SMOOTH DIFFEOMORPHISMS WITH HOMOGENEOUS SPECTRUM AND DISJOINTNESS OF CONVOLUTIONS

A BSTRACT . On any smooth compact connected manifold M of dimension m ≥ 2 admitting a smooth non-trivial circle action S = { S t } t ∈ (cid:83) 1 and for every Liouville number α ∈ (cid:83) 1 we prove the existence of a C ∞ -diffeomorphism f ∈ A α = (cid:169) h ◦ S α ◦ h − 1 : h ∈ Diff ∞ ( M , ν ) (cid:170) C ∞ with a good approximation of type ( h , h + 1), a maximal spectral type disjoint with its convolutions and a homogeneous spectrum of multiplicity two for the Cartesian square f × f . This answers a question of Fayad and Katok ([10, Problem 7.11]). The proof is based on a quantitative version of the approximation by conjugation-method with explicitly deﬁned conjugation maps and tower elements.


INTRODUCTION
Let M be a smooth compact connected manifold of dimension m ≥ 2 admitting a smooth non-trivial circle action S = {S t } t ∈S 1 preserving a smooth volume ν. In the case of a manifold with boundary by a smooth diffeomorphism we mean infinitely differentiable in the interior and such that all the derivatives can be extended to the boundary continuously. In this setting we consider the closure of conjugates A = h • S t • h −1 : h ∈ Diff ∞ (M , ν) , t ∈ S 1 C ∞ and more precisely for α ∈ S 1 the restricted spaces In [10,Problem 7.11], the following question is posed: QUESTION 1. Given a circle action S and the corresponding space A , is there a diffeomorphism f ∈ A with any of the following properties: 1. a good approximation of type (h, h + 1); 2. a maximal spectral type disjoint with its convolutions; 3. a homogeneous spectrum of multiplicity two for the Cartesian square f × f ?
This question takes up problems in the category of measure-preserving transformations on a Lebesgue space X , µ . For instance, there is extensive research on the spectral multiplicity problem about the construction of transformations possessing specific essential values M U T of the spectral multiplicities:

