NONSTANDARD SMOOTH REALIZATIONS OF LIOUVILLE ROTATIONS

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1 NONSTANDARD SMOOTH REALIZATIONS OF LIOUVILLE ROTATIONS BASSAM FAYAD, MARIA SAPRYKINA, AND ALISTAIR WINDSOR Abstract. We augment the method of C conjugation approximation with explicit estimates on the conjugacy map. This allows us to construct ergodic volume preserving diffeomorphisms measure-theoretically isomorphic to any apriori given Liouville rotation on a variety of manifolds. In the special case of tori the maps can be made uniquely ergodic.. Introduction We call a diffeomorphism f of a compact manifold M that preserves a smooth measure µ a smooth realization of an abstract system (X, T, ν) if they are measuretheoretically isomorphic. A diffeomorphism of a compact manifold has finite entropy with respect to any Borel measure. The natural question therefore becomes whether every finite entropy automorphism of a Lebesgue space has a smooth realization. This problem remain stubbornly intractable and there remain abstract examples that have no known smooth realizations. We seek to find smooth realizations of one of the simplest types of automorphisms; aperiodic automorphisms with pure point spectrum with a group of eignevalues with a single generator. Such automorphisms are measure theoretically isomorphic to irrational rotations of the circle. They therefore have a natural smooth realization. We seek smooth realizations on manifolds other than T. Such relizations are called non-standard smooth realizations. We extend the conjugation approximation method of Anosov and Katok [] to construct non-standard smooth realizations of a given Liouville rotation on T on a variety of manifolds M. Indeed, in the special case that the manifold is T d for d 2, we can produce uniquely ergodic realizations of the given Liouville rotation. The crucial new ingredient is an explicit construction of the conjugating maps that allows us to estimate their derivatives. This allows us to ensure that the construction converges for a predetermined Liouville number α. The approach parallels that taken in [3]. The original paper of Anosov and Katok paper constructed nonstandard smooth realizations of a dense set of Liouville rotations. However, without estimates, it was not possible to identify which Liouville rotations could be realized. Definition. A number α R\Q is a Liouville number if for all k > 0 we have () lim inf q qk qα = 0 where qα = inf p Z qα p. M. Saprykina would like to thank the University of Texas at Austin and Rafael de la Llave for support and hospitality. A. Windsor gratefully acknowledges the generous support of the Université Paris 3 and the CNRS as well as the hospitality of the Laboratoire de Probabilités et Modèles Aléatoires.

2 2 B. R. FAYAD, M. SAPRYKINA, AND A. WINDSOR Let T d := R d /Z d denote the d-dimensional torus. Let R θ : T T be the rotation of the circle, taken with the Haar probability measure, given by R θ (x) = x + θ mod. Denote by Diff (M, µ) the class of C diffeomorphisms of M that preserve a C smooth volume µ. Throughout this paper we will use λ for the probability measure induced by the standard Lebesgue measure. Theorem. Let M be a compact connected manifold, possibly with boundary, of dimension at least 2 that admits an effective C action of T preserving a C smooth volume µ. For every α R\Q Liouville there exists an ergodic T Diff (M, µ) measure-theoretically isomorphic to the rotation R α. In the special case M = T d we can strengthen the result to obtain unique ergodicity. Theorem 2. For every Liouville α R\Q, and every d 2 there exists a uniquely ergodic transformation T Diff (T d, λ) such that T is measure-theoretically isomorphic to the rotation R α. It remains open whether there are C realizations of Diophantine rotations on any manifold other than T. 2. Construction 2.. Outline. The required measure preserving diffeomorphism T is constructed as the limit of a sequence of periodic measure preserving diffeomorphisms T n. For each of the properties that we wish the limiting diffeomorphism T to possess, we establish an appropriate finitary version possessed by the periodic diffeomorphism T n. Let S : T M M denote an effective C action of T on M that preserves the volume and denote by S α the diffeomorphism S(α, ). The diffeomorphism T n is given by (2) T n := H n S αn Hn where α n Q and H n Diff (M, λ). We choose a sequence α n := p n/q n such that α n α 0 monotonically. This choice defines a sequence of intermediate scales by q n = qn d q n satisfying q n < q n < q n+ which are geometrically natural for all the previous transformations. Fixing q n determines H n+ via the iterative formula (3) H n+ = H n h n,qn. Defining the family of maps h n,q and investigating their properties will form the bulk of this paper Reduction. Though Theorem appears considerably more general than Theorem 2 they follow from nearly identical arguments. We are able to reduce the case of a general M admitting a smooth C action of T to the case of M = I d T, where I = [0, ] is the standard unit interval, with S θ : I d T I d T given by S θ (x,...,x d ) = (x,..., x d, x d + θ mod ). Let σ denote the effective T action on M. For q we denote by F q the set of fixed points of the map σ(/q, ) and let B := M q F q be the set of exceptional points.

