Anti-classification results for weakly mixing diffeomorphisms

We extend anti-classification results in ergodic theory to the collection of weakly mixing systems by proving that the isomorphism relation as well as the Kakutani equivalence relation of weakly mixing invertible measure-preserving transformations are not Borel sets. This shows in a precise way that classification of weakly mixing systems up to isomorphism or Kakutani equivalence is impossible in terms of computable invariants, even with a very inclusive understanding of ``computability''. We even obtain these anti-classification results for weakly mixing area-preserving smooth diffeomorphisms on compact surfaces admitting a non-trivial circle action as well as real-analytic diffeomorphisms on the $2$-torus.


Introduction
One of the oldest and most influential problems in ergodic theory is the classification of systems up to appropriate equivalence relations. Dating back to the foundational paper [Ne32] by J. von Neumann the isomorphism problem asks to classify measure-preserving transformations (MPT's) up to isomorphism. By an MPT we mean a measure-preserving automorphism of a standard non-atomic probability space and we let X denote the set of all MPT's of a fixed standard non-atomic probability space (Ω, M, µ). We endow X with the weak topology (see Section 2.1). This topology is compatible with a complete separable metric and hence makes X into a Polish space. We also recall that two automorphisms S, T ∈ X are isomorphic (written S ∼ = T ) if there exists ϕ ∈ X such that S • ϕ and ϕ • T agree µ-almost everywhere.
The isomorphism problem has been a guiding light for directions of research within ergodic theory, and has been solved only for some special classes of transformations. Two great successes are the classification of ergodic MPT's with pure point spectrum by the spectrum of the associated Kooppman operator [HN42] and the classification of Bernoulli shifts by their measure-theoretic entropy [Or70]. Many properties of transformations like mixing of various types or finite rank have been In the current paper we obtain anti-classification results for weakly mixing transformations (addressing Problem 4 in [Fopp]). We recall that (Ω, M, µ, T ) is said to be weakly mixing if there is no nonconstant function h ∈ L 2 (Ω, µ) such that h(T x) = λ · h(x) for some λ ∈ C. Equivalently, (Ω, M, µ, T ) is weakly mixing iff for every pair A, B ∈ M we have The collection WM of weakly mixing transformations is a dense G δ subset of X endowed with the weak topology [Ha44]. Hence, the topology induced on WM is Polish as well. By genericity of WM and the turbulence result from [FW04] we know that there are no complete algebraic invariants for the isomorphism relation on WM (we refer to [FW04] or [Fopp,section 5.5] for details on the concept of turbulence developed by G. Hjorth to show that an equivalence relation is not reducible to an S ∞ -action). It is an open problem if the isomorphism relation on ergodic diffeomorphisms is turbulent (see [FW22, Problem 1]).
In this paper we obtain stronger anti-classification results for weakly mixing transformations as well as diffeomorphisms. To state our result precisely we let Then we show the unclassifiability of weakly mixing transformations with respect to isomorphism, Kakutani equivalence, and any equivalence relation between them.
Theorem A. Let R be any equivalence relation on X satisfying R iso ⊆ R ⊆ R Kak . Then the collection {(S, T ) : S and T are weakly mixing and R-equivalent} ⊂ X × X is a complete analytic set. In particular, it is not Borel.
For instance, our result holds for even equivalence and α-equivalence introduced in [FJR94]. It is also worth to mention that our weakly mixing systems have measure-theoretic entropy zero.
In a forthcoming paper with M. Gerber we obtain analogous anti-classification results for the even more restricted class of K-automorphisms (that clearly have positive measure-theoretic entropy). It is still an open problem whether those results also hold for smooth systems. However, the methods of the current paper allow us to obtain anti-classification results for weakly mixing diffeomorphisms.
Theorem B. Let M be the disk, annulus or torus with Lebesgue measure λ. Furthermore, let R be any equivalence relation on Diff ∞ λ (M ) satisfying Since R-equivalence for measure-preserving diffeomorphisms is reducible to Requivalence on X , Theorem A immediately follows from Theorem B.