QUESTION 2. Given a subset E ⊂ N ∪ {∞}, is there an ergodic transformation T such that M U T = E ?
This question is a weak version of one of the main problems in the spectral theory of dynamical systems at the interface of unitary operator theory and ergodic theory: QUESTION 3. What are possible spectral properties for a Koopman operator associated with a measure-preserving transformation?
These two problems are open and no restrictions (except for the obvious ones) are known. However, there is an impressive progress concerning Question 2 (see [9] for a survey on spectral multiplicities of ergodic actions) and there exist two standard points of view: to consider the spectrum of T (and in particular M U T ) either on L 2 X , µ or on the orthogonal complement L 2 0 X , µ of the constant functions. In [18] it was proved that all possible subsets of N ∪ {∞} can be realized as M U T for some ergodic transformation T in the first case (since 1 is always an eigenvalue because of the constant functions, "possible" means any subset of N ∪ {∞} with 1 as an element). In the second case the Cartesian powers of a generic transformation provide a good opportunity for the construction of examples with the infimum of essential spectral multiplicities larger than 1. Although it seems very unlikely that these Cartesian powers have finite maximal spectral multiplicity, this is the generic case: Independently, Ageev and Ryzhikov proved the celebrated result that for a generic automorphism T the Cartesian square T × T has homogeneous spectrum of multiplicity 2 (see [1] resp. [25]). Ageev was even able to show that for the n-th power T n = T ×· · ·×T of a generic transformation T it holds M (T n ) = {n, n · (n − 1) , . . . , n!} (cf. [1,Theorem 2]). He also proved for every n ∈ N the existence of an ergodic transformation with homogeneous spectrum of multiplicity n in the orthogonal complement of the constant functions ([2, Theorem 1]) solving Rokhlin's problem on homogeneous spectrum in ergodic theory.
Here the property of admitting an approximation of type (h, h + 1) is often used to find an upper bound for M U T ×T like in [17,Proposition 3.6]. Moreover, it was used in [21] to construct homeomorphisms with continuous spectrum. In particular, this will enable us to conclude the ergodicity of f × f in the case of our constructions.
The second part of Question 1 is linked to a conjecture of Kolmogorov respectively Rokhlin and Fomin (after verifying that the property held for all dynamical systems known at that time, especially large classes of systems of probabilistic origin like Gaussian ones), namely that every ergodic transformation possesses the so-called group property, i.e., the maximal spectral type σ is symmetric and dominates its square σ * σ. This conjecture is an analogue of the well-known group property of the set of eigenvalues of an ergodic automorphism and was proven to be false. Indeed, in [28] A. M. Stepin gave the first example of a dynamical system without the group property. V. I. Oseledets constructed an analogous example with continuous spectrum ( [24]). Later Stepin showed that for a generic transformation all convolutions σ k 0 , k ∈ N, of the maximal spectral type σ 0 on L 2 0 X , µ are mutually singular (see [29]). In the smooth category there are only few results in this direction. In general, it is one of the most important problems in ergodic theory to find smooth models of ergodic transformations. Explicitly, Danilenko asks which subsets E = {1} admit a smooth ergodic transformation T with M U T = E ([9, Section 10]). Blanchard and Lemanczyk showed that every set E containing 1 as well as lcm(e 1 , e 2 ) for e 1 , e 2 ∈ E is realizable as the set of essential spectral multiplicities for a Lebesgue measure-preserving analytic diffeomorphism of a finite dimensional torus ( [6]). In [29,Section 4], Stepin constructed diffeomorphisms, for which the convolutions of the maximal spectral type on L 2 0 M , µ are mutually singular, on manifolds M as above using a smooth variant of the method of approximation by periodic transformations.
In order to extend this result of Stepin and answer the previously cited Question 1 affirmatively we prove the following theorem: THEOREM 1. Let M be a smooth compact connected manifold of dimension m ≥ 2 admitting a smooth non-trivial circle action S = {S t } t ∈S 1 preserving a smooth volume ν and α a Liouvillean number. Then the set of smooth diffeomorphisms that have a maximal spectral type disjoint with its convolutions, a homogeneous spectrum of multiplicity 2 for f × f , and admit a good approximation of type (h, h + 1), is residual (i.e., it contains a dense G δ -set) in A α in the Diff ∞ (M )-topology.
The proof is based on the so-called "approximation by conjugation-method" introduced in [3]: The diffeomorphisms are constructed as limits of conjugates f n = H n •S α n+1 •H −1 n , where α n+1 = p n+1 q n+1 ∈ Q, H n = H n−1 •φ n and φ n is a measurepreserving diffeomorphism satisfying S 1 qn • φ n = φ n • S 1 qn . In each step of the construction, the conjugation map φ n as well as specific partial partitions of the manifold have to be chosen in such a way that f n imitates the aimed properties with a certain precision. Moreover, the conjugation maps must allow explicit norm estimates. Then we will exploit the fact that α is a Liouville number in order to prove convergence of the sequence f n n∈N in A α .
Further applications of this method are the construction of smooth diffeomorphisms with specific ergodic properties (e.g., weak mixing ones in [3,Section 5] or [13]) or non-standard smooth realizations of measure preserving systems (e.g., [3,Section 6], [4], and [12]). See [10] for more details and other results of this method.