3 NONSTANDARD SMOOTH REALIZATIONS OF LIOUVILLE ROTATIONS 3 We quote the following proposition of [2] that is similar to other statements in [, 6] Proposition. [2, proposition 5.2] Let M be an d-dimensional compact connected C manifold with an effective circle action σ preserving a smooth volume µ. Then here exists a continuous surjective map Γ : I d T M with the following properties () The restriction of Γ to (0, ) d T is a C diffeomorphic embedding; (2) µ(γ( (I d T)) = 0; (3) Γ( (I d T)) B; (4) Γ (λ) = µ; (5) σ Γ = Γ S. An application of Proposition at each step allows us to conclude Theorem from the special case M = I d T. Thus the construction need only be carried out for two specific manifolds; M = T d or M = I d T. For both we take the action S θ : M M given by S θ (x,..., x d ) = (x,...,x d, x d + θ mod ) that preserves the smooth unit volume λ induced by the usual Lebesgue measure on R d Partitions and Measure-Theoretic Isomorphism. The most difficult property to define on a finite scale is that of measure-theoretic isomorphism to a circle rotation. We use the abstract theory of Lebesgue spaces. Given an isomorphism of measures space (M, B, µ ) and (M 2, B 2, µ 2 ) there is a natural isomorphism of the associated measure-algebras. If both the measure-spaces are Lebesgue spaces then the converse is true; every isomorphism of the measure-algebras arises from a point isomorphism of the measure spaces. This is the crucial observation that leads to the follwing abstract lemma, which appears as [, Lemma 4.]. Given a partition ξ of a space M we write ξ(x) for the atom of the partition which contains x. We say that a sequence of partitions ξ n generates if there is a set F of full measure such that for every x F we have {x} = F ξ n (x). Lemma. Let M and M 2 be Lebesgue spaces. Let (ξ n (i) ) n= be a monotone sequence of finite measurable partitions of M i that generates. Let (T n (i) ) n= be a sequence of automorphisms of M i such that () (T n (i) ) n= converges in the weak topology to an automorphism T (i) of M i. (2) T n (i) ξ n (i) = ξ n (i). Suppose that for each n there exists a measure-theoretic isomorphism K n : M /ξ n () M 2 /ξ n (2) of the probability vectors such that: () Kn T n (2) (2) ξ n (2) for all ξ () n K n = T () n ξ () n. n= K n = K n. Then the automorphisms T () and T (2) are measure-theoretically isomorphic.

4 4 B. R. FAYAD, M. SAPRYKINA, AND A. WINDSOR Consider the partition of T given by (4) η q := { i,q : 0 i < q d } where i,q := [iq d, (i + )q d ). This partition is preserved under the action R p/q. For any increasing sequence of q n the sequence of partitions η qn generates. Let M 2 = T, ξ n (2) = η qn and T n (2) = R αn. Since q n divides q n+ we have η qn < η qn+. Let π d : M T denote the projection onto the last component of M. We obtain a partition of M by (5) η q = π d η q = { i,q : 0 i < q d } where i,q := {x : x d [iq d, (i + )q d )}. Since π d S α = R α π d the partition η q is preserved under the action of S p/q and, moreover, the action of S p/q on η q is conjugated with that of R p/q on η q. Unfortunately the sequence of partitions η qn does not generate. 9,3 9,3 π d,3,3 Figure. The partition η 3 of either I T or T 2 and the partition η 3 of T. Let M = M and define the sequence of partitions (6) ξ () n := H n+ η qn = H n h n,qn η qn. Unlike the sequence η qn, the sequence ξ n () can be made to generate. We construct h n,q as a diffeomorphism of π d [0, q ] and extend it to all of M by requiring that it commute with S q. Then () Since q d n divides q n we have for 0 i < q d n (2) Since q n divides q n we have h n,qn i,qn = i,qn. h n,qn S αn = S αn h n,qn. As η qn < η qn we have H n+ η qn < H n+ η qn. By the first of our two properties we have that H n+ η qn = H n η qn and hence ξ () n < ξ() n. Thus {ξ n () } is a monotone sequence of partitions as required by Lemma. The second property ensures that T n ξ n () = ξ n (). Define the map K n = π d H n+.