Theorem B is related to another major question in ergodic theory dating back to the pioneering paper [Ne32]: The smooth realization problem asks whether there are smooth versions to the objects and concepts of abstract ergodic theory and whether every ergodic measure-preserving transformation has a smooth model. Here, a smooth model of an MPT (Ω, µ, T ) is a smooth diffeomorphism f of a compact manifold M preserving a measure λ equivalent to the volume element such that the MPT (M, λ, f ) is isomorphic to the MPT (Ω, µ, T ). The only known general restriction is due to Kushnirenko who proved that such a diffeomorphism must have finite entropy. There are restrictions in low dimension: Any circle diffeomorphism with invariant smooth measure is conjugated to a rotation and any weakly mixing surface diffeomorphism of positive measure-theoretic entropy is Bernoulli by Pesin theory [Pe77]. Thus, weakly mixing surface diffeomorphism of positive measuretheoretic entropy are classifiable by entropy.
Apart from Kushnirenko's result excluding smooth models of infinite entropy MPT's, there is a lack of general results on the smooth realization problem. One of the most powerful tools of constructing smooth volume-preserving diffeomorphisms of entropy zero with prescribed ergodic or topological properties is the so-called approximation by conjugation method (also known as the AbC method or Anosov-Katok method) developed by D. Anosov and A. Katok in a highly influential paper [AK70]. We refer to the survey articles [FK04] and [Kpp] for expositions of the AbC method and its wide range of applications in dynamics. In particular, it provided the first examples of weakly mixing C ∞ diffeomorphisms on the disk D in [AK70, section 5]. The AbC method is also used to construct weakly mixing diffeomorphisms preserving additional properties like a measurable Riemannian metric [GK00] or a prescribed Liouville rotation number [FS05].
Furthermore, the AbC method plays a key role in transfering the anti-classification results for ergodic MPT's in [FRW11] and [GKpp] to the smooth setting. In [FW19a], Foreman and Weiss found a class of symbolic systems (the so-called circular systems) that are realizable as smooth diffeomorphisms using the untwisted version of the AbC method (i. e. the conjugation map in the AbC construction maps its fundamental domain into itself). Then they showed in [FW19b] that there is a functor between the class of MPT's with an odometer factor and the class of circular systems that preserves factor and isomorphism structure. This functor allows them in [FW22] to propagate the odometer-based systems from [FRW11] to circular systems which are then realized as ergodic diffeomorphisms using the untwisted AbC method. Since untwisted AbC transformations cannot be weakly mixing [Ka03,Proposition 8.1], we design a specific twisted version of the AbC method that allows us to produce weakly mixing systems with a manageable symbolic representation. This takes on a project proposed in [FW19a] to find symbolic representations for other versions of the AbC method. In our case, the associated twisted symbolic systems will serve as counterpart of the circular systems in the Foreman-Weiss' series of papers. After some small modificatios to the constructions in [GKpp] we propagate those systems to the twisted symbolic systems that we can realize as weakly mixing diffeomorphisms. This allows us to deduce Theorem B. We refer to Section 1.2 for a more detailed outline of the proof.
Beyond C ∞ , the next natural question is the setting of real-analytic diffeomorphisms. Our weakly mixing AbC constructions can also be realized as real-analytic diffeomorphisms on T 2 using the concept of block-slide type of maps introduced in [Ba17]. This allows us to obtain our anti-classification results in the real-analytic category (see Subsection 2.6 for the definition of the space Diff ω ρ (T 2 , λ)). Theorem C. Let ρ > 0 and λ be the Lebesgue measure on T 2 . Furthermore, let R be any equivalence relation on Diff ω ρ (T 2 , λ) satisfying R iso ∩ Diff ω ρ (T 2 , λ) × Diff ω ρ (T 2 , λ) ⊆ R ⊆ R Kak ∩ Diff ω ρ (T 2 , λ) × Diff ω ρ (T 2 , λ) . Then the collection {(S, T ) : S and T are weakly mixing diffeomorphisms and R-equivalent} in Diff ω ρ (T 2 , λ) × Diff ω ρ (T 2 , λ) is a complete analytic set and, hence, not a Borel set with respect to the Diff ω ρ topology. We emphasize that all real-analytic constructions in this article are done on the torus. It is a challenging problem to extend them to other real-analytic manifolds.
To prove Theorem B we actually show the following stronger result.