for n ∈ Z, where either l = n or l = −n. Moreover, for i ∈ {2, . . . , m} the coordinate function f i has to be Z-periodic in the first component, i.e., for every n ∈ Z.
The subsequent notation will be useful in the following, especially for defining explicit metrics on Diff k S 1 × [0, 1] m−1 .
Diffeomorphisms on S 1 × [0, 1] m−1 can be regarded as maps from [0, 1] m to R m . In this spirit the expressions f i 0 as well as D a f i 0 for any multiindex a with | a| ≤ k have to be understood for f = f 1 , . . . , f m ∈ Diff k S 1 × [0, 1] m−1 . Since such a diffeomorphism is a continuous map on the compact manifold and every partial derivative can be extended continuously to the boundary, all these expressions are finite. Thus, the subsequent definition makes sense: 2. Using the definitions from 1, we define for f , g ∈ Diff k S 1 × [0, 1] m−1 : Obviously d k describes a metric on Diff k S 1 × [0, 1] m−1 measuring the distance between the diffeomorphisms as well as their inverses. As in the case of a general compact manifold the following definition connects to it: It is a general fact that Diff ∞ S 1 × [0, 1] m−1 is a complete metric space with respect to this metric d ∞ .
Again considering diffeomorphisms on S 1 × [0, 1] m−1 as maps from [0, 1] m to R m we add the adjacent notation: where a is a multiindex. REMARK 1.5. By the above-mentioned observations for every multiindex a with | a| ≥ 1 and every i ∈ {1, . . . , m} the derivative D a h i is Z-periodic in the first variable. Since in case of a diffeomorphism g = g 1 , . . . , 1.1.2. Partial partitions. Furthermore, we introduce the notion of a partial partition of a measure space X , µ , which is a pairwise disjoint countable collection of measurable subsets of X . DEFINITION 1.6.
• A sequence of partial partitions ν n converges to the decomposition into points if and only if for a given measurable set A and for every n ∈ N there exists a measurable set A n , which is a union of elements of ν n , such that lim n→∞ µ (A A n ) = 0. We often denote this by ν n → ε. • A partial partition ν is a refinement of a partial partition η if and only if for every C ∈ ν there exists a set D ∈ η such that C ⊆ D. We write this as η ≤ ν.
Using the notion of a partition we can introduce the weak topology in the space of measure-preserving transformations on a Lebesgue space: 1. For two measure-preserving transformations T, S and for a finite partition ξ the weak distance with respect to ξ is defined by 2. The base of neighborhoods of T in the weak topology consists of the sets where ξ is a finite partition and ε is a positive number.
By the same reasoning as in [12,Section 2.2], this proposition allows us to carry a construction from S 1 × [0, 1] m−1 , R, µ to the general case (M , S , ν).
n , where f n = R α n+1 in a neighborhood of the boundary (in Proposition 1.9 we will see that these conditions can be satisfied in the constructions of this article). Then we define a sequence of diffeomorphisms: Constituted in [10, Section 5.1] (which is based on [16, Proposition 1.1]), this sequence is convergent in the C ∞ -topology to the diffeomorphism provided the closeness from f to R α in the C ∞ -topology.
We observe that f andf are metrically isomorphic (recall that two measure preserving dynamical systems X 1 , B 1 , µ 1 , T 1 and X 2 , B 2 , µ 2 , T 2 are metrically isomorphic if there exist B 1 ∈ B 1 and B 2 ∈ B 2 such that T 1 B 1 ⊆ B 1 , T 2 B 2 ⊆ B 2 , µ 1 (B 1 ) = 1, µ 2 (B 2 ) = 1 and there exists an automorphism φ : . Thus,f admits a good approximation of type (h, h + 1) because the speed and the type of a periodic approximation are invariant under isomorphisms. Moreover, f andf are unitarily equivalent (see Remark 3.1). Since the spectral types and multiplicities are spectral invariants, we conclude thatf has the aimed properties.
Hence, it is sufficient to prove Theorem 1 in case of S 1 × [0, 1] m−1 , R, µ . In this setting we will show the subsequent statement: PROPOSITION 1.9. For every Liouvillean number α there are a sequence (α n ) n∈N of rational numbers α n = p n q n converging monotonically to α and a sequence of measure-preserving smooth diffeomorphisms φ n , that coincide with the identity in a neighborhood of the boundary and satisfy such that the diffeomorphisms f n = H n • R α n+1 • H −1 n , where H n = H n−1 • φ n , converge in the Diff ∞ -topology to a limit f = lim n→∞ f n , which satisfies f ∈ A α and admits a good linked approximation of type (h, h + 1) as well as a good cyclic approximation.
Furthermore, for every ε > 0 the parameters in the construction can be chosen in such a way that d ∞ f , R α < ε.
In Section 9 we will deduce Theorem 1 from this Proposition which is proven in Section 8. Using results on periodic approximation as well as spectral theory of dynamical systems stated in the successive two sections we will be able to reduce this task to the construction of a diffeomorphism admitting a good linked approximation of type (h, h + 1) and a good cyclic approximation (see Section 9). For this purpose, we will use the "approximation by conjugation"-method and obtain the aimed diffeomorphism as the limit of conjugates . These conjugation maps φ n are constructed very explicitly in Section 4. We will sketch the construction of φ n as well as its action on the explicitly defined tower elements in subsection 4.4.
Subsequently, we prove that the sequence f n = H n • R α n+1 • H −1 n converges in the Diff ∞ -topology to a measure-preserving smooth diffeomorphism f ∈ A α under some conditions on the sequence (α n ) n∈N of rational numbers (cf. Lemma 5.9). Here, we require precise norm estimates on the conjugation maps. In the adjacent two sections (6 and 7) we show that this constructed limit f admits the required types of approximation with the respective speeds of approximation. Hereby, we will proof Proposition 1.9 in Section 8.