5 NONSTANDARD SMOOTH REALIZATIONS OF LIOUVILLE ROTATIONS 5 Using the two properties we have that K n T n () = T n (2) K n K n (H n i,qn ) = K n (H n i,qn ) as required by Lemma. This completes the proof of the main theorem except for the proof that the sequence T n converges in Diff (M, λ) and the proof that ξ n () generates Construction of the Conjugating Maps. We will carry out the constructions for M = T d and M = I d T simultaneously. The proof of unique ergodicity in the case M = T d will appear in a later section. Lemma 2. Let n > 2d and q N. There exists a map h n,q Diff (M, λ) and a set E n,q M such that: ( () h n,q S q = S q h n,q and h n,q π d [0, q ] ) = π d [0, q ]. (2) λ(e n,q ) > 4 d n. 2 (3) for each 0 i < q d, diamh n,q ( i,q E n,q ) < dq Heuristic Construction. In order to motivate the construction of the family of conjugacy maps we first construct a family of measure-preserving discontinuous maps h q such that h q commutes with S q and carries each i,q into a d-dimensional cube with side-length q. 9,3 8,3 7,3 6,3 5,3 4,3 3,3 2,3,3 φ 3 φ 3 7,3 φ3 8,3 φ3 9,3 φ 3 4,3 φ3 5,3 φ3 6,3 φ 3,3 φ3 2,3 φ3 3,3 Figure 2. Action of φ 3 = h 3 on the partition η 3. Let φ q be defined on [0, ] [0, q ] by letting it act on the interior by φ q (x, y) := (qy, q ( x)) and extend it to all of [0, ] [0, ] by requiring φ q (x, y +q ) = φ q (x, y)+(0, q ). (i) Define φ q by [ φ q ] (x i, x i+ ) j = i (7) [ φ (i) q ] j (x,...,x d ) = [ φ q ] 2 (x i, x i+ ) j = i + otherwise x j The map h q is defined by h q := () (d ) φ q φ q.

6 6 B. R. FAYAD, M. SAPRYKINA, AND A. WINDSOR Each i,q is mapped, by h q, into a cube of side-length q. The map h q commutes (d ) (i) with S q since φ q commutes with S q by construction and the other φ q don t affect x d Proof of Lemma 2. Our family of conjugating maps h n,q is constructed using the same process as h q above. Clearly control of some of the space must be relinquished in order to be able to produce a C volume preserving map. One additional complication arises ensuring that we retain sufficient control over every orbit. Let ϕ n denote a C map of the unit square satisfying () ϕ n is C flat on the boundary. (2) ϕ n acts as a pure rotation by π 2 on [ ] [ n, 2 n ] 2 n, 2 n. 2 (3) ϕ n preserves Lebesgue measure. Let C q (x, y) := (x, q y) and define φ n,q on [0, ] [0, q ] by (8) φ n,q := C q ϕ n C q. Extend φ n,q to the entire unit square by requiring that φ n,q (x, y + q ) = φ n,q (x, y) + (0, q ). This agrees with φ n,q on a set of volume ( 2/n 2 ) 2 which we estimate from below by 4/n 2 (i). Analogously to our earlier definition of φ q we define φ n,q. (i) [φ q ] (x i, x i+ ) j = i [φ (i) q ] j (x,...,x d ) = [φ q ] 2 (x i, x i+ ) j = i + otherwise x j M = I d T Case. We define the conjugating map h n,q : I d T I d T by h n,q := φ () n,q φ(d ) n,q. This map agrees with h q on a set E n,q given by (9) E c n,q = d i= ( [0, π ) ( i n 2 d n 2, ]) q j j= k= Treating the sets on the right as disjoint we can estimate (0) λ(e n,q ) > 4 d n 2. π d ( k q j n 2 q j, k q j + n 2 q j ) M = T d Case. In order to produce a unique ergodic diffeomorphism T it is necessary to control all orbits. The set E n,q constructed above for the case of M = I d T excludes entire orbits. In order to rectify this requires one more map. Let ψ q : T d T d denote the translation () ψ q (x,...,x d, x d ) := (x,..., x d, x d ) + x d (q,..., q, 0) mod. Obviously ψ q commutes with S q and preserves the Lebesgue measure. Furthermore, since ψ q does not affect the last coordinate, it preserves each i,q. For the uniquely ergodic case we define (2) h n,q := φ () n,q φ n,q (d ) ψ q