Theorem 6. Let M be the disk, annulus or torus with Lebesgue measure λ. There is a continuous one-to-one map Φ : T rees → Diff ∞ λ (M ) such that for every T ∈ T rees the diffeomorphism T = Φ(T ) is weakly mixing and we have: (1) If T has an infinite branch, then T and T −1 are isomorphic.
(2) If T and T −1 are Kakutani equivalent, then T has an infinite branch.
We now show how Theorem B follows from Theorem 6.
Proof of Theorem B. Since the equivalence relation R is finer than or equal to isomorphism and coarser than or equal to Kakutani equivalence, Theorem 6 yields the existence of a continuous one-to-one map Φ : T rees → Diff ∞ λ (M ) such that for T ∈ T rees and weakly mixing T = Φ(T ): T has an infinite branch if and only if T and T −1 are R-equivalent.
Using the terminology from above, this says that Φ is a continuous reduction from the complete analytic set of ill-founded trees to the collection of weakly mixing diffeomorphisms T such that T and T −1 are R-equivalent. In the next step, we apply the map i(T ) = T, T −1 . It is a continuous mapping from Diff The proof of Theorem C will follow the same strategy as in the smooth setting. In Remark 72 we comment on the modifications.
1.2. Outline of the paper. In their proof of the anti-classification result for ergodic measure-preserving transformations in [FRW11], Foreman, Rudolph, and Weiss construct a continuous function F from the space T rees to the invertible measure-preserving transformations assigning to each tree T an ergodic transformation T = F (T ) of zero-entropy such that T ∼ = T −1 just in case T has an infinite branch. The assembly of such a transformation T can be viewed as cutting&stacking construction or as the construction of a symbolic shift. When taking the second viewpoint, then the symbolic system is built with a strongly uniform and uniquely readable construction sequence (W n (T )) n∈N which implies its ergodicity (see Section 2.2 for terminology). Related to the structure of the tree, equivalence relations Q n s (T ) on the collections W n (T ) of n-words and group actions on the equivalence classes in W n (T )/Q n s (T ) are specified in [FRW11]. Then (n + 1)-words are built by substituting finer equivalence classes of n-words into coarser classes using a probabilistic substitution lemma. The constructed transformations have an odometer as a non-trivial Kronecker factor and, hence, are Odometer-based Systems in up-todate terminology. Then a complete analysis of joinings over the odometer base is used in [FRW11] to find possible isomorphisms between transformations and their inverses. Since weakly mixing transformations have trivial Kronecker factors, we cannot use this method of joinings in our proof. Instead, we use a finite coding argument as in the proof of anti-classification results for Kakutani equivalence in [GKpp]. (As announced in [FRW06], the coding approach and d-estimates could also be used to exclude an isomorphism between T and T −1 in case that the tree has no infinite branch.) The words in [GKpp] are built using a deterministic procedure by substituting so-called Feldman patterns of finer classes into Feldman patterns of coarser classes. We refer to Section 2.4 for a description of Feldman patterns and their properties. In particular, different Feldman patterns cannot be matched well in f even after a finite coding. We review further important properties of the construction from [GKpp] in Section 5.1. The resulting transformations are odometer-based systems and are not weakly mixing.
One also meets the odometer obstacle when looking for smooth versions of the aforementioned anti-classification results, because it is a persistent open problem to find a smooth realization of transformations with an odometer-factor (see [FK04,Problem 7.10]). Foreman and Weiss circumvent that obstacle by showing that the collection of odometer-based systems has the same global structure with respect to joinings as another collection of transformations, the so-called circular systems that are extensions of particular circle rotations. For this purpose, they show in [FW19b] that there is a functor F between these classes that takes specific types of isomorphisms between odometer-based systems to isomorphisms between circular systems.