PERIODIC APPROXIMATION IN ERGODIC THEORY
This section provides a short introduction to the concept of periodic approximation in Ergodic Theory. A more comprehensive presentation can be found in [17].
Let X , µ be a Lebesgue space. A tower t of height h(t ) = h is an ordered sequence of disjoint measurable sets t = {c 1 , . . . , c h } of X having equal measure, which is denoted by m (t ). The sets c i are called the levels of the tower, especially c 1 is the base. Associated with a tower there is a cyclic permutation σ sending c 1 to c 2 , c 2 to c 3 ,. . . and c h to c 1 . Using the notion of a tower we can give the next definition: DEFINITION 2.1. A periodic process is a collection of disjoint towers covering the space X together with an equivalence relation among these towers identifying their bases.
There are two partial partitions associated with a periodic process: The partition ξ into all sets of all towers and the partition η consisting of the union of bases of towers in each equivalence class and their images under the iterates of σ, where when we go beyond the height of a certain tower in the class we drop this tower and continue until the highest tower in the equivalence class has been exhausted. Obviously, we have η ≤ ξ.
A sequence ξ n , η n , σ n of periodic processes is called exhaustive if η n → ε. Such an exhaustive sequence of periodic processes is consistent if for every measurable subset A ⊆ X the sequence σ n (A) converges to a set B , i.e., µ (σ n (A) B ) → 0 as n → ∞. Moreover, we will call a sequence of towers t (n) from the periodic process ξ n , η n , σ n substantial if there exists r > 0 such that h t (n) · m t (n) > r for every n ∈ N. DEFINITION 2.2. Let T : X , µ → X , µ be a measure-preserving transformation. An exhaustive sequence of periodic processes ξ n , η n , σ n forms a periodic approximation of T if d (ξ n , T, σ n ) = c∈ξ n µ (T (c) σ n (c)) → 0 as n → ∞.
Given a sequence g (n) of positive numbers we will say that the transformation T admits a periodic approximation with speed g (n) if for a certain subsequence (n k ) k∈N there exists an exhaustive sequence of periodic processes ξ k , η k , σ k such that d (ξ k , T, σ k ) < g (n k ).
In order to define the type of the periodic approximation we need the notion of equivalence for sequences of periodic processes: DEFINITION 2.3. Two sequences of periodic processes P n = ξ n , η n , σ n and P n = ξ n , η n , σ n are called equivalent if for every n ∈ N there is a bijective correspondence θ n between subsets S n and S n of the sets of towers of P n respectively P n such that • For t ∈ S n : h (θ n (t )) = h (t ).
• If two towers from S n are equivalent in P n , then their images under θ n are equivalent in P n .
There are various types of approximation. We introduce the most important ones: 1. A cyclic process is a periodic process which consists of a single tower of height h. An approximation by an exhaustive sequence of cyclic processes is called a cyclic approximation. More specifically we will refer to a cyclic approximation with speed o 1 h as a good cyclic approximation. 2. An approximation generated by periodic processes equivalent to periodic processes consisting of two substantial towers whose heights differ by one is said to be of type (h, h + 1). Equivalently the heights of the two towers t 1 and t 2 with base B 1 resp. B 2 are equal to h and h +1 and for some r > 0 we have µ (B 1 ) > r h as well as µ (B 2 ) > r h+1 . We will call the approximation of 3. An approximation of type (h, h + 1) will be called a linked approximation of type (h, h + 1) if the two towers involved in the approximation are equivalent. This insures that the sequence of partitions η n generated by the union of the bases of the two towers and the iterates of this set converges to the decomposition into points.