7 NONSTANDARD SMOOTH REALIZATIONS OF LIOUVILLE ROTATIONS 7 Figure 3. The set E nq for the case M = I T (left) and for the case M = T 2 (right). Exactly as for the ergodic case h n,q agrees with h q on a set E n,q with λ(e n,q ) > 4 d n 2. The map ψ q ensures that E n,q contains most of every orbit.

8 8 B. R. FAYAD, M. SAPRYKINA, AND A. WINDSOR 2.5. Analytic Properties Notation. All of our diffeomorphisms h : I d T I d T are identity in a neighborhood of the boundary and hence can be identified with a diffeomorphism h : T d T d. Defining a topology on Diff k (T d, T d ) defines a topology on the closure of the space of diffeomorphisms h : I d T I d T that are identity in a neighborhood of the boundary. Let f, g C 0 (T d, T d ). We define ˆd 0 (f, g) = max x M d( f(x), g(x) ). Let f C k (R d, R). Given a N d we denote a := a + + a d and Using this we can define For f C k (R d, R d ) we define D a f := a f.... xa d x a f k = max a k max x M D af(x). f k = max max max i d a k d x M D af i (x). For h : T d T d we can define a natural lift ĥ : Rd R d. Now given f, g C k (T d, T d ) we define ˆd k (f, g) = max{d 0 (f, g), ˆf ĝ k } Finally, for f, g Diff k (T d, T d ) we define d k (f, g) = max{ ˆd k (f, g), ˆd k (f, g )} The metric defined in this way is equivalent to the usual one defined via the operator norms but is easier to work with for explicit estimates. For further details consult [5] Estimates. Lemma 3. We have the following estimate: (3) h n,q k < C q dk where C depends on d, k, and n but is independent of q. Proof. By direct computation we obtain (4) φ (i) n,q k < q k ϕ n k and (5) ψ q k < q. We claim that partial derivatives with a = k consist of sums of products of at most (d )k terms of the form ( (6) Db [φ n,q] (i) ) j (φ (i+) n,q... φ n,q (d ) ψ q ) with b k and at most k terms of the form (7) D c [ψ q ] j

