constant function assuming the identity value 1 in G) we say f is acoboundary.

These notions (which can be extended to other categories of functions)arise as follows:

For f ∈ Cα(X,G) we define the skew-product φf : X ×G −→ X ×G by

φf (x, y) = (φx, yf(x)) .

For g0 ∈ G, the map g0(x, y) = (x, g0y) defines an action of G on X × Gwhich commutes with φf (φf is equivariant). Cohomological considerationsarise when we look for a G-equivariant isomorphism ψ between two skew-products φf , φg with the restriction that ψ factors to the identity on X, i.e.ψ is a skew-product of the form ψ(x, y) = (x, yh(x)). The existence of suchan isomorphism is precisely the requirement that f, g are cohomologous.

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2 Measure-Preserving Transformations

Let φ be a measure-preserving transformation of the probability space (X,B,m).φ is said to be ergodic if φ−1B = B ( B ∈ B) implies m(B) = 0 or 1. Astronger condition is (weak) mixing which can be defined in several ways. Inparticular φ is weak-mixing if the equation

F ◦ φ = αF (α ∈ C, F ∈ L2(X))

implies that F is constant almost everywhere (i.e. apart from the one-dimensionalspace of constants, the induced isometry of L2(X) has no discrete spectrum.)

Keynes and Newton [16], generalising earlier work of Anzai and Fursten-berg established the following criteria for the ergodicity or weak-mixing ofskew-products in the measure-category.

Theorem 2.1 Let φ be an ergodic measure-preserving transformation of theprobability space (X,B,m) and let φf : X ×G −→ X ×G be a skew-product

φf (x, y) = (φx, yf(x))

where f : X −→ G is measurable. Then φf is ergodic with respect to themeasure m×mG (mG Haar measure) if and only if the equation

F (φx) = Rχ(f(x))F (x)

a.e. (where F ∈ L2(X,Cd) and Rχ is an irreducible unitary representationof G ⊂ U(d)) has only the trivial solutions: d = 1, Rχ is trivial and F isconstant a.e.

Theorem 2.2 Let φ be a weak mixing measure-prserving transformation ofthe probability space (X,B,m) and let φf : X × G −→ X × G be a skew-product. Then φf is weak-mixing if in addition to being ergodic (see abovetheorem), the equation

F (φx) = αγ(f(x))F (x)

(where F ∈ L2(X,C), α ∈ C, γ a one-dimensional unitary representation)has only the trivial solution: F is constant.

The equations in Theorems 2.1, 2.2 are suggestive of the cohomologyconditions of the previous section. This will be made explicit later whenwe consider measure-spaces which are also metric spaces. More precisely weshall be looking exclusively at so-called hyperbolic sets of Axiom A diffeo-morphisms [29]. The interplay of measure and topology will be a recurrenttheme in our exposition.

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3 Hyperbolicity and Shifts of Finite Type

Let φ : M −→ M be a C∞ diffeomorphism of a compact Riemannian mani-fold whose metric we denote by d.

A closed φ-invariant set X is said to have hyperbolic structure if thetangent bundle TXM over X can be written as a direct sum TXM = Eu⊕Es

(continuously split) such that

1. Dφ(Eu) = Eu and Dφ(Es) = Es

2. there exists constants c > 0, 0 < λ < 1 such that

‖Dφnv‖ ≤ cλn‖v‖ for v ∈ Es

‖Dφ−nv‖ ≤ cλn‖v‖ for v ∈ Eu

The restriction φ = φ |X is called a hyperbolic map if

1. X has hyperbolic structure

2. φ is transitive i.e. there exists x ∈ X such that {φnx : n ∈ Z} is densein X

3. the periodic points of φ are dense in X

4. there exists an open set U ⊃ X such that

∞⋂n=−∞

φnU = X

Very often we shall take φ to be topologically mixing which means thatthe condition 2 is strengthened to

2′. for all open U, V there exists N such that

φnU ∩ V 6= ∅ for all n > N .

If we simply assume that M itself has hyperbolic structure (without theother assumptions) then φ : M −→M is called an Anosov diffeomorphism.

The prototype of a diffeomorphism which is Anosov is an automorphismof a torus whose defining matrix does not have eigenvalues of modulus 1.

Examples of hyperbolic systems (which are not Anosov) are provided byshifts of finite type, since Williams [30] has shown that every such shift canbe embedded as a hyperbolic system in S3, and Bowen [5] proved that every

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zero-dimensional hyperbolic system is (homeomorphic to) a shift of finitetype.

Shifts of finite type (topological Markov shifts) are defined as follows: LetA be a k × k 0− 1 irreducible matrix and let

Σ = ΣA = {x ∈ (1, . . . , k)Z : A(xn, xn+1) = 1 for all n ∈ Z}

(We shall always consider aperiodic shifts, i.e. there exists N such thatAN(i, j) > 0 for all i, j.) The shift transformation σ (which is defined ondoubly infinite sequences x = {xn} by (σx)n = xn+1) leaves Σ invariant. Itsrestriction σ = σ |Σ is a shift of finite type.

The underlying space Σ is compact (with respect to the Tychonov topol-ogy), zero-dimensional, and metrizable. A convenient metric is given byd(x, y) = 1/2N , where xn = yy for |n| ≤ N and either xN 6= yN or x−N 6= y−N .

Shifts of finite type are connected with hyperbolic systems in another,perhaps more fundamental way. They provide symbolic models for all hy-perbolic maps in the following sense [3]:

Theorem 3.1 (Bowen) Let φ : X −→ X be a hyperbolic map. Then thereexists a shift of finite type, σ : Σ → Σ and a Holder continuous surjectivemap π : Σ −→ X such that πσ = φπ. Moreover, except for points in a set offirst category, each x ∈ X has a unique inverse image and in any case thereexists a bound b on the number of inverse images for all x ∈ X.

Briefly, the theorem comes from the hyperbolic structure of X as fol-lows: for each x ∈ X define the stable set through x as W s(x) = {y ∈X : d(φnx, φny) −→ 0} and the unstable set as W u(x) = {y ∈ X :d(φ−nx, φ−ny) −→ 0}. The stable sets foliate (partition) X as do the unsta-ble sets. Rectangles (with precise properties) can be defined by these setsand X can be covered by a finite union of rectangles, R1, . . . , Rk in such away that the interiors of these sets are pairwise disjoint, each Ri is the closureof its interior, and for each i = 1, . . . , k

φ(W s(x) ∩Ri) ⊂ W s(φx) ∩Rj if x ∈ int(Ri ∩ φ−1Rj)

φ−1(W u(x) ∩Ri) ⊂ W u(φ−1x) ∩Rj if x ∈ int(Ri ∩ φRj)

Such a partition (strictly speaking covering) into rectangles is called a Markovpartition.

The existence of arbitrary small Markov partitions was proved in increas-ing order of generality by Adler and Weiss [2](for ergodic automorphisms ofa 2-torus), Sinai [27] (for Anosov diffeomorphisms) and Bowen [3](for hy-perbolic maps). From this data one can now produce a shift of finite type

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(Σ, σ) as follows: Define A(i, j) = 1 if Ri∩φ−1Rj has non-empty interior andA(i, j) = 0 otherwise. When the Markov partition is small enough, there isat most one point x for which φnx ∈ intRxn for all n ∈ Z which implies thatthe map

π(x) =∞⋂

n=−∞

φ−nRxn (x = {xn})

is well defined and maps ΣA = Σ onto X with the exceptional points ywith more than one inverse image being those for which there exists n ∈ Nsuch that φny belongs to the boundary of some rectangle. These exceptionalpoints form a set of first category.

4 Equilibrium States and One-Sided Shifts

Let φ : X −→ X be a hyperbolic map and let f ∈ Cα(X,R). The pressureof f is defined by

P (f) = sup{h(µ) +

∫f dµ}

where the supremum is taken over all φ invariant probabilities µ and h(µ)denotes the µ measure theoretic entropy of φ. If P (f) = h(m) +

∫f dm we

say thatm is an equilibrium state associated with f . A key fact for hyperbolicsystems is that for each Holder continuous function there is one and only oneequilibrium state [4]. In much of what is to follow there will be a (Holder)equilibrium state in the background, which frequently need not be specified.If m is the equilibrium state of f and π : Σ −→ X is the projection defined inBowen’s theorem, then m = µπ−1 where µ is the equilibrium state of f ◦ π.Moreover π is a measure-theoretic isomorphism between (Σ, σ) and (X,φ)with respect to these measures, a fact which frequently allows problems forhyperbolic systems to be transfered to problems for shifts.

In addition to the two-sided shifts of finite type, defined above, there areone-sided versions. Again we begin with the matrix A and define

Σ+ = Σ+A = {x ∈ (1, . . . , k)Z+

: A(xn, xn+1) = 1 for all n ∈ Z+}

and σ(x)n = xn+1. (We retain the same notation for the one-sided shift). Theshifts σ : Σ −→ Σ and σ : Σ+ −→ Σ+ are related by the obvious projectionwhich deletes negatively indexed coordinates. Using this projection it is notdifficult to see that any probability measure which is invariant for the one-sided shift gives rise to a corresponding invariant probability for the two-sidedshift — and vice versa. Moreover the corresponding systems enjoy the sameergodic or mixing properties.