REMARK 2.5.
As noted in [27] a good linked approximation of type (h, h + 1) implies the convergence · Id in the weak operator topology for every k ∈ N and some r ∈ (0, 1), where U T is the Koopman-operator of T (see subsection 3.1).
From the different types of approximations various ergodic properties can be derived. For example in [20, Corollary 2.1], the subsequent Lemma is proven.

preserving transformation. If T admits a good cyclic approximation, then T is ergodic.
In [21] Katok and Stepin proved the genericity of automorphisms having a continuous spectrum in the set of measure-preserving homeomorphisms (recall that a transformation has a continuous spectrum, i.e., the corresponding operator U T in the space L 2 M , µ has no eigenfunctions other than constants, if and only if it is weak mixing). For this purpose, they deduced the following result ([21, Theorem 5.1]): LEMMA 2.7. Let T : X , µ → X , µ be a measure preserving transformation. If T is ergodic and admits a good approximation of type (h, h + 1), then T has continuous spectrum.
Moreover, the theory of periodic approximation can be used to prove genericity of constructed properties. The applied statement can be summarized as follows (cf. [17, Theorem 2.1]): LEMMA 2.8. Given a type τ and a speed g (n), the set of all measure-preserving transformations of a Lebesgue space which admit a periodic approximation of type τ with speed g (n) is a residual set (i.e., it contains a dense G δ -set) in the weak topology.

SPECTRAL THEORY OF DYNAMICAL SYSTEMS
Besides the concept of periodic approximation we will need further mathematical tools. We refer to [22] and [14] for more details.
3.1. Spectral types. Let X , µ be a Lebesgue space and T : X , µ → X , µ be an automorphism. Then we define the induced Koopman-operator U T : If two measure-preserving dynamical systems X 1 , µ 1 , T 1 and X 2 , µ 2 , T 2 are metrically isomorphic, their isomorphism h : X 1 → X 2 induces an isomorphism of Hilbert spaces V h : h and this relation is called unitary equivalence of operators. Hence, any invariant of unitary equivalence defines an invariant of isomorphisms. Such invariants are said to be spectral invariants or spectral properties. Moreover, we note that 1 is always an eigenvalue of U T because of the constant functions. So when we discuss the spectral properties of U T we refer to its spectral properties that are restricted to the orthogonal complement of the constants. Hence, we consider the properties of U T in the space L 2 0 X , µ of all L 2 -functions with zero integral.
Some important spectral invariants are the so-called spectral measures: Let . Using Bochner's theorem one can prove the existence of a finite Borel measure σ f defined on the unit circle S 1 in the complex plane satisfying Then σ f is called the spectral measure of f with respect to U T . Moreover, by the Hahn-Hellinger Theorem, there is a sequence of functions f n ∈ L 2 0 X , µ , n ∈ N, for which These measures are unique in the sense that for any other family of functions g n ∈ L 2 0 X , µ , n ∈ N, for which L 2 0 X , µ = n∈N Z g n and σ g 1 σ g 2 · · · we have σ f n ∼ σ g n for every n ∈ N.
According to this we say that U T has a continuous spectrum if σ f 1 is a continuous measure and U T has a discrete spectrum if σ f 1 is a discrete measure.

Spectral multiplicities.
Besides the maximal spectral type, an important characterization of U T is the multiplicity function M U T : Here Using this multiplicity function we establish the set M U T of essential spectral multiplicities, which is the essential range of M U T with respect to σ f 1 . Then we define the maximal spectral multiplicity m U T as the essential supremum (with respect to σ f 1 ) of M U T .
The multiplicity function is defined on S 1 by the relation m (λ) = m for λ ∈ A m . Note that the measures σ (m) are not the spectral measures, but the spectral type of the measure σ (m) is called a spectral type of multiplicity m.
In connection with the previous chapter, we state the following result ([20, Theorem 3.1]):

then the spectrum of U T is simple.
For automorphisms with simple spectrum we have the subsequent theorem of Ryzhikov ([26, Theorem 2.1]): LEMMA 3.5. Let X , µ be a Lebesgue space with µ (X ) = 1 and T : X , µ → X , µ be an automorphism with simple spectrum. Suppose that the weak convergence holds for some a ∈ (0, 1) and some strictly increasing sequence (k n ) n∈N of natural numbers. Then the Cartesian square T × T has a homogeneous spectrum of multiplicity 2.

Disjointness of convolutions.
In this section we study the convolutions of the maximal spectral type σ. Therefore, we state the definition of a convolution of measures: DEFINITION 3.6. Let G be a topological group and µ, ν finite Borel measures on G. Then their convolution µ * ν is defined by If all the convolutions σ k = σ * · · · * σ for k ∈ N are pairwise mutually singular, one speaks about disjointness of convolutions. To guarantee this pairwise singularity of convolutions of the maximal spectral type of a measure-preserving transformation the following property is useful: An automorphism T of a Lebesgue space X , µ is said to be κ-weak mixing, κ ∈ [0, 1], if there exists a strictly increasing sequence (k n ) n∈N of natural numbers such that the weak convergence holds, where P c is the projection on the subspace of constants. REMARK 3.8. By [29, Proposition 3.1], we can characterize this property in geometric language: A transformation T is κ-weak mixing if and only if there is an increasing sequence (k n ) n∈N of natural numbers such that for all measurable sets A and B We recognize that 0-weak mixing corresponds to rigidity and 1-weak mixing to the usual notion of weak mixing.
As announced this property has connections with certain properties of the maximal spectral type (see [29, Theorem 1]): LEMMA 3.9. If the transformation T is κ-weak mixing for some 0 < κ < 1 and σ is the maximal spectral type for U T | L 2 0 (X ,µ) , then σ and all its convolutions σ k = σ * · · · * σ are pairwise mutually singular.