9 NONSTANDARD SMOOTH REALIZATIONS OF LIOUVILLE ROTATIONS 9 with c =. This is true for a = by computation and, by the product and chain rules, if it is true for a = k then it is true for a = k +. By induction it is therefore true for all k. Now suppose the estimate (3) holds for k we wish to show it holds for k+. We use our structure theorem for k. Differentiating a term of the from (6) we get a sum of products of d+ i terms. The first is of the form (6) but with the power of the derivative raised by. The next d i terms are first partial derivatives. The final term is a first partial derivative of ψ q. Applying the estimates (4) we see that the required power of q has been increased by at most d. Differentiating (7) gives zero since ψ q is linear. of φ (i+) n,q,..., φ (d ) n,q By an application of the Faà di Bruno s formula we obtain the following corollary. Corollary. We have the following estimate (8) H n h n,q k < C 2 q kd where C 2 depends on H n, n, and k but is independent of q Completing the Construction. Having now constructed the family of maps h n,q from which the maps H n are assembled it remains only to explain how we choose the sequence q n. The choice of q n determines α n as the best approximation to α with denominator q n.the choices of q,..., q n completely determines H n. We show how given H n we choose q n so that T n has the desired properties. In the original Anosov and Katok method of construction the choice of α n in the definition of T n (2) determined the distance between the already determined T n and T n in Diff n. The observation there was that if α n could be chosen arbitrarily close to α n then the transformation T n could be made arbitrarily close to T n. The advantages of this approach are that no estimates on the maps H n are required. Unfortunately this approach is inconsistent with ensuring that the sequence α n converges to an a priori given number α. In the approach we take the choice of q n (and hence of α n ) determines the distance between T n and, the as-yet undetermined transformation, T n+. Since the choice of q n fixes the conjugacy map H n+ the only undetermined quantity in T n+ is the choice of α n+. Supposing only that the choice of α n+ will be a better approximation to α than α n we are able to estimate the distance between T n and T n+ knowing only the choice of α n. Lemma 4. Let k N. For all h Diff k (M) and all α, β R we obtain d k (h S α h, h S β h ) C 3 h k+ k+ α β where C 3 depends only on k. Proof. For k = 0 we have the estimate d 0 (h S α h, h S β h ) h α β by the mean value theorem. We claim that for a N d with a = k the partial derivative D a [h i S α h h i S β h ] will consist of a sum of terms with each term being the product of a single partial derivative ( ) (9) Db h i (Sα h ) ( ) D b h i (Sβ h )

10 0 B. R. FAYAD, M. SAPRYKINA, AND A. WINDSOR with b k, and at most k partial derivatives of the form (20) D b h j with b k. For k = we have x j [h i S α h h i S β h ] = d ( h i S α h h i S β h ) h l. x l x l x j l= We proceed by induction. By the product rule we need only consider the effect of differentiating (9) and (20). Differentiating (9) with respect to x j we obtain d l= ( D b h i x l S α h D bh i x l S β h ) h l. x j which increases the number of terms of the form (20) by. Differentiating (20) we get another term of the form (20) but with b k +. We estimate D a h i S α h D a h i S β h 0 h a + α β D a h l 0 h a These estimates together with claimed structure of the partial derivatives, and the fact that the inverse maps have the same structure, completes the proof. The constant C 3 is the number of terms in the sum which depends only on k and not on the map h. Define F n := H n+ (E n,qn ) and let F := liminf F n. Clearly, from Lemma 2, we have that λ(f) lim ( 4(d ) n m 2 ) =. m=n We will show that any point in F has a unique coding relative to the sequence of partitions ξ n. Proposition 2. Let ǫ n be a summable sequence of positive numbers. There is a choice of {q n} such the transformations T n defined by (2) satisfy () d n (T n, T n+ ) < ǫ n. (2) for A ξ n diam(a F n ) < ǫ n Proof. By the definition of a Liouville number for any polynomial P(q ) we can find q n > q n such that α n := p n/q n is a better approximation to α than α n and such P(q n ) p n q n α < ǫ n We will define q = qn d q to ensure that h n,q satisifes. Since q < (q ) d+ we that for any polynomial P(q) we can find find q n such that α n := p n/q n is a better approximation to α than α n and such P(q n ) p n q n α < ǫ n