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Let σ : Σ −→ Σ be a shift of finite type and let f ∈ Cα(Σ,Cd). Thefollowing theorem enables one to transfer problems to a one-sided setting.

Theorem 4.1 [28] If f ∈ Cα(X,Cd) then there exist functions f+, h ∈Cα/2(X,Cd) such that f+ = f + hσ − h where f+ is independent of thepast, i.e. f+(x) = f ′(πx) where f ′ ∈ Cα/2(X+,Cd).

To establish properties of f it is frequently useful to prove them for f ′ inthe context of one-sided shifts and then to invoke the theorem.

The proof of the following theorem mimics the proof of Sinai’s. (In factthe crucial property used is the bi-invariance of the group metric.)

Theorem 4.2 If f ∈ Cα(X,G), (G ⊂ U(d)) then there exist functionsf+, h ∈ Cα/2(X,G) such that f+ = h−1fhσ, where f+ is independent ofthe past.

5 The Ruelle Transfer Operator

In this section we concentrate on a one-sided shift of finite type σ : Σ+ −→Σ+. For u ∈ Cα(Σ+,R), we define the transfer operator,

Lu : Cα(Σ+,R) −→ Cα(Σ+,R)

by

(Luw)(x) =∑σy=x

eu(y)w(y) ,

which is a continuous linear operator.

Theorem 5.1 (Ruelle) If P (u) is the pressure function of section 4, eP (u)

is the maximum eigenvalue of Lu; it is simple and its eigenfunctions h ∈Cα(Σ+,R) may be taken to be strictly positive. The rest of the spectrum ofLu is contained in a disc of radius strictly less than eP (u).

(cf. [26]) For Luh = eP (u)h, define u′ = u + log h − log hσ − P (u), (sothat u′ − u is cohomologous to a constant) then this eigenfunction equationtransforms to Lu′1 = 1, when we say that u′ is normalised (the normalisationof u).

The equilibrium state m defined by u (or any other function which iscohomologous to it plus a constant) can be defined using the normalisedoperator Lu′ . It is the unique measure satisfying∫

(Lu′w) dm =

∫w dm

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for all continuous w (i.e. L∗u′m = m). In view of Ruelle’s theorem, Lu′

has 1 as a simple eigenvalue corresponding to the constant function 1 andthe rest of the spectrum is contained in a disk of radius less than 1. Thislatter fact plays a crucial role in establishing mixing properties of both theone-sided and two-sided shifts. In particular it is used to show that σ isexponentially mixing meaning: there exists c > 0, 0 < λ < 1 such that for allv, w ∈ Cα(Σ+,R),∣∣∣∣∫ v ◦ σnw dm−

∫v dm

∫w dm

∣∣∣∣ ≤ c.λn‖v‖‖w‖ .

We can define a transfer operator for complex valued functions u + iv ∈Cα(Σ+,C) in the same way as above,

(Lu+ivw)(x) =∑σy=x

eu(y)eiv(y)w(y)

= Lu(eivw).

Or more generally if f ∈ Cα(Σ+, G) where G ⊂ U(d) we can define

Lf : Cα(Σ+,Cd) −→ Cα(Σ+,Cd)

by Lfw = Lu(fw). (Here u is taken to be a fixed element of Cα(Σ+,R)).Proposition 5.4 to follow was first proved by Pollicott (for the case d = 1,

which amounts to the complex operator case). We shall present a generalisedversion. Crucial to the proof is the inequality given by:

Lemma 5.2 If Lu is normalised and f ∈ Cα(Σ+, U(d)), then there exists aconstant C such that for all w ∈ Cα(Σ+,Cd),∣∣Ln

fw∣∣α≤ C|w|∞ + αn|w|α .

(See [23] for a proof when d = 1. The general case is similar.)We shall also need

Lemma 5.3 The function w ∈ Cα(Σ+,Cd) satisfies Lfw = αw (|α| = 1) ifand only if fw = αw ◦ σ.

ProofThe second equation implies the first equation by a simple application of theoperator Lu. Assuming Lfw = αw, we see that Lu|w| ≥ |w| and itegrationthen shows that we must have Lu|w| = |w|. This means that |w| takes a

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constant value, which we may take to be 1. For each x, αw(x) is a point ofthe unit sphere in Cd and ∑

σy=x

eu(y)f(y)w(y)

is a convex combination of points f(y)w(y) on the unit sphere since∑

σy=x eu(y) =

1. The equation therefore implies that for each y ∈ σ−1x, f(y)w(y) = αw(x),i.e. f(y)w(y) = αw(σy) for all y ∈ Σ+. �

Proposition 5.4 Let V ⊂ Cα(X+,Cd) be a closed subspace such that LfV ⊂V and Lf = Lf |V has no eigenvalues of modulus 1. Then ρ(Lf |V ) < 1 whereρ is the spectral radius.

ProofBy Lemma 5.2, we see that for M = C + 1, Ln

fB1 ⊂ BM for all n ∈ N, whereB1, BM are the closed balls of radius 1 and M respectively. In particular,this means that

AN(w) =1

N

N−1∑n=0

Lnfw ∈ BM for all w ∈ B1

and since BM is uniformly compact, AN(w) converges uniformly throughsome subsequence to, say, w∗ ∈ V . Moreover, Lfw

∗ = w∗ so that w∗ = 0 (forotherwise 1 would be an eigenvalue). The sequence |A2N (w)|∞ is decreasingsince ∣∣∣∣∣∣ 1

2N+1

2N+1−1∑n=0

Lnfw

∣∣∣∣∣∣∞

≤

∣∣∣∑2N−1n=0 Ln

fw∣∣∣∞

2N+1+

∣∣∣L2N

f

(∑2N−1n=0 Ln

fw)∣∣∣∞

2N+1

i.e. |A2N+1(w)|∞ ≤ |A2N (w)|∞. Since B1 is uniformly compact we thereforeconclude that for every ε > 0 there exists N such that |AN(w)|∞ < ε for allw ∈ B1. The basic inequality gives∣∣Lk

fAN(w)∣∣α≤ |AN(w)|∞ .C + αk |AN(w)|α

and

|AN(w)|α =

∣∣∣∣∣ 1

N

N−1∑n=0

Lnfw

∣∣∣∣∣α

≤ 1

N

N−1∑n=0

∣∣Lnfw

∣∣α

≤ 1

N

N−1∑n=0

(|w|∞C + αn|w|α)

≤ ‖w‖(C + 1) .

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Thus ∣∣LkfAN(w)

∣∣α≤ C. |AN(w)|∞ + αk(C + 1)‖w‖

and ∣∣LkfAN(w)

∣∣∞ ≤ |AN(w)|∞

which added gives∥∥LkfAN(w)

∥∥ ≤ (C + 1)(|AN(w)|∞ + αk‖w‖

).

If we choose ε = 1/(2C + 2) and N such that |AN(w)|∞ < ε for all w ∈ B1

and k such that αk < 1/(2C+2) we will have ‖LkfAN(w)‖ < 1 for all w ∈ B1.

Thus 1 is not an eigenvalue of Lf and hence ρ(Lkf ◦ AN) = sup{|p(λ)| : λ ∈

spLf} < 1 where

p(λ) =1

N

k+N−1∑n=k

λn .

Therefore 1 /∈ spLf .Repeating this argument with f replaced by αf where |α| = 1 we conclude

that α is not an eigenvalue of Lf . Hence sp(Lf ) is contained in the open unitdisc. �

Theorem 5.5 The space Cα(X+,Cd) = Cα decomposes as Cα = V0 + V1

where Lf : V1 −→ V1, Lf : V0 −→ V0 and

1. ρ(Lf |V0) < 1,

2. dim(V1) ≤ d

3. V1 is spanned by (generalized) eigenvectors corresponding to eigenvaluesof modulus 1.

ProofLet V1 be the linear space of finite linear combinations of eigenvectors whoseeigenvalues have modulus 1. We equip this space with the inner prod-uct 〈〈v, w〉〉 =

∫〈v(x), w(x)〉 dm where 〈v(x), w(x)〉 is the Euclidean in-

ner product on Cd. If Lfv = αv and Lfw = βw where |α| = |β| = 1,then fv = αv ◦ σ, fw = βw ◦ σ so that (using the fact that f is U(d)-valued), 〈v(x), w(x)〉 = αβ〈v(σx), w(σx)〉 . Since σ is mixing we concludethat 〈v(x), w(x)〉 is constant and α = β if this constant is non-zero. Choos-ing a fixed point x0 = σx0, we therefore have 〈v(x), w(x)〉 = 〈v(x0), w(x0)〉for all x ∈ X+. Since V1 is spanned by Lf eigenfunctions we see that themap V1 → Cd given by v → v(x0) is an isometry. Hence dim(V1) ≤ d.