CONSTRUCTION OF THE CONJUGATION MAPS
We fix an arbitrary Liouvillean number α and present step n in our inductive process of construction. Hence, we assume that we have already defined the rational numbers α 1 , . . . , α n−1 ∈ S 1 and the conjugation map In order to construct the conjugation map φ n we will need two types of maps which we will introduce in the subsequent subsections. Proof. Without loss of generality, we prove the statement in case of i < j . We Furthermore, let τ ε be a smooth diffeomorphism with the following properties: Then the diffeomorphismφ ε coincides with the identity on R m ∆ ε 2 and with the rotation in the . With the aid of "Moser's trick" we want to construct a diffeomorphism ϕ ε that is measure-preserving on the whole R m and agrees withφ ε on U ε . Therefore, we consider the canonical volume form Ω 0 on R m : Additionally we introduce the volume form Ω 1 :=φ * ε Ω 0 .
At first we note thatφ ε preserves the m − 1-form ω 0 on U ε : Clearly this holds Furthermore, we introduce Ω := Ω 1 − Ω 0 . Since the exterior derivative commutes with the pull-back, it holds Ω = d φ * ε ω 0 − ω 0 . In addition we consider the volume form Ω t := Ω 0 + t · Ω and note that Ω t is non-degenerate. Thus, we get a uniquely defined vector field X t such that Ω t (X t , ·) = ω 0 −φ * ε ω 0 (·). Since ∆ is a compact manifold, the non-autonomous differential equation Consequently ν * 1 Ω 1 = ν * 0 Ω 0 = Ω 0 using ν 0 = id in the last step. As we have seen, Since Ω t is nondegenerate, we conclude X t = 0 on U ε and hence ν 1 = ν 0 = id on U ε ∩ ∆. Now we can extend ν 1 smoothly to R m as the identity. Denote Using the transformation formula we compute for an arbitrary measurable set A ⊆ R m : We have det (Dν 1 ) > 0 (because ν 0 = id and all the maps ν t are diffeomorphisms) as well as det Dφ ε > 0 and thus det Dϕ Consequently ϕ ε is a measure-preserving diffeomorphism on R m satisfying the aimed properties.
. . , x m ). Hereby, we define a smooth measure-preserving diffeomorphism Sinceφ (i ) λ,ε coincides with the identity on a neighborhood of the boundary, we can proceed using the description The map ψ k,q, a, Let k, q ∈ Z, ε > 0 and ρ : R → R be a smooth increasing function that equals 0 for x ≤ −1 and 1 for x ≥ 0. Moreover, for every 0 ≤ i ≤ k − 1 we have a(i ) ∈ Z with 0 ≤ a(i ) ≤ q − 1. This set of parameters is denoted by a. With it we define the mapψ k,q, a,ε : Note that for every 0 ≤ i ≤ k − 1 we haveψ k,q, a,ε i k , i +1 k −ε = a(i ) q and we can estimate D lψ k,q, a,ε 0 ≤ 1 ε l · D l ρ 0 . In our constructions we will have a(0) = 0 as well as a(k − 1) = 0.
We emphasize that the maps σ ε are introduced to guarantee that ψ k,q, a,ε coincides with the identity in a neigbourhood of the boundary. Moreover, we observe ψ k,q, a,ε • R 1 q, a,ε and ψ k,q, a,ε l ≤ C (ε, l ).

4.3.
The conjugation map φ n . Using the maps from the preceding subsections we construct the conjugation map φ n : • ψ q n ,q n , a n , 1 where the parameters a n = a n (0) , . . . , a n q n − 1 with 0 ≤ a n (i ) ≤ q n − 1 will be determined later (see the end of Section 5).

DEFINITION 4.2.
By "good domain" G n of φ n and φ −1 n we denote the domain where all the occuring maps ϕ ε,1, j of φ ( j ) λ j ,ε act as the particular rotation and the map ψ k,q, a,ε acts as one of the translations by a(i ) q . In order to guarantee that a strip of our partition element is contained in the "good domain" completely we chooseε n := 1 2q n−1 slightly larger than ε n = 1 4q n−1 . Hereby, we observe 4.4. Sketch of the construction. As announced we would like to sketch the construction and its combinatorics.
In the subsequent section we will see that the constructed conjugation maps φ n and H n = H n−1 •φ n allow explicit norm estimates (independent of the choice of the parameters a n of our map ψ q n ,q n , a n , 1 4q n−1 ). Exploiting the fact that α is a Liouville number, this will enable us to prove convergence of the sequence f n = H n • R α n+1 • H −1 n in A α in Lemma 5.8 and 5.9. This proof will give us a sequencẽ α n =p ñ q n ∈ Q withp n ,q n relatively prime. In order to avoid technicalities in the positioning of the tower elements we want q n of α n−1 = p n q n to be a multiple of 2 · q n−2 · q m n−1 . Hence, we put q n = 2 · q n−2 · q m n−1 ·q n , p n = 2 · q n−2 · q m n−1 ·p n and note that α n−1 =α n−1 . Moreover, α n+1 = p n+1 q n+1 can be written in the form α n+1 = α n ± γ n q n+1 with γ n ∈ N. Hereby, we define m n := q n+1 γ n ·q 2 n . Additionally, we introduce two setsc (n) 0,i , i = 1, 2, in Section 6.1. With the aid of these we define the bases c (n) 0,i = H n−1 c (n) 0,i as well as the heights m n and m n + 1 of the towers which will be used to prove that f = lim n→∞ f n admits a good linked approximation of type (h, h + 1). In particular, each setc (n) 0,i is contained in a cube of edge length 1 q n such that the diameter of c (n) 0,i is small. Moreover,c (n) • ψ −1 q n ,q n , a n , 1 is built as a union of sets due to different reasons which we will explain in the following. First of all, the union over s i and t as well as the corresponding "gaps" are used in order to positionc (n) 0,i in the "good domain" of the map φ −1 n+1 . This will be very useful in the calculations of the speed of approximation, especially in the proof of Lemma 6.7.
In order to see that f admits an approximation of type (h, h + 1) the socalled "j -stripes", i.e., sets obtained by the union over s i and t , are very im- yields a j -dependent translation by a n ( j ) maps each of these sets to a set of almost full length in the r 1 -coordinate as well. Altogether, under φ −1 n each "j -stripe" is mapped on a set of almost full length in the r 2 , . . . , r m−1 -coordinates and θ-width of approximately γ nqn q n+1 (which motivates the name "stripe"). In the schematic visualisation of the action of φ −1 n in the figures the "j -stripes" are drawn without the "gaps". As seen this action of the conjugation map φ −1 n depends on the parameters a n ( j ) in the construction of the map ψ k,q, a,ε . In this connection we note that the number m n is defined such that m n · |α n+1 − α n | is approximately 1 q 2 n . Hence, under R m n α n+1 ≈ R mn ·pñ qn ± 1 q 2 n a "j -stripe" is mapped into another 1 q n -sector and is shifted by approximately 1 q 2 n . In order to get recurrence in the base we need another stripe to be positioned there, which will be fulfilled by our choice of the parameters a n ( j ) at the end of subsection 5.2: These parameters will be determined in such a way that a great portion of R m n for the prescribed height m n (resp. m n + 1) of the particular tower.
Since both towers have to be substantial, the measure of each base element has to be about 1 2m n . We ensure this by the union over k and l . Additionally, we require the union over k because p n and q n are not relatively prime as mentioned before and so our rotation R α n is an effective rotation byp ñ q n . The proof that f admits a good cyclic approximation is considerably easier. Since fq n n = id we can use a subset with measure about 1 q n of our tower base in the (h, h + 1)-approximation.