11 NONSTANDARD SMOOTH REALIZATIONS OF LIOUVILLE ROTATIONS Now combining (8) and Lemma 4 we have Similarly for H n+ i,qn ξ n we have d n (T n, T n+ ) < P(q n ) α n α n+ < 2P(q n ) α n α. diam(h n+ i,qn F n ) = diam(h n h n,qn ( i,qn E n,qn )) H n diamh n,qn ( i,qn E n,qn ) H n dq n using Lemma 2. Thus we see that we can choose α n such that the required two properties hold. Since ǫ n is summable we have that {T n } is a Cauchy sequence in Diff (M, λ) and hence converges to some T Diff (M, λ). For any x F we have x F n for all but finitely many n. Thus, by Proposition 2, we have for all x F ξ n (x) F = {x}. n= This shows that {ξ n } is a generating partition and hence completes the proof of Theorem. 3. Unique Ergodicity When M = T d we wish to prove unique ergodicity. We will use the following abstract lemma, also used in [6]. Lemma 5. Let q n be an increasing sequence of natural numbers and T n : X X a sequence of transformations which converge uniformly to a transformation T. Suppose that for each continuous function ϕ from a dense set of continuous functions Φ there is a constant c such that (2) and q n q n i=0 (22) d (qn) (T n, T) := max x Then T is uniquely ergodic Proof. Condition (22) implies that ϕ(tn i x) c uniformly n max d(tn i x, T i x) 0 0 i<q n q n ϕ(t n x) q n ϕ(tx) 0 0 q n q n i=0 and then condition (2) becomes the standard result that if the Birkhoff sums converge uniformly then the map is uniquely ergodic [4]. To establish condition (2) it is insufficient to know only that E n,q has large measure, we also need to know that most of every S θ orbit intersects E n,q. For each x T d define σ x : T T d by σ x θ = S θ x. i=0

12 2 B. R. FAYAD, M. SAPRYKINA, AND A. WINDSOR σ x (T) Figure 4. The orbit of x T 2, indicated by the arrow on the left, combines with E n,q, indicated by the shaded region on the left, to produce the set J (x) n,q, indicated by the shaded region on the right. Lemma 6. Let q > dn 2. For each x T d there is a set J (x) n,q T d, measurable with respect to η q, with measure (23) λ(j (x) n,q) > 4d n 2 such that if i,q J (x) n,q then (24) (25) Proof. It is immediate that (26) (E n,q )c = d i= σ x ( i,q E c n,q ) =, λ( i,q E n,q ) > ( πi ( n 2, d n 2 ) j= k= 2(d ) n 2 ) λ( i,q ). q j π d ( k q j n 2 q j, k q j + n 2 q j ) Let x be arbitrary. We compute σx ψ q(e n,q )c using (26) and (). σx ψ q πi ( ) q ( l n 2, = n 2 q n 2 q x d x i q, l q + n 2 q x d x ) i q σx ψ q π d l= ( k q j n 2 q j, k q j + n 2 q j ) = ( k q j n 2 q j x d, k q j + n 2 q j x d This excluded set of τ consists of at most (d )q + q d intervals. Expanding these intervals to make them measurable with respect to σx η q excludes an additional set of measure at most 2 ( (d )q + q d ) q d < 4 n 2. Let E denote the measurable hull of σx Ec n,q in σ x η q. We have λ(e) = 4d/n 2. Define the set J n,q (x) to be the η q measurable set satisfying σ x J (x) n,q = E c. )

13 NONSTANDARD SMOOTH REALIZATIONS OF LIOUVILLE ROTATIONS 3 Note that the proportion in (23) is lower than the proportion in (0). We have had to give up control over parts of each orbit in order to gain control over all orbits. The set J n,q (x) consists of those atoms of η q where we have control over the behaviour of all of S θ x under h n,q. Using the geometric information contained in these lemmas we can prove a distribution result. Proposition 3. Let ǫ > 0, q N, and ϕ be a ( dq d, ǫ)-uniformly continuous function, i.e For all q N and for all x T d, ϕ(b dq d (x)) B ǫ (ϕ(x)). (27) q q i=0 ϕ(h n,q S i /q x) ϕdλ < 4d n 2 ϕ 0 + 2qd q ϕ 0 + 2ǫ. Proof. For x, y i,q E n,q we have d(h n,q x, h n,q y) diamh n,q ( i,q E n,q ) dq d. By the hypothesis on ϕ we have ϕ(h n,q x) ϕ(h n,q y) < 2ǫ. Averaging over all y i,q E n,q we obtain for any x i,q E n,q, (28) ϕ(h n,qx) ϕdλ λ( i,q E n,q ) < 2ǫ. h n,q( i,q E n,q) Let O (x) consist of q q d q d points of the orbit of x under S /q that are equidistributed among the atoms of the partition η q. There are at most q d exceptional points outside of O (x). By (24) for i,q J n,q (x) the number of points from O (x) in i,q E n,q is q. q d Let I := {0 i < q : S/q i x J (x) n,q O (x) } be the equidistributed points in good atoms. Using this count and (28) we obtain q ϕ(h n,q S/q i x) i I q i,q J (x) n,q q q d λ( i,q E n,q ) h n,q( i,q E n,q) ϕdλ < 2ǫ. The remaining estimates just formalize the observation that since J (x) n,q is nearly full measure and since I is nearly all of the orbit the above estimate implies (27).