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Now let V0 = {v ∈ Cα : v ⊥ V1} . It is clear that V0 is a closed subspaceof Cα. If v ∈ V0 and Lfv0 = αv0 ∈ V1 where |α| = 1 (so that fv0 = αv0σ)then 〈Lfv, v0〉 = 〈L(fv), v0〉 = 〈fv, v0σ〉 = 〈v, fv0σ〉 = 〈v, αv0〉 = 0, i.e.LfV1 ⊂ V1. Furthermore, Cα = V0 + V1 since V1 is a closed subspace ofL2(m). By Proposition 5.4, ρ(Lf |V0) < 1. �

6 Regularisation of Functions

In this section we shall show how the spectral properties of transfer operatorsdiscussed in the last section can be used to prove Livsic type theorems.Typically, one wants to show that measurable solutions to certain functionalequations are necessarily (Holder) continuous.

Theorem 6.1 Let Lfw = αw a.e. where |α| = 1, w ∈ L2(Σ+,Cd), f ∈Cα(Σ+, U(d)). Then there exists w1 ∈ Cα(Σ+, U(d)) such that w = w1 a.e.

ProofReplacing f by αf if necessary we can assume Lfw = w. Let L2 = V ⊥

1 ⊕ V1

then V ⊥1 ⊃ V0 and Cα = V0 ⊕ V1. Choose ε > 0 and v ∈ Cα such that∫

|v − w|2 dm < ε2. Then v = v0 + v1, w = w0 + w1 where v0 ∈ V0, v1 ∈V1, w0 ∈ V ⊥

1 , w1 ∈ V1. Thus ε2 ≥∫|v −w|2 dm =

∫|v0 −w0|2 dm+

∫|v1 −

w1|2 dm and hence∫|v0 − w0|2 dm ≤ (

∫|v0 − w0|2 dm)1/2 ≤ ε. Furthermore

LfV⊥1 ⊂ V ⊥

1 . To see this suppose 〈u, u1〉 = 0 whenever Lfu1 = αu1 (|α| = 1)or equivalently fu1 = αu1 ◦ σ then 〈u, αf−1u1 ◦ σ〉 = 0 which implies that〈Lfu, αu1〉 = 0. This shows that Lfu ⊥ V1 when u ⊥ V1.

Hence∫|Ln

fv0 −Lnfw0| dm ≤

∫|v0 −w0| dm ≤ ε and since Ln

fv0 −→ 0 wehave limn→∞

∫|Ln

fw0| dm ≤ ε, i.e. w0 = 0 a.e. and w = w1 a.e. �

For a proof of the following cf. [23].

Lemma 6.2 If f ∈ Cα(Σ, U(d)) depends only on future coordinates. If w ∈L2(Σ) satisfies

fw = αw ◦ σ a.e. (|α| = 1)

then w depends (essentially) only on future coordinates.

This can now be used in conjunction with Theorem 6.1 and Theorem 4.1to prove:

Theorem 6.3 Let σ : Σ −→ Σ be a (two-sided) shift of finite type and letf ∈ Cα(Σ, U(d)). If w ∈ L2(Σ,Cd) satisfies

fw = αw ◦ σ a.e. (|α| = 1)

then there exists w1 ∈ Cα/2(Σ,Cd) such that w = w1 a.e.

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ProofBy Theorem 4.1 there exists f+, h ∈ Cα/2(Σ,Cd) such that f = h(σ)−1f+hwhere f+ depends only on future coordinates. Hence

f+(hw) = α(hw) ◦ σ .

By Lemma 6.2 it follows that hw depends only on future coordinates andapplying Theorem 6.1 we see that hw = w1 a.e. for some w1 ∈ Cα/2(Σ,Cd),i.e. w = h−1w1 a.e. �

We are now in a position to prove the following generalisation of a theoremdue to Livsic (who considered the case where one of the functions f, g istrivial).

Theorem 6.4 Let f, g ∈ Cα(Σ, G), (G ⊂ U(d)) and let h : Σ −→ G bemeasurable where

f = h−1ghσ a.e

Then there exists h′ ∈ Cα/2(Σ, G) such that h = h′ a.e. (and f = h−1ghσeverywhere).

ProofWe consider the Euclidean space M(d) endowed with the inner product〈A,B〉 = TraceAB∗. The linear map A 7→ UAV , (where U, V ∈ U(d))is a unitary operator on this d2-dimensional space since

〈UAV,UBV 〉 = Tr(UAV V ∗B∗U∗)

= Tr(UAB∗U−1)

= TrAB∗

= 〈A,B〉 .

We can therefore apply Theorem 6.1 to the unitary valued function x 7→φ(x)−1Af(x) applied to A. In other words the equation

g(x)−1h(x)f(x) = h(σx) a.e.

can be represented in the form required by Theorem 6.1. We conclude thatthe M(d)-valued (in fact G-valued) function h equals a.e. a function h′ ∈Cα/2(Σ, G) . �

Using Bowen’s Theorem (Theorem 3.1) the above result can be used toprove the following.

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Theorem 6.5 If φ : X −→ X is hyperbolic and f, g ∈ Cα(X,G) whereG ⊂ U(d) and if

f = h−1ghφ a.e.

then there exists h′ ∈ Cα/2(X,G) such that h = h′ a.e.

The basic idea is to lift the given equation to the setting of a shift of finitetype using the projection π in Bowen’s theorem. The functions f ◦π, g◦π areHolder continuous on the shift space and h◦π is measurable. Thus h◦π maybe replaced by a Holder continuous function h′′ on the shift space. The nextstep is to show that h′′ = h′ ◦ π is the lift of a continuous function on X (c.f.[22]). Hence f = h′−1gh′ ◦ φ where h′ is continuous. One can now appeal tothe use of Markov partitions in Theorem 3.1 to show that h′ ∈ Cα/2(X,G).

Theorem 6.5 may be thought of as a ‘rigidity’ theorem. It has the conse-quence that if φf , φg are equivariant, e.g. isomorphic (in the measure theoreticsense) via an extension of the identity then this isomorphism is (essentially)Holder continuous.

7 Ergodic Components

Let φ : X −→ X be hyperbolic and let f ∈ Cα(X,G), G ⊂ U(d). InTheorem 2.1 we gave criteria for φf to be ergodic with respect to m ×mG

where m is an equilibrium state. The situation there was purely measure-theoretic. In view of Theorem 6.5 we can get a clearer picture regardingergodic components.

Theorem 7.1 The skew-product is ergodic if and only if the equation

R(f)w = wφ (everywhere)

where R is an irreducible d-dimensional representation and w ∈ Cα(X,Cd)has only the trivial solution given by the trivial d = 1 dimensional represen-tation and w is constant.

For the case of a shift of finite type, σ : Σ −→ Σ we can give a descriptionof the ergodic components of φf (where φf is non-ergodic). For if R(f)w =w ◦ σ is a non-trivial equation we see that the function R(y−1)w(x) is anon-constant σf invariant function.

Choose x0 ∈ Σ define E = {(x, y) : R(y−1)w(x) = w(x0)}. If (x, y) ∈ Eand (x, yg0) ∈ E then R(g0)w(x0) = w(x0) and vice versa. Let H = {g0 :R(y0)w(x0) = w(x0)} then H is a closed proper subgroup of G for otherwiseR would be the trivial representation. It is clear that E ∩ Eg0 = ∅ if g0 /∈ H

13

and E ≡ Eg0 if g0 ∈ H. Furthermore E is φf invariant since w(σx) =R(f(x))w(x). Choosing x0 ∈ Σ to be a point with a dense orbit one seesthat the closed set E projects onto Σ.

Lemma 7.2 If φf is not ergodic then there exists a closed proper subgroupH and a Cα(Σ, H) function f ′ such that f, f ′ are cohomologous.

ProofDefine H as above. Since Σ is zero-dimensional one can choose a Holdercross section in E, i.e. h : Σ −→ G where (x, h(x)) ∈ E for all x ∈ Σ.Hence (σx, f(x)h(x)) ∈ E. This means that there exists f ′(x) ∈ H such thatf(x)h(x) = h(σx)f ′(x) which was to be proved. �

Theorem 7.3 If φf is not ergodic then there exists a minimal closed sub-group H ⊂ G for which f is cohomologous to some f ′ ∈ Cα(Σ, H). Moreoverσf : Σ×H −→ Σ×H is ergodic with respect to m×mH .

This follows from the lemma on noting that if σf ′ is not ergodic thenwe conclude that there is a closed proper σf ′ invariant E ′ ⊂ E which is H ′-invariant where H ′ ⊂ H (proper) and where E ′ projects to Σ. This wouldcontradict the minimality of H.