CONVERGENCE OF f n n∈N IN DIFF ∞ (M )
5.1. Properties of the conjugation maps φ n . In order to estimate the norm of the conjugating maps we will need the next technical result which is an application of the chain rule: . . , m} and k ∈ N. For any multiindex a with | a| = k the partial derivative D a φ j consists of a sum of products of at most (m − 1) · k terms of the form where l ∈ {1, . . . , m}, i ∈ {2, . . . , m}, and b is a multiindex with b ≤ k.
In the same way we can show a similar statement holding for the inverses: . . , m} and k ∈ N. For any multiindex a with | a| = k the partial derivative D a ψ j consists of a sum of products of at most (m − 1) · k terms of the form where l ∈ {1, . . . , m}, i ∈ {2, . . . , m}, and b is a multiindex with b ≤ k.
With these we can prove the following norm estimates: where C 1 m, k, q n−1 as well as C 2 m, k, q n−1 are constants depending on m, k and q n−1 but are independent of q n .
Start: k = 1. Let l ∈ {1, . . . , m} be arbitrary. By Lemma 5.1 a partial derivative of φ l of first order consists of a sum of products of at most m − 1 first order partial derivatives of functionsφ ( j ) λ j ,ε . Then we obtain using φ ( j ) With the aid of Lemma 5.2 we obtain the same statement for φ −1 = φ (2) Induction step k → k + 1: In the proof of Lemma 5.1 one observes that at the transition k → k + 1 in the product of at most (m − 1) · k terms of the form with j ∈ {1, . . . , m} and at most m − 2 partial derivatives of first order. Because of φ (i ) ≤ C · λ k+1 max and φ ( j ) λ j ,ε 1 ≤ C · λ max the λ max -exponent increases by at most 1 + (m − 2) · 1 = m − 1.
In the same spirit one uses the proof of Lemma 5.2 to show that also in case of φ −1 the λ max -exponent increases by at most m − 1.
Using the assumption we conclude So the proof by induction is completed.
In the setting of our explicit construction of the map φ (m) where C m, k, q n−1 is a constant independent of q n .
In the same spirit we obtain the estimate on φ (2) q n , 1

REMARK 5.4.
In the proof of the following lemmas we will use the formula of Faà di Bruno in several variables. It can be found in [8] for example. For this purpose, we introduce an ordering on N d 0 : For multi-indices µ = µ 1 , . . ., µ d and ν = ν 1 , . . ., ν d in N d 0 we will write µ ≺ ν, if one of the following properties is satisfied: Additionally we will use these notations: Then we get for the composition h(x 1 , . . . , Here D l j g denotes D l j g (1) , . . . , D l j g (m) and p s ν, λ := k 1 , . . . , k s , l 1 , . . . , l s : With the aid of these technical results we can prove an estimate on the norms of the map φ n : where C is a constant depending on m, k, n and q n−1 but is independent of q n .
JOURNAL OF MODERN DYNAMICS VOLUME 10,2016, Proof. First of all, we consider the mapφ (1) := ψ q n ,q n , a n , 1 4q n−1 •φ (1) , at which we use the notationφ (1) According to subsection 4.2 we have ψ q n ,q n , a n , 1 whereC m, k, q n−1 is a constant independent of q n . Let r ∈ {1, . . . , m} and ν be any multiindex with | ν| = k. With the aid of the formula of Faà di Bruno mentioned in remark 5.4 we compute: 1≤| λ|≤k D λ ψ q n ,q n , a n , 1 1≤| λ|≤k D λ ψ q n ,q n , a n , 1 , whereĈ is independent of q n . By definition of the set p s ν, λ we Altogether, we obtain φ (1) k ≤ C · q 4k n with a constant C independent of q n . In the next step we denoteφ (2) (1) . By the same calculations as above we obtain for any multiindex ν with | ν| = k: where we used Lemma 5.3 in the last step and the constants C ,C are independent of q n . Analogously we show the same estimate on φ (1) −1 • φ (2) −1 . Finally, we conclude: where C m, k, q n−1 is a constant independent of q n .
Again using the formula of Faà di Bruno we are able to prove an estimate on the norms of the map H n : LEMMA 5.6. For every k ∈ N we get: whereC is a constant depending solely on m, k, q n−1 and H n−1 . Since H n−1 is independent of q n in particular, the same is true forC .
In the same way we prove an analogous estimate on D ν H −1 n r 0 and verify the claim.
In particular, we see that this norm can be estimated by a power of q n .