14 4 B. R. FAYAD, M. SAPRYKINA, AND A. WINDSOR First we produce estimates that account for the fact that q d does not divide q and hence we do not have equidistribution of the entire orbit. q q q d ϕdλ λ( i,q E n,q ) i,q J (x) n,q q q i=0 i,q J (x) n,q h n,q( i,q E n,q) q d λ( i,q E n,q ) h n,q( i,q E n,q) ϕdλ < qd q ϕ 0 ϕ(h n,q S/q i x) q ϕ(h n,q S/q i x) < qd q ϕ 0 i O (x) Second we produce estimates using (23) and (24) q ϕ(h n,q S/q i x) q ϕ(h n,q S/q i x) < 4d n 2 ϕ 0, i O (x) i I ϕdλ ϕdλ < 4d n 2 ϕ 0 h n,qj (x) n,q Finally we produce estimates using (25) ϕdλ ϕdλ 2(d ) < h n,q(j n,q E (x) n,q) h n,qj n,q (x) n 2 ϕ 0, q d ϕdλ λ( i,q E n,q ) h n,q(j (x) n,q E n,q) h n,q(j n,q E (x) n,q) Combining these estimates gives us exactly (27) as required. < 4(d ) n 2 ϕ 0. Let Φ = {ϕ n } be a set of Lipshitz functions that is dense in C 0 (M, R). Let L n be a Lipshitz constant for ϕ H n,...,ϕ n H n. At step n we can choose q n so that L n dq n < n 2 and q n > n2 q n. Then applying Proposition 3 we see that for ϕ {ϕ,, ϕ n } we have q n q n i=0 ϕ(t i n+ x) ϕdλ < 7d n 2 ϕ 0. This establishes (2) from Lemma 5. To establish (22) from 5 observe that d (qn) (T n, T n+ ) H n+ q n α n α n+ P(q n ) α n α and hence we can choose q n so that this is less than /n. In actual fact this estimate is weaker than those that arise in the proof of Proposition 2 and so is automatic. This verifies the hypotheses of Lemma 5 and hence we conclude that T is uniquely ergodic. References [] D. V. Anosov and A. B. Katok. New examples in smooth ergodic theory. Ergodic diffeomorphisms. Trudy Moskov. Mat. Obšč., 23:3 36, 970. [2] Bassam Fayad and Anatole Katok. Constructions in elliptic dynamics. Ergodic Theory Dynam. Systems, 24(5): , 2004.

15 NONSTANDARD SMOOTH REALIZATIONS OF LIOUVILLE ROTATIONS 5 [3] Bassam Fayad and Maria Saprykina. Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary. Ann. Sci. École Norm. Sup. (4), 38(3): , [4] Anatole Katok and Boris Hasselblatt. Introduction to the modern theory of dynamical systems, volume 54 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 995. [5] Maria Saprykina. Analytic nonlinearizable uniquely ergodic diffeomorphisms on mathbbt 2. Ergodic Theory Dynam. Systems, 23(3): , [6] Alistair Windsor. Minimal but not uniquely ergodic diffeomorphisms. In Smooth ergodic theory and its applications (Seattle, WA, 999), volume 69 of Proc. Sympos. Pure Math., pages Amer. Math. Soc., Providence, RI, 200. Bassam Fayad, LAGA, UMR 7539, Université Paris 3, Villetaneuse, France Maria Saprykina, Department of Math & Stats, Jeffery Hall, University Ave. Kingston, ON Canada, K7L 3N6 Alistair Windsor, Department of Mathematics, University of Texas at Austin, University Station, C200, Austin, TX , USA