We have therefore arrived at a complete ergodic decomposition of Σ×G.If Γ consists of choices from each H coset above, then the σf invariant setsEg, g ∈ Γ are disjoint and σf invariant and Σ × G =

⋃g∈ΓEg. Further-

more Eg is g−1Hg invariant and φf |Eg is ergodic. These observations inconjunction with Bowen’s theorem (Theorem 3.1) lead to:

Theorem 7.4 Let φf : X ×G −→ X ×G where φ : X −→ X is hyperbolicand f ∈ Cα(X,G). Then there exists a closed φf invariant set E ′ whichprojects to X and a closed subgroup H ⊂ G such that E ′g0∩E ′ = ∅ if g0 /∈ Hand E ′g0 = E ′ if g0 ∈ H. Furthermore φf |E′ is ergodic with respect to themeasure m×mH .

ProofLet π : Σ −→ X be the projection of Bowen’s theorem and let π(x, g) =(πx, g). Then it is easy to see that πE = E ′ and H satisfies our requirementswhere E,H are defined in the theorem. �

RemarkAlthough this complete description of the ergodic decomposition exists forthe general case of a skew-product φ where φ is hyperbolic, we cannot assert

14

(in general) the existence of a function f ′ corresponding to the one defined inLemma 7.2 since its construction depended on the existence of a cross-sectionh to the set E, and this is not always true.

For an orbit segment, ` = (y, φy, . . . , φny) we define f(`) = f(y) · · · f(φny)and we define f−n(x) = f(φ−nx) · · · f(φ−1x), fn(x) = f(φx) · · · f(φnx). Let zbe a fixed point and let W u(z) = {x : φ−nx −→ z},W s(z) = {x : φnx −→ z}and W (z) = W u(z)∩W s(z). Points in W (z) are called homoclinic (with re-spect to z). Since f(z) is not necessarily 1 we choose, and fix, a sequence {Nn}such that f(z)Nn −→ 1 and define f−(x) = limn→∞ f−Nn(x), when x ∈ W u(z)and f+(x) = limn→∞ fNn(x) when x ∈ W s(x). These limits exist since bothf(z)−nf−n(x) and fn(x)f(z)−n converge due to the fact that the convergencesφ−nx −→ z and φnx −→ z are exponentially fast and f is Holder continuous.If x ∈ W (z) then we refer to the homoclinic orbit `(x) = {φnx} as a loopbased at z (generated by x), and we define f`(x) = f−(x)f(x)f+(x). Notethat we must distinguish between the loop generated by x and the loop gen-erated by φx since f`(φx) = f(z)−1f`(x)f(z). We also note that z ∈ W (z)and `(z) = {z} so that f`(z) = f(z).

Let Γ denote the free group generated by loops `(x), x ∈ W (z) a typicalelement being a formal product `±1

1 `±12 · · · `±1

k of loops, some of which maybe empty i.e. the identity element of Γ. With these definitions it is clear thatf : Γ −→ G (defined by f(`±1

1 `±12 · · · `±1

k ) = f(`1)±1f(`2)

±1 · · · f(`k)±1) is a

homomorphism.

Theorem 7.5 For f ∈ Cα(X,G) the skew-product φf is ergodic if and onlyif f(Γ) is dense in G.

ProofSuppose φf is ergodic and let K = {f(z)n}. Define E = {(x, kf−(x)) :k ∈ K and x ∈ W (z)} then E is φf - invariant and projects to X. Observethat points homoclinic to z are dense. Hence E contains a transitive pointso that E = X × G. Let g ∈ G be arbitrary and approximate (z, g) by(x, kf−(x)) ∈ E with x so close to x that fn(x)f(z)−n is close to 1 for alln > N and f(x) is close to f(z). Evidently kf−(x)f(x)fnf(z)−(n+1) is closeto g as is kf−(x)f(x)f+(x)f(z)−1 = kf`(x)f(z)−1. Thus g is arbitrarily closeto f(Γ), i.e. f(Γ) is dense in G.

Conversely, if Γ is dense and (z, 1) ∈ E, an ergodic component, then theorbit of (x, f−(x)) ∈ E since φ−Nn

f (x, f−(x)) =(φ−Nnx, f−(x) (f−Nn(x))−1) −→

(z, 1). Furthermore,

φNn(x, f−(x)) = (φNnx, f−(x)f(x)fNn−1(x)) −→ (z, f(`(x))f(z)−1) ∈ E.

15

We conclude that f`(x)f(z)−1 ∈ H, the closed subgroup H of Theorem 7.3.Hence f`(x) ∈ H for all x ∈ W (x), i.e. H = G and φf is ergodic. �

8 Weighted Periodic Points

Up to now the main theme has been regularisation theorems. In other words,given an equivariant measure-theoretic isomorphism between two skew-products(with appropriate conditions), we showed that the isomorphism is (essen-tially) Holder continuous.

In this section we shall not be provided with an initial isomorphism. Ourconditions, instead are given independently for each skew-product and areexpressed in terms of periodic points.

As before we are interested in the cocycle equation. So let φ : X −→ Xbe hyperbolic and let f, g ∈ Cα(X,G), G ⊂ U(d). The question we shalladdress is when are f, g cohomologous? If there exists h ∈ Cα(X,G) suchthat

f = h−1gh ◦ φ

then for every periodic point x (φnx = x) we have

f(x) · · · f(φn−1x) = h(x)−1g(x) · · · g(φn−1x)h(x)

i.e. the weights f(x) · · · f(φn−1x), g(x) · · · g(φn−1x) of the closed orbit x, φx,. . . , φn−1x, are conjugate in G.

We shall see in the next section, conversely, that if h ∈ Cα(X,G), givenin advance provides such a conjugacy of weights for all periodic orbits, then fand g are cohomologous. By modifying one of these functions appropriately,one may as well take h ≡ 1 and require that the weights are equal.

The identity of weights (or equivalently the provision of the above functionh) is a strong condition. We shall concentrate on the more general problem:How are f, g related if f -weights and g-weights are conjugate in G, i.e. whenno function h, as above, is given?

When f (or g) is trivial (consistently equal to 1, the identity of G) thenthe above condition is simply that all weights equal 1 and in this case Livsic’scelebrated theorem says that the non-trivial function is cohomologous to 1,i.e. it is a coboundary. We are interested in the general situation which doesnot reduce to Livsic’s case when G is non-abelian.

Our answer to the above question will be sketched, with less detail thanin earlier sections.

Let φz = z be fixed throughout. (If no fixed points exist, our argumentmay be modified using a periodic point.) In what follows f and g will be

16

Cα(X,G) functions (G ⊂ U(d)) and f(x) · · · f(φn−1x), g(x) · · · g(φn−1x) willbe conjugate whenever φnx = x and in particular f(z), g(z) will be conjugate.In fact, by conjugating one of f, g by an element of G there is no harm inassuming f(z) = g(z). We need the following closing lemmas (cf. [3]).

For two orbit segments, ` = (x, φx, . . . , φnx), `′ = (x′, φx′, . . . , φnx′) ofthe same length we define,

dα(`, `′) =n∑

i=0

d(φix, φix′)α .

Lemma 8.1 (Anosov’s closing lemma) There exists ε > 0, C(α) depend-ing only on φ and α such that if d(φNx, x) < ε then there exists a closed orbit`′ = (w, φw, . . . , φN−1w), (φNw = w) such that

dα(`, `′) ≤ C(α)d(x, φNx)α

where ` = (x, φx, . . . , φN−1x).

This can be used to prove

Lemma 8.2 If `1, . . . , `k are orbit segments such that (cyclically) the dis-tance di between the last point of `i and the first point of `i+1 is less than εthen there exists a periodic orbit whose consecutive segments `′1, . . . , `

′k satisfy

k∑i=1

dα(`i, `′i) ≤ C(α) max

idα

i .

Let φz = z and let W s(z),W u(z),W (z) = W s(z) ∩ W u(z) be the setsdefined in the last section.

Lemma 8.3 If x ∈ W u(z) and y ∈ W s(z) and d(x, y) < ε then there existsw ∈ W (z) such that

0∑i=−∞

d(φix, φiw)α +∞∑i=1

d(φiy, φiw)α ≤ C(α)d(x, y)α .

Lemma 8.4 If `(x) is a loop based at z then f(`(x)), g(`(x)) are conjugate.

We approximate `(x) by the finite orbit φ−Nnx, . . . x, . . . , φNnx = `n(x),where n is large and therefore d(φ−Nnx, φNnx) is small. This almost closedorbit can then be approximated by a genuinely closed orbit,

`n(x) = φ−Nnw, . . . , w, . . . φNnw (φNn+1w = φ−Nnw) ,

17

and therefore

f`n(w) ≡ f(φ−Nnw) · · · f(w) · · · f(φNnw)

andg`n(w) ≡ g(φ−Nw) · · · g(w) · · · g(φNw)

are conjugate, say, f`n(w) = λ−1n g`n(w)λn. This means that f`n(x) and

g`n(x) are almost conjugate becoming closer to true conjugacy as n −→ ∞and w chosen accordingly. Using compactness (and possibly a subsequenceof {Nn}), we conclude that f`(x) and g`(x) are conjugate.