Proof of convergence.
For the proof of the convergence of the sequence f n n∈N in the Diff ∞ (M )-topology, the next result, which can be found in [12,Lemma 4], is very useful.

LEMMA 5.7. Let k ∈ N 0 and h be a C ∞ -diffeomorphism on M . Then we get for every
where the constant C k depends solely on k and m. In particular C 0 = 1.
In the following lemma (similar to [11,Lemma 5.7]) we show that under some assumptions on the sequence (α n ) n∈N the sequence f n n∈N converges to f ∈ A α in the Diff ∞ (M )-topology. Afterwards, we will show that we can fulfil these conditions (see Lemma 5.9). LEMMA 5.8. Let ε > 0 be arbitrary and (k n ) n∈N be a strictly increasing sequence of natural numbers satisfying ∞ n=1 1 k n < ε. Furthermore, we assume that in our constructions the following conditions are fulfilled: where C k n are the constants from Lemma 5.7.

Then the sequence of diffeomorphisms f n
Proof.
1. According to our construction it holds φ n • R α n = R α n • φ n and hence Applying Lemma 5.7 we obtain the following for every k, n ∈ N: In Equation B we will assume |α − α n | n→∞ −→ 0 monotonically. Using the triangle inequality we obtain |α n+1 − α n | ≤ |α n+1 − α|+|α − α n | ≤ 2·|α − α n | and therefore Equation 1 becomes By the assumptions of this lemma it follows for every k ≤ k n In the next step we show, that for arbitrary k ∈ N f n n∈N is a Cauchy sequence in Diff k (M ), i.e., lim n,m→∞ d k f n , f m = 0. For this purpose, we calculate We consider the limit process m → ∞, i.e., we can assume k ≤ k m and obtain from equations 2 and 3 lim n,m→∞ Since Diff k (M ) is complete, the sequence f n n∈N converges consequently in Diff k (M ) for every k ∈ N. Thus, the sequence converges in Diff ∞ (M ) by definition. Furthermore, we estimate where we used the notation f 0 = R α 1 .
By explicit calculations we obtain d k R α , As seen above d k f n , f n−1 ≤ 1 k n for every k ≤ k n . Hereby, it follows further Hence, using Equation 4 we obtain the aimed estimate d ∞ f , R α < 3 · ε. 2. We have to show:f n → f in Diff ∞ (M ). For it we compute with the aid of Lemma 5.7 for every n ∈ N and k ≤ k n Fix some k ∈ N.
Proof. Let δ > 0 be given. Since f n → f in Diff ∞ (M ) we have f n → f in Diff k (M ) in particular. Hence, there is n 1 ∈ N, such that d k f , f n < δ 2 for every n ≥ n 1 . Because of k n → ∞ we conclude the existence of n 2 ∈ N, such that 1 k n < δ 2 for every n ≥ n 2 , as well as the existence of n 3 ∈ N, such that k n ≥ k for every n ≥ n 3 . Then we obtain for every n ≥ max {n 1 , n 2 , n 3 } Hence, the claim is proven.
In the next step we show lim n→∞ d ∞ f n , f = 0. For this purpose, we examine Consequently lim n→∞ d ∞ f n ,f n = 0. Hereby, we compute As asserted we obtain lim n→∞ d ∞ f n , f = 0.
As announced we show that we can satisfy the conditions from Lemma 5.8 in our constructions: LEMMA 5.9. Let (k n ) n∈N be a strictly increasing sequence of natural numbers with ∞ n=1 1 k n < ∞ and C k n be the constants from Lemma 5.7. For any Liouvillean number α there exists a sequence α n = p n q n of rational numbers with 2 · q n−2 · q m n−1 divides q n (A) (α n ) n∈N converges to α monotonically (B) such that our conjugation maps H n constructed in Section 4 fulfill the following conditions: 1. For every n ∈ N,
In particular, we have q n > n · 16 · q n−2 · q m n−1 .
Proof. The sequence of rational numbers α n = p n q n will be created out ofα n =p ñ q n withp n ≤ p n andq n ≤ q n relatively prime.
In the proof of the (h, h + 1)-property the number m n = q n+1 γ n ·q 2 n (see Equation   7) will play a decisive role. This number m n is known, when α n+1 =α n ± γ n q n+1 is determined guaranteeing the convergence of the sequence f n n∈N in Diff ∞ (M ) with the help of Lemma 5.9. Then we can compute m n ·α n . Let r q n := m n ·α n mod 1. Hereby, in "case +" we defineã n l ·q n + i to be In "case −" the parameterã n l ·q n + i is chosen as follows: In both cases we define a n l ·q n + i = 2 · q n−2 · q m n−1 ·ã n l ·q n + i .
where the union is taken over all j , k, l , s 2 , t ∈ Z satisfying , k q n + s 1 q n ·q n+1 + t +1 where the union is taken over all j ,k,l ,s 1 ,t ,s 2 ∈ Z satisfying q n+1 where the union is taken over all j , k, l , s 2 , s i , t ∈ Z satisfying q n+1 With these the base of the first tower is defined to be the set c (n) 0,1 := H n−1 c (n) 0,1 . Moreover, we observe that φ −1 n c (n) 0,1 is always contained in the left half of domains of the form u q n , u+1 q n × [0, 1] m−1 . The setsc (n) 0,2 are defined similarly tõ c (n) 0,1 . This time, φ −1 n c (n) 0,2 is supposed to be contained in the right half of the aforementioned domains. For example, in case of dimension m = 2 letc (n) 0,2 be the set where the union is taken over all j , k, l , s 2 , t ∈ Z satisfying Analogously, we definec (n) 0,2 in higher dimensions. Then c (n) 0,2 := H n−1 c (n) 0,2 is the base of the second tower. which is smaller than 1 n because of condition 2 in Lemma 5.9.
In the next step we will construct a sequence (m n ) n∈N of natural numbers in such a way that m n · (α n+1 − α n ) = ±m n · γ n q n+1 is approximately ± 1 With this sequence we define the following sets: It should be noted that these sets are disjoint by construction. Hence, we are able to define these two towers: In the rest of this subsection we check that these towers satisfy the requirements of the definition of a (h, h + 1)-approximation. First of all, we notice that both towers are substantial because we have Using the notation from Section 2 we consider the partial partition  Proof. The lemma is proven if we show that the partial partitionsξ n := c ∈ ξ n : diam (c) < 1 n satisfy µ c∈ξ n c → 1 as n → ∞. For this purpose, we examine which tower elements satisfy the condition on their diameter. Since we have to check for how many iterates i the set R i α n+1 • φ −1 n c (n) 0, j is contained in the "good domain" of φ n . Note that the bases of both towers are positioned in this "good domain". By the 1 q n -equivariance of the map φ n and i · α n+1 = i ·p n q n ± i ·γ n q n+1 we consider the displacement i ·γ n q n+1 , which is at most 1 So the restrictions will come from the maps φ (2) ·m n partition elements c (n) i , j inξ n and this corresponds to a measure LEMMA 6.6. We have c∈ξ n µ f n (c) σ n (c) ≤ 8 · γ n · q n−2 · q m n−1 · q n q n+1 .
In the next step we consider c∈ξ n µ f n+1 (c) f n (c) .
Proof. We have to compare the sets n c (n) 0,1 are positioned in the "good domain" of the map φ n+1 for i < m n by definition ofc (n) 0,1 , the deviation |α n+2 − α n+1 | = γ n+1 q n+2 on the θ-axis for every of the at most 1 2 causes the following measure difference: This difference occurs for every i ∈ {0, . . . , m n − 1} and allows us to estimate Similarly we estimate Thus, we obtain Lastly we consider LEMMA 6.8. We have Proof. We compute for every c ∈ ξ n using the notationc := H −1 n+1 (c): Since we have no control on φ −1 n+2 (c) for these areas ofc that do not belong to the "good domain" of the map φ n+2 , they will be part of the measure difference in our estimates. On the other hand, for the part ofc belonging to the "good domain" of the map φ n+2 the difference is caused by the deviation |α n+3 − α n+2 |. Using Lemma 4.3 the "good domain" of the map φ n+2 on an θinterval of the form l q n+2 , l +1 q n+2 has length at least 1 − m q n+1 · 1 q n+2 . It follows that the measure difference of Moreover, we recall that H −1 n+1 (c) consists of at most 1 2 Each of the (2m n + 1) elements c ∈ ξ n contributes and so we obtain Analogously estimating the other summands we observe Proof of Proposition 6.5. Using Equation 8 and the preceding three lemmas we conclude c∈ξ n µ f (c) σ n (c) ≤ 8 · γ n · q n−2 · q m n−1 · q n q n+1 + 2 · q m n+1 · γ n+1 q n+2 + 10 · m q n+1 .
Since this converges to 0 as n → ∞ (in particular because of Equation 6), we have a good linked approximation of type (h, h + 1). Recall the relations q n+1 = 2 · q n−1 · q m n ·q n+1 as well as α n+1 = p n+1 q n+1 =p n+1 q n+1 , wherẽ p n+1 andq n+1 are relatively prime. Hence, the tower levels are disjoint sets of equal measure not less than In order to see that this provides a cyclic approximation of the constructed map f we show that the partial partition Γ n := d (n) i : i = 0, . . . ,q n+1 − 1 converges to the decomposition into points. • For all the sequences f n n∈N built by our constructions from the previous sections the respective limit diffeomorphism f ∈ A α belongs to Θ because it belongs to U n f n for every n ∈ N by construction. So Θ contains all the constructed diffeomorphisms with the aimed properties. Hence, it is dense in A α due to the above considerations. • In the next step we want to show that f ∈ Θ admits a good linked approximation of type (h, h + 1) as well as a good cyclic approximation: for any f ∈ n∈N s≥n Θ s there is a sequence (n k ) k∈N with n k → ∞ as k → ∞, such that f ∈ Θ n k . So there is a sequence f n k k∈N of diffeomorphisms, at which f n k is the n k -th element of one of the above mentioned sequences of constructed diffeomorphisms, such that f ∈ U n k f n k . We observe that ξ n k → ε as well as Γ n k → ε as k → ∞, where ξ n k and Γ n k are the partitions belonging to the diffeomorphism f n k . Then f admits a good linked approximation of type (h, h + 1) as well as a good cyclic approximation by the definition of the neighborhoods U n k f n k .

PROOF OF GOOD
Thus, the set of diffeomorphisms in A α admitting a good linked approximation of type (h, h + 1) as well as a good cyclic approximation contains a dense G δ -set. Since these types of approximation imply the aimed properties, we conclude that the set of diffeomorphisms f ∈ A α with the following properties • a good approximation of type (h, h + 1); • a maximal spectral type disjoint with its convolutions; • a homogeneous spectrum of multiplicity two for the Cartesian square f × f is a residual subset in the C ∞ -topology. So Theorem 1 is deduced.