The definition of these circular systems is inspired by a symbolic representation for circle rotations by certain Liouville rotation numbers found in [FW19a]. Then Foreman and Weiss use the AbC method to show that these circular systems can be realized as area-preserving ergodic C ∞ -diffeomorphisms on torus or disk or annulus (under some assumptions on the circular coefficients). In the AbC method one constructs diffeomorphisms as limits of conjugates T n = H n • R αn • H −1 n with α n+1 = pn+1 qn+1 = α n + 1 kn·ln·q 2 n ∈ Q and H n = H n−1 • h n , where the h n 's are measure-preserving diffeomorphisms satisfying R 1 qn • h n+1 = h n+1 • R 1 qn . In each step the conjugation map h n+1 and the parameter k n are chosen such that the diffeomorphism T n+1 imitates the desired property with a certain precision. In a final step of the construction, the parameter l n is chosen large enough to guarantee closeness of T n+1 to T n in the C ∞ -topology, and this way the convergence of the sequence (T n ) n∈N to a limit diffeomorphism is provided. The resulting realization map R of circular systems allows to propagate aforementioned anti-classification results from [FRW11] and [GKpp] for measure-preserving transformations to the setting of smooth area-preserving diffeomorphisms via the reduction R • F • F . The AbC constructions in the series of papers [FW19a,FW19b,FW22] by Foreman-Weiss and in [GKpp] were untwisted, that is, the conjugation map h n+1 maps the fundamental domain [0, 1/q n ] × [0, 1] into itself. By [Ka03, Proposition 8.1] an untwisted AbC transformation has a factor isomorphic to a circle rotation. Hence, an untwisted AbC transformation cannot be weakly mixing. Accordingly, we have to design a specific twisted version of the AbC method in Sections 4.1-4.3 to produce weakly mixing transformations.
On the one hand, our AbC constructions are complicated enough to produce weak mixing behaviour. On the other hand, they are sufficiently manageable to still allow a simple symbolic representation. This symbolic representation is described in Section 4.4. It motivates the introduction of so-called twisted symbolic systems. We present their definition and basic properties in Section 3. The concept of twisted systems is our counterpart of the circular systems in [FW19a,FW19b,FW22]. It is an interesting task to explore properties of twisted systems in parallel to the analysis of circular systems which culminated in a global structure theory in [FW19b].
In Section 4.5, we obtain realizations of our weakly mixing abstract AbC transformations as C ∞ diffeomorphisms on T 2 , D, and A = S 1 × [0, 1]: If the parameter sequence (l n ) n∈N growths sufficiently fast, then there is an area-preserving C ∞ diffeomorphism measure-theoretically isomorphic to a given twisted symbolic system.
In order to prove our Theorem 6 we undertake some small modifications to the inductive construction of words in [GKpp] such that the resulting odometerbased construction sequence allows the propagation to a weakly mixing system via our twisting operator. We present the general substitution step in Section 5.2 with emphasis on the small modifications. In Section 5.3 we execute the inductive construction process of the odometer-based construction sequence and the associated twisted construction sequence. In this way we build the continuous reduction Φ : T rees → Diff ∞ λ (M ). Finally, we verify in Section 6 that Φ satisfies the properties stated in Theorem 6.

Preliminaries
We review some terminology from ergodic theory and symbolic dynamics.
2.1. Some basics in Ergodic Theory. A separable non-atomic probability space (Ω, M, µ) is called a standard measure space. For two measure spaces (Ω, M, µ) and (Ω , M , µ ) a map φ defined on a set of full µ measure of Ω onto a set of full µ measure of Ω is called a measure-theoretic isomorphism if φ is one-to-one and both φ and φ −1 are measurable and measure-preserving. Since every standard measure space is isomorphic to the unit interval with Lebesgue measure λ on the Borel sets B, every invertible MPT (also called an automorphism) of a standard measure space is isomorphic to an invertible Lebesgue measure-preserving transformation on [0, 1]. We let X denote the group of MPT's on [0, 1] endowed with the weak topology, where two MPT's are identified if they are equal on sets of full measure. Recall that T n → T in the weak topology if and only if µ(T n (A) T (A)) → 0 for every A ∈ M.
In order to give a concrete description of the weak topology we also recall that a (finite) partition P of a standard measure space (Ω, M, µ) is a collection P = {c σ } σ∈Σ of subsets c σ ∈ M with µ(c σ ∩c σ ) = 0 for all σ = σ and µ σ∈Σ c σ = 1, where Σ is a finite set of indices. Each c σ is called an atom of the partition P. For two partitions P and Q we define the join of P and Q to be the partition P ∨ Q = {c ∩ d | c ∈ P, d ∈ P}, and for a sequence of partitions {P