Now let `(x1), . . . , `(xk) be k loops based at z. We can approximatethese by almost closed orbits `n(x1), . . . , `n(xk) as above. The lemmas canthen be used to allow the introduction of an approximating closed orbit`1(w1), `2(w2), . . . , `k(wk) (these are contiguous segments of the closed orbit).As before we have f`(w1) · · · f`(wk) and g`(w1) · · · g`(wk) are conjugate fromwhich one concludes that f`n(x1) · · · f`n(xk) and g`n(x1) · · · g`n(xk) are al-most conjugate. As long as our limits are taken through our fixed sequence,{Nn} we conclude as before that

Lemma 8.5 If `(x1), . . . , `(xk) are k loops based at z then f`(x1) · · · f`(xk)is conjugate to g`(x1) · · · g`(xk).

Now using the compactness of G and the fact that for each g1, . . . , gm ∈ Gthere is a subsequence Mn such that gMn

i −→ g−1i , (i = 1, . . . ,m) we arrive

at the following result.

Theorem 8.6 For all ` = `±11 `±1

2 · · · `±1k ∈ Γ we have f(`) and g(`) are

conjugate.

The kernel of the map f : Γ −→ G consists of those ` = (`±11 `±1

2 · · · `±1k )

such that f(`) = 1 and since f(`), g(`) are conjugate, this is the case ifand only if g(`) = 1. Hence ker(f) ≡ ker(g) and therefore f(Γ), g(Γ) areisomorphic, whose isomorphism is denoted by α. Evidently α is unambigu-ously defined by α(f(`)) = g(`) and it is immediate that α is an isometryand stabilises conjugacy classes. Hence the same is true of its extension,α : f(Γ) −→ g(Γ). When f(Γ) = G, a simple exercise shows that α issurjective, i.e. α : G −→ G is an automorphism.

9 Conjugate Weights Theorem

We can now use the results from the previous section to address the questionsposed at the beginning of that section.

18

Theorem 9.1 Let φ : X −→ X be hyperbolic and let f, g ∈ Cα(X,G) (G ⊂U(d)) have conjugate weights for all periodic points. If φf is ergodic thenthere exists an automorphism α : G −→ G (stabilising conjugacy classes)such that αf and g are cohomologous.

ProofSince f(Γ) is dense α is defined on G. The fact that α is an isometry enablesus to conclude that α : G −→ G is surjective. Clearly

f−(x)f(x) = f(z)−1f(φx)

g−(x)g(x) = g(z)−1g(φx) (for x ∈ W (z))

and since α is defined on all of G,

αg−(x)αg(x) = αg(z)−1 · αg(φx) .

However, f(z) = g(z) = αf(z) and therefore

g(x)−1g−(x)−1αf−(x)αf(x) = g(φx)−1αf(φx) .

Defining h(x) = g−(x)−1αf−(x) we have

g(x)−1h(x)α(f(x)) = h(φx)

i.e. α(f(x)) = h(x)−1g(x)h(φx) for x ∈ W (z). Since W (z) is dense in X(since one can easily see that by irreducibility homoclinic points on Σ aredense, and then take the image under π), it suffices to prove that h definedon W (z) is uniformly Holder continuous.

Let x, y ∈ W (z), d(x, y) < ε, ε > 0 and w the homoclinic point associatedwith {φ−nx}, n ≤ 0 and {φny}, n ≥ 0. Then

|h(x)− h(y)| ≤ |h(x)− h(w)|+ |h(w)− h(y)| .

Evidently

|h(x)− h(w)| = |g−(x)−1αf−(x)− g−(w)−1αg−(w)|≤ |g−(x)− g−(w)|+ |f−(x)− f−(w)|

since α is an isometry. Furthermore,

|h(w)− h(y)| = |g−(w)−1αf−(w)− g−(y)−1αf−(y)|= |g(w)g+(w)αf+(w)−1αf(w)−1 − g(y)g+(y)αf+(y)−1αf(y)−1|≤ |g(w)g+(w)− g(y)g+(y)|+ |f(w)f+(w)− f(y)f+(y)| .

19

From these estimates it follows that

|h(x)− h(y)| ≤−1∑

n=−∞

(|g(φnz)− g(φny)|+ |f(φnx) + f(φnw)|)

+∞∑

n=0

(|g(φnw)− g(φny)|+ |f(φnw)− f(φny)|)

≤−1∑

n=−∞

(|g|α + |f |α) d(φnx, φnw)α +∞∑

n=0

(|g|α + |f |α)

+∞∑

n=0

(|g|α + |f |α) d(φny, φnw)α

≤ C(α)d(x, y)α

and therefore h is uniformuly Holder on W (z). �

Corollary 9.2 Under the above conditions there exist a compact Lie groupG′ ⊃ G such that G′/G is finite and f, g are cohomologous with respect toG′, i.e. there exists h ∈ Cα(x,G′) such that f = h−1ghφ.

The proof depends on the facts that

1. α is an isometry

2. the group of automorphisms of a compact simple group modulo innerautomorphisms is finite, and

3. every compact Lie group is the quotient by a finite group of a directproduct of a semi-simple Lie group with a torus.

Above we used the ergodic hypothesis to ensure that α was defined ev-erywhere. In principle, this assumption is not needed, or rather it is only apossible topological obstruction which prevents a complete characterisationof the conjugate weights condition. More precisely the proof in the generalcase would work if we could assume that an ergodic component had a trivialbundle structure, or equivalently that there was a Holder cross-section to thiscomponent. This is the case, for example, if X is zero-dimensional so thatwe have:

Theorem 9.3 If φ : X −→ X is hyperbolic, where X is zero-dimensionaland if f, g ∈ Cα(X,G) (G ⊂ U(d)) satisfy the conjugate weights condition forperiodic orbits then there exists a closed subgroup H ⊂ G, an isometric iso-morphism α : H −→ α(H) which stabilises conjugacy classes, and a functionf ′ ∈ Cα(X,H) such that f, g are cohomologous to f ′, α(f ′) respectively.

20

If we replace the conjugate weights condition by the much stronger as-sumption that f -weights and g-weights of closed orbits are identical then theanalysis is greatly simplified. No discussion of the group Γ is needed and ineffect the isomorphism α is the identity.

Theorem 9.4 If f, g ∈ Cα(X,G) and f(x), · · · f(φn−1x) = g(x) · · · g(φn−1x)whenever φnx = x then f and g are cohomologous.

The proof follows the main lines of the proof of theorem 9.1.Our final remarks on the conjugate weights theorem concern shifts of finite

type (i.e. the zero-dimensional case). If we suppose our functions f, g arelocally constant, there is no harm in supposing that f(x) = f(x0, x1), g(x) =g(x0, x1) where {xn} ∈ X. One can then show that if h ∈ Cα(X,G) satisfies

f(x) = h(x)−1g(x)h(φx)

then h is a function of the single coordinate x0, i.e. f(x) = f(x0).The problems we have discussed so far now specialise to the context of

graph theory, and theorems 9.3 and 9.4 become (with much simpler proofs):

Theorem 9.5 Let f, g be two edge functions (with values in G) of a finitestrongly connected oriented graph. Suppose that there is at most one edgeleading from one vertex to another.

(i) If the products in G of the edge weights around each closed loop areconjugate then there exists a closed subgroup H, an isomorphism α : H → Hwhich stablises conjugacy classes, an H valued edge function f ′, and a vertexfunction h such that f ′(v0, v1) = h(v0)

−1f(v0, v1)h(v1) and α(f ′(v0, v1)) =h(v0)

−1g(v0, v1)h(v1).(ii) If the products in G of the edge weights around each closed loop agree

then there exists a vertex function h such that f(v0, v1) = h(v0)−1g(v0, v1)h(v1).

10 Stable Mixing

Our final problem concerns the question as to whether skew-products whichare ‘near’ to mixing skew-products are themselves mixing. More preciselyif φ : X −→ X is hyperbolic, m is a (Holder) equilibrium state, and f ∈Cα(X,G) (G ⊂ U(d)) we say that φf is stably mixing if there exists ε > 0such that φg is mixing for all g with ‖g − f‖ < ε. Our aim is to prove thatthe set of mixing f ∈ Cα(X,G) (i.e. φf is mixing) contains a dense open setwhen

1. G is semi-simple, or

21

2. X is a connected hyperbolic attractor and G is a torus.

Here we shall consider the semi-simple case. In the next section we dealwith the case where G is a torus and X is an attractor. Putting thesetwo cases together using basic Lie group structure theory we can claim thefollowing result.

Theorem 10.1 If X is a connected hyperbolic attractor and G ⊂ U(d) thenthe set of f ∈ Cα(X,G) for which φf is mixing contains a dense open set.

For the rest of this section we assume G ⊂ U(d) is semi-simple. We shallneed the following result due to Field (c.f. [11])

Lemma 10.2 The set U = {(g, h) ∈ G × G : 〈g, h〉 = G} contains a denseopen set, where 〈g, h〉 means the smallest closed subgroup of G which containsg and h.

Using this it is straight forward to prove that there exists a dense set off ∈ Cα(X,G) for which φf is mixing. We choose x, y ∈ W (z) whose orbitsare distinct. For any f ∈ Cα(X,G) we consider the elements f(`(x)) andf(`(y)) of G. Together they may or may not generate G. However it is clearfrom the above lemma that we can make arbitrarily small perturbations off in neighbourhoods of x and y so that for the resulting function g we have〈g`(x), g`(y)〉 = G. By Theorem 2.2 it follows that φg is mixing.

The following lemma will be used to prove that {f ∈ Cα(X,G) : 〈f`(x), f`(y)〉 =G} is open.

Lemma 10.3 The map f 7→ f`(x), Cα(X,G) −→ G is continuous.

ProofSince f`(x) = f−(x)f(x)f+(x) it will suffice to prove f −→ f−(x) (and inthe same way f −→ f+(x)) is continuous. Let f, g ∈ Cα(X,G) and chooseN so that d(φnx, z) ≤ Kλn (0 < λ < 1) for |n| ≥ N . Evidently

|f(z)−nf−n(x)− g(z)−ng−n(x)|≤

∣∣f(z)−N(f(z)−(n−N)f(φ−nx) · · · f(φ−N−1x)

)f−N(x)− f(z)−Nf−N(x)

∣∣+

∣∣g(z)−N(g(z)−(n−N)g(φ−nx) · · · g(φ−N−1x)

)g−N(x)− g(z)−Ng−N(x)

∣∣+

∣∣f(z)−Nf−N(x)− g(z)−Ng−N(x)∣∣

=∣∣f(φ−nx) · · · f(φ−(N−1)x)− f(z)−(n−N)

∣∣ +∣∣g(φ−nx) · · · g(φ−(N−1)x)− g(z)−(n−N)

∣∣+

∣∣f(z)−Nf−N(x)− g(z)−Ng−N(x)∣∣

22

≤−(N−1)∑i=−n

(|f |α + |g|α) d(φix, z)α +∣∣f(z)−Nf−N(x)− g(z)−Ng−N(x)

∣∣≤ K (|f |α + |g|α)

λαN

(1− λα)+

∣∣f(z)−Nf−N(x)− g(z)−Ng−N(x)∣∣

which is independent of n > N . Hence |f`(x) − g`(x)| is dominated by thesame expression. Given ε > 0 we can choose N so that the first term isless than ε/2, when ‖g − f‖ < 1 (say) and then we can choose δ so that‖g − f‖ < δ implies that the second term is less than ε/2. �

If G is semi-simple then each f for which 〈f`(x), f`(y)〉 = G gives rise toa mixing skew-product φf . Hence

Theorem 10.4 When G is semi-simple, the set of f ∈ Cα(X,G) such thatφf is mixing contains a dense open set.

ProofWe need only note that {f : 〈f`(x), f`(y)〉 ∈ U} where U is the openset specified in Lemma 10.2 is an open subset of Cα(X,G) since f −→〈f`(x), f`(y)〉, Cα(X,G) −→ G×G is continuous. �

11 Stability of Mixing — the Torus Case

Let φ : X −→ X be hyperbolic map of a connected set, preserving the(Holder) equilibrium state, m. Here we consider functions f ∈ Cα(X,G)where G is a k-dimensional torus. In fact we shall take k = 2 since proofsare essentially unaltered for the general case k ≥ 2.

Consider the Bruschlinsky group H1(X,Z) of continuous maps of X tothe circle K modulo null-homotopic maps. Since the latter form a divisiblesubgroup H of C(X,K) we can write Cα(X,K) = H ×H0 where now H0 isthe subgroup of Cα null homotopic maps. Evidently

Cα(X,K) =⋃η,ζ

H(η, ζ)

where

H(η, ζ) = {(f, g) ∈ Cα(X,K)×Cα(X,K) : f is homotopic to η and g is homotopic to ζ}

Here we take η, ζ ∈ H. We fix η, ζ and concentrate on H(η, ζ) which we noteis a closed and open set.

23

For each F ∈ Cα(X,K) we write F = γF · e2πirF where γF ∈ H, rF ∈Cα(X,R) and for each γ ∈ H we write γ◦φ = φ∗γe2πisγ where sγ ∈ Cα(X,R)and φ∗ is the induced automorphism on H. Suppose (fn, gn) ∈ H(η, ζ)define non-mixing skew-products φfn,gn : X ×K ×K −→ X ×K ×K and(fn, gn) −→ (f, g), then we have non-trivial equations

Fn ◦ φ/Fn = e2πian · fknn gln

n

= e2πian(knun+lnvn)ηknζ ln

where an ∈ R, un, vn ∈ Cα(X,R), (kn, ln) 6= (0, 0) fn = ηe2πiun , gn =ζe2πivn , f = ηe2πiu, g = ζe2πiv and un −→ u, vn −→ v, (u = rf , v = rg).

Moreover, Fn◦φ/Fn = (γFn◦φ/γFn)e2πi(fFn◦φ−fFn ) = (φ∗γFn/γFn)e2πi(sγFn+fFn◦φ−rFn ) .

Thus (φ∗γFn/γFn) = ηknζ ln and sγFn+ rFn ◦φ− rFn = an + knun + lnvn where

an is a′n modified by an integer. We note that s : H −→ Cα(X,R)/Z is ahomomorphism and we let V be the closed subspace of Cα(X,R) spannedby {sγ : γ ∈ H} and 1. Hence knun + lnvn ∈ V mod Bα where Bα is theclosed (by an application of Livsic’s theorem c.f. [22]) subspace of Cα(X,R)consisting of coboundaries. When H has finite rank V is finite dimensionaland therefore V + Bα is closed. Clearly u, v are linearly dependent moduloV +Bα.

Theorem 11.1 If H has finite rank then φf,g is mixing if rf , rg are linearlyindependent modulo V +Bα.

Corollary 11.2 {(f, g) ∈ Cα(X,K ×K)} contains an open dense set whenH is of finite rank.

ProofIt is not difficult to see that Cα(X,R)/Bα and therefore Cα(X,R)/(Bα +V )is finite dimensional. It follows that

{(u, v) ∈ Cα(X,R× R) : u, v are linearly independent modBα + V }

is open and dense, from which the corollary follows. �

In conclusion it should be noted that Theorem 11.1 applies to the case ofa hyperbolic attractor:

Proposition 11.3 If X is a hyperbolic attractor then H has finite rank.

ProofLet γi, . . . , γn be integrally independent members of H and let U ⊃ X be a

24

manifold with boundary such that φU ⊂ interior of U and⋂∞

n=1 φnU = X.

Each γi can be extended to a circle valued function γ′i defined on an openneighbourhood V ⊃ X so choosing N large enough we have φNU ⊂ V sothat γ′1, . . . , γ

′n are integrally independent when restricted to φNU . Clearly

rankH1(UZ) = rankH1(φNU,Z) <∞ so that n ≤ rankH1(U,Z), i.e., H hasfinite rank. �

A somewhat different approach to this and related problems (based onthe geometric behaviour near homoclinic orbits) was introduced in [12]. Thishas the advantage of giving results for quite general attactors. Moreover,Pugh and Shub have studied stable ergodicity in the context of quite gen-eral partially hyperbolic systems [25]. We shall briefly discuss both of thesedevelopments in the next section.

12 Related results

12.1 Stable ergodicity and partially hyperbolic systems

In this subsection, we shall consider diffeomorphisms that preserve the nor-malised Riemannian volume. In recent work Pugh and Shub studied stableergodicity, in a more general sense, whereby any nearby volume preservingdiffeomorphism on the manifold is required to be ergodic. We shall brieflyreview the basic results. A very complete survey appears in [8].

Let φ : M −→ M be a C∞ diffeomorphism of a compact Riemannianmanifold whose metric we denote by d. The diffeomorphism φ : M −→M issaid to be partially hyperbolic if the tangent bundle TM can be written asa direct sum TM = Eu ⊕ E0 ⊕ Es (continuously split) such that

1. Dφ(Eu) = Eu, Dφ(E0) = E0, and Dφ(Es) = Es.

2. there exists constants c > 1, α < β < 1 < γ < δ such that

‖Dφnv‖ ≤ cαn‖v‖ for v ∈ Es

1

cδn‖v‖ ≤ ‖Dφnv‖ for v ∈ Eu

1

cβn‖v‖ ≤ ‖Dφnv‖ ≤ cγn‖v‖ for v ∈ E0

for n ≥ 0.

This is a natural generalization of the case of skew products, where φ isno longer necessarily an isometry in the direction of the bundle E0, but

25

instead the distortion associated to Dφ|E0 is dominated by the contractionand expansion in the stable and unstable directions, respectively.

Standard examples of partially hyperbolic diffeomorphisms include notonly the skew products we have focused on, but also other familiar examplessuch as quasi-hyperbolic toral automorphisms, time-one geodesic flows andframe flows.

Recall that we can associate to the bundles Es, Eu stable and unsta-ble manifolds defined by W s(x) = {y ∈ M : d(φnx, φny) → 0, as n → 0},W u(x) = {y ∈M : d(φ−nx, φ−ny) → 0, as n→ 0}, for each x ∈M . Follow-ing Pugh and Shub, we say that the diffeomorphism φ has the accessibilityproperty if for any points x, y ∈M we can find a path connecting x and y andlying in the union of finitely many pieces of stable and unstable manifolds.

Let us make the following assumptions on φ.

1. ‖Dxf |E0x‖.‖(Dxf |E0

x)−1‖ is sufficiently close to 1 (center bunching);

2. The distributions E0, E0 ⊕ Eu and E0 ⊕ Es are all integrable andtangent to the associated foliations (dynamically coherent); and

3. Any diffeomorphism sufficiently close to φ in the C2 topology has theaccessibility property (stable accessibility property).

Assume that φ preserves the normalised volume µ on M . In this moregeneral context, we say that φ is stably ergodic (in the general sense) ifany sufficiently C2-close volume preserving diffeomorphism φ′ to φ is alsonecessarily ergodic. The following remarkable result was shown by Pugh andShub.

Theorem 12.1 Assume that φ : M → M is a volume preserving partiallyhyperbolic diffeomorphism satisfying hypotheses 1, 2 and 3. Then φ is stablyergodic (in the general sense).

The proof of this theorem is based on a delicate version of the classicalHopf argument. The basic idea is to consider the forward (or backward)Birkhoff averages of continuous functions along orbits, which exist almosteverywhere by the pointwise ergodic theorem. Observe that these are equalfor points lying on the same stable (or unstable manifolds). The aim is toshow that these averages exist and are equal to a constant value for almostevery pair of points x, y ∈ M by linking them by suitable paths lying instable and unstable manifolds (using the accessibility property). One canthen deduce that the Birkhoff averages are equal for almost every point, andthus conclude µ is ergodic. However, the main technical problem is that

26

the continuous foliation corresponding to E0 is not necessarily absolutelycontinuous (an essential ingredient in the original Hopf argument). Oneof the key ideas of Pugh and Shub was to replace absolute continuity bythe more general concept of julienne quasi-conformality along stable (andunstable) leaves [8], which suffices for the Hopf argument and yet holds evenfor φ′ in a small C2 neighbourhood of φ.

In practise, there are effective estimates on the approximation requiredin hypothesis 1. Moreover, one immediately sees that centre bunching holdsautomatically if Dφ is an isometry on E0. For example, in the particular casethat φ is a compact group extension of an Anosov diffeomorphism we see thatit is partially hyperbolic and hypotheses 1 and 2 hold. In this context, Burnsand Wilkinson used Theorem 12.1 to prove the following interesting resultfor skew products [9]

Theorem 12.2 Assume φ is a skew product with respect to a compact semi-simple Lie group and a volume preserving Anosov diffeomorphism. Then φis stably ergodic (in the general sense) if and only if φ is ergodic.

Although such results show ergodicity is stable in a much broader sensethan we considered earlier, it does require more restrictive hypotheses. Firstly,it requires diffeomorphisms to be close in a stronger sense (i.e., in the C2

topology rather than the Holder topology). Secondly, it applies only to thenormalized volume, rather than more general equilibrium measures.

12.2 Skew products and homoclinic points

Recently, Field, Melbourne and Torok developed another approach to show-ing stability of mixing for skew products over hyperbolic sets X [12]. Inparticular, their approach has the distinct advantage that it works withoutrequiring X to be a connected attractor. Furthermore, they are also able toconsider the case of Cr skewing functions.

Let φ : M → M be a C2 diffeomorphism and let G be a compact Liegroup. Let X ⊂ M be a φ-invariant hyperbolic set and let m be an equilib-rium measure.

Theorem 12.3 Let r > 0. There is an open and dense set of Cr functionsf : M → R such that the associated skew product φf : X × G → X × G isergodic (and mixing).

We shall briefly explain the main ideas in the proof. Let us assume fordefiniteness that G = K (the circle) [14]. The starting point is the analogueof the results we described in section 11. Let us write f = e2πiF , say. In

27

particular, φf is ergodic provided F 6∈ V +B where, in the present setting, Bis the space of Cr boundaries and V is the space of locally constant functions;and we take the closure in the Cr topology.

A key ingredient in the proof is to show good asymptotic bounds on theaverage of functions along a sequence of periodic points approximating ahomoclinic point. More precisely, consider F ∈ Cr(M,R). Let p0 denote aperiodic orbit (which, for simplicity, we may assume to be a fixed point). Wecan also assume, for convenience, that F (p0) = 0. Assume that x ∈ W s(p0)∩W u(p0)− {p0} is an associated (transverse) homoclinic point. In particular,since we have assumed F (p) = 0, and F is Lipschitz, the summation of Falong the orbit of x, which we can denote by F∞(x) :=

∑∞i=−∞ F (φix), is

finite. One can choose a sequence of periodic orbits pN (N ≥ 1) with periodsqN (i.e., φqN (pN) = pN), say , such that pN → x as N → +∞. In particular,the difference

F∞(x)− F qN (pN)

can be made arbitrarily small, by choosing N sufficiently large. If we assumethat F ∈ V +B then for N sufficiently large we also know that F qN (pN)vanishes. In particular, this implies that F∞(x) = 0. Using very detailedestimates on the behaviour of orbits, Field, Melbourne and Torok show thatfor generic F (in the Cr topology) one has that F∞(x) 6= 0. Thus, for suchF we deduce that F 6∈ V +B and, in particular, φf is ergodic.

12.3 Different function spaces

A central idea in these notes has been the regularisation of functions usingtransfer operators. If we start in the context of hyperbolic maps, this involvesusing Markov partitions (and Bowen’s Theorem) to first introduce a two sidedsubshift of finite type, and then a one sided version. However, a recent idea ofGoetszel and Liverani is suggestive of an alternative approach, which we shallbriefly explore [13]. Let φ : M → M be a transitive Anosov diffeomorphismon a compact manifold which preserves the normalised Riemannian volume.Consider the operator Uφ : L2(µ) → L2(µ) given by Uφw = |Jac(φ)|w ◦ φ.

Theorem 12.4 [13] There exists an invariant Banach space (of Schwartziandistributions B) for which the operator Uφ : B → B is quasi-compact.

RemarkThe Banach space B is quite complicated to describe, but its elements arechosen to exploit the hyperbolicity of φ. A function in B has more regularitythan a typical element of L2(µ). Let W denote a piece of smooth manifold

28

with the same dimension as, and which lies at a small angle θ, say, to, theunstable manifolds. Consider functions f : M → R which are smooth, andconsider all derivatives, up to order k, say, when restricted to W . We canthen associate linear functionals on C l(M), for l ≤ k − 1, say, by takingthe inner product with these derivatives. Finally, B is the completion of thesmooth functions with respect to norm given by the supremum over f in theunit ball of Ck(W ) of linear functionals.

The analogue of Theorem 5.5 is the following.

Theorem 12.5 The space B decomposes as B = V0 + V1 where Uφ : V1 −→V1, Lf : V0 −→ V0 and

1. ρ(Uφ |V0) < 1,

2. dim(V1) ≤ d

3. V1 is spanned by (generalized) eigenvectors corresponding to eigenvaluesof modulus 1.

This suggests the possibility of repeating the proof of Theorem 6.1 toshow that any measurable solution to certain coboundary identities mustalso be in B. More precisely, assume that

w ◦ φ = |Jac(φ)|w.Theorem 12.6 Let Uφw = w a.e., where w ∈ L2(µ,C). Then there existsw1 ∈ B such that w = w1 a.e.

ProofLet L2 = V ⊥

1 ⊕ V1 then V ⊥1 ⊃ V0 and B = V0 ⊕ V1. Choose ε > 0 and

v ∈ B such that∫|v − w|2 dµ < ε2. Then v = v0 + v1, w = w0 + w1 where

v0 ∈ V0, v1 ∈ V1, w0 ∈ V ⊥1 , w1 ∈ V1. Thus ε2 ≥

∫|v − w|2 dµ =

∫|v0 −

w0|2 dµ+∫|v1−w1|2 dµ and hence

∫|v0−w0|2 dµ ≤ (

∫|v0−w0|2 dµ)1/2 ≤ ε.

Furthermore UφV⊥1 ⊂ V ⊥

1 . To see this suppose 〈u, u1〉 = 0 whenever Uφu1 =αu1 (|α| = 1) or equivalently fu1 = αu1 ◦ σ then 〈u, αf−1u1 ◦ σ〉 = 0 whichimplies that 〈Uφu, αu1〉 = 0. This shows that Uφu ⊥ V1 when u ⊥ V1.

Hence∫|Un

φ v0 − Unφw0| dµ ≤

∫|v0 − w0| dµ ≤ ε and since Un

φ v0 −→ 0 wehave limn→∞

∫|Un

φw0| dµ ≤ ε, i.e. w0 = 0 a.e. and w = w1 a.e. �

The above identity sometimes occurs quite naturally. For example, ifan Anosov diffeomorphism φ preserves an absolutely continuous invariantmeasure then the density w ∈ L1(µ) satisfies this type of identity. (In theparticular case that φ preserves the volume then, of course, we have the trivialsolution w = 1.) However, in this case de la Llave, Marco and Moriyan haveused Sobolev regularity techniques to show that if φ is smooth then so is thedensity w.

29

13 Non-uniformly hyperbolic systems

One can ask if there are Livsic theorems under weaker conditions on φ thanhyperbolicity. One approach is to consider the broader class of non-uniformlyhyperbolic systems. Here, the hyperbolicity is typically restricted to a set offull measure, where it is characterized by non-zero Lyapunov exponents. Oneapproach, which has been successful in particular settings, is to replace thesubshift of finite type by a tower, in the sense of L.-S. Young [31]. This allowsthe system to be modelled by a countable state subshift. This approach toLivsic theorems was considered by Bruin, Nicol and Holland [7]. We shalldescribe an approach which is closer in spirit to the original work of Livsicfor hyperbolic systems (cf also [18]).

Let φ : M → M be a C∞ diffeomorphism of a compact Riemannianmanifold. Assume that

1. φ preserves the normalised volume µ;

2. the Lyapunov exponents of µ are non-zero.

In the particular case that M is a surface and h(µ) > 0 then condition 2.holds automatically.

Let G be a compact Lie group. Given f ∈ Cα(M,G) consider thecoboundary identity

f = h−1h ◦ φ a.e.,

where h : M → R is measurable. Observe that by Luzin’s theorem, for anyε > 0 one can find a (compact) set X ⊂M with µ(M−X) < ε and such thath|X is continuous. The following theorem improves this to a Holder bound.

Theorem 13.1 For any ε > 0 one can find a set Y ⊂M with µ(M−Y ) < εand h|Y is Holder continuous.

When we have non-zero Lyapunov exponents, many of the propertiesfamiliar in the hyerbolic theory remain true in a weaker form.

The proof of Theorem 13.1 is based on the following 3 lemmas. The firstis due to Pesin [24].

Lemma 13.2 There exists a family Λ1 ⊂ Λ2 ⊂ · · ·M of compact sets suchthat

1. µ(Λn) ↗ 1 as n→ +∞;

30

2. For each n ≥ 1 exists δn > 0 such that for x ∈ Λn the local stablemanifold W s

loc(x) = {y ∈ M : d(φnx, φny) ≤ δn, n ≥ 0} and the localunstable manifold W u

loc(x) = {y ∈ M : d(φ−nx, φ−ny) ≤ δn, n ≥ 0} areembedded disks of uniform size which meet transversely at x;

3. There is a Cα dependence Λn 3 x 7→ W sloc(x) and Λn 3 x 7→ W u

loc(x);

4. The laminations {W uloc(x)}x∈Λn and {W s

loc(x)}x∈Λn are both absolutelycontinuous.

This lemma naturally leads to non-uniform “local product structure”.More precisely, for n ≥ 1 there exists εn > 0 such that for x, y ∈ Λn withd(x, y) ≤ εn the local stable and unstable manifolds intersect at a single pointz = z(x, y), say, i.e., W s

loc(x) ∩W uloc(y) = {z}.

Let us assume ε < 14. By part 1 of Lemma 13.2, we can choose n suffi-

ciently large that µ(Λn) ≥ 1 − ε. In particular, if we write X0 = X ∩ Λn,then we have µ(X0) ≥ 1− 2ε > 1

2, say.

Lemma 13.3 For a.e. x ∈ X0, we have that

limn→∞

1

n

n∑i=1

χX0(φix) = µ(X0) >

1

2.

Moreover, the same estimate holds for a.e. y ∈ W sloc(x) and a.e. z ∈ W u

loc(x)

ProofThis follows immediately from the Birkhoff ergodic theorem. �

Without loss of generality, we can replace X0 by its density points (forthe measure µ). The following lemma is essentially due to Livsic [17].

Lemma 13.4 1. For a.e. x, y ∈ X0 with d(x, y) < εn, say, one canfind x1, x2, x3 such that x1 ∈ W u

loc(x), x2 ∈ W sloc(x1) ∩ X0 and x3 ∈

W uloc(x2) ∩W s

loc(y);

2. There exists K > 0 such that for any Kd(x, y) ≥ d(x, x1) + d(x1, x2) +d(x2, x3) + d(x3, y).

We can now modify the standard argument of Livsic ([17], p.1299) toprove Theorem 13.1. Let x ∈ X0 ∩W s

loc(y), say. For any n ≥ 1 exists D > 0

31

and 0 < θ < 1 such that d(φnx, φny) ≤ Dθnd(x, z). For any n ≥ 1 we canwrite

|h(x)− h(y)| = |f(x)−1 . . . f(φn−1x)−1h(φnx)− f(y)−1 . . . f(φn−1y)−1h(φny)|≤ |[f(x)−1 − f(y)−1]f(φx)−1 . . . f(φn−1x)−1h(φnx)|+ |f(y)−1[f(φx)−1 − f(φy)−1]f(φ2x)−1 . . . f(φn−1x)−1h(φnx)|

· · ·+ |f(y)−1 · · · f(φn−2y)−1[f(φn−1x)−1 − f(φn−1y)−1]h(φnx)|+ |f(y)−1 · · · f(φn−1y)−1[h(φnx)− h(φny)]|≤ Cd(x, y)α + |h(φnx)− h(φny)|

where C = |f−1|αDα/(1 − θα). By Lemma 13.3 we can find a subsequencenk such that φnk(x), φnk(y) ∈ X0. Thus, letting k → +∞ we see that

|h(x)− h(y)| ≤ Cd(x, y)α

Similarly, if x ∈ X0∩W uloc(y) then we can show that |h(x)−h(y)| ≤ Cd(x, y)α.

Finally, by Lemma 13.4, for x, y ∈ X0 we can associate x1, x2, x3 and applythe above bounds to write:

|h(x)− h(y)| ≤ |h(x)− h(x1)|+ |h(x1)− h(x2)|+ |h(x2)− h(x3)|+ |h(x3)− h(y)|≤ Cd(x, x1)

α + Cd(x1, x2)α + Cd(x2, x3)

α + d(x3, y)α

≤ 4CKd(x, y)

This completes the proof of Theorem 13.1.

14 Some open questions

Despite progress in recent years (or, perhaps, even because of it) there remaina number of interesting open questions and conjectures. In this final sectionwe shall briefly recall a few of these.

Question 14.1 When can we replace the compact Lie group G by other non-compact groups in the stable ergodicity results?

In simplest case G = Rd, Nitica and Pollicott gave conditions (which areopen, but far from dense) on the skewing function f : X → Rd to make theskew product φf transitive [19]. This work has been extended in [12]. On theother hand, a natural candidate as a generalization of ergodicity is exactness,which has been studied in the context of Rd extensions of the one-sided shiftσ : Σ+ → Σ+ by Guivarc’h and Aaronson and Denker.

32

Another direction in which one could try to extend the stable ergodicityresults is by generalizing the class of base transformations φ : M → M . Forexample, an obvious question is the following.

Question 14.2 If we replace the base transformation φ : X → X by apartially hyperbolic diffeomorphism, what are the analogous results on stableergodicity for skew products?

There has been work on Livsic type theorems for partially hyperbolicmaps by Katok and Kononenko which may have a bearing on this question.

The results of Pugh and Shub depend on various technical hypothesesthat may eventually prove to be unnecessary. Indeed, they have formulatedthe following natural conjecture [8].

Conjecture 14.3 On any compact manifold M , ergodicity holds for an openand dense set of C2 volume preserving hyperbolic diffeomorphisms.

There are partial results in this direction in low dimensions.In [12], Field, Melbourne and Torok show that show that for a C2 hyper-

bolic diffeomorphism φ and an equilibrium state the family {f ∈ Cr(M,G) : φf is ergodic}is open in the C2 topology. In [12] the following question is also posed:

Question 14.4 Do the corresponding results hold in the case that f ∈ Cs,for 1 ≤ s < 2?

A closely related concept to stable ergodicity is that of robust transitiv-ity. We say that a diffeomorphism φ : M → M is robustly transitive if anysufficiently close φ′ : M →M (in a suitable topology) is also transitive. (Al-though transitivity is a weaker property than ergodicity, we no longer restrictto volume preserving diffeomorphisms). This property has been studied bya number of people (most notably by Bonatti and Diaz) but there remain anumber of natural open problems.

Question 14.5 When is a partially hyperbolic diffeomorphism robustly tran-sitive? For example, are time one geodesic flows and frame flows robustlytransitive? When are skew products robustly transitive?

Finally, we conclude with a very well known conjecture which has beenopen for several decades, but whose positive solutions would have a strongbearing on the problems of stable ergodicity.

Conjecture 14.6 Let φ : M → M be a transitive Anosov diffeomorphismon a compact manifold. The linear action φ∗ : H1(M,R) → H1(M,R) on thefirst real homology does not have 1 as an eigenvalue.

33

An equivalent formulation of this conjecture is that any (Holder) contin-uous function f : M → R which sums to integer values around closed orbitsmust necessarily be cohomologous to a constant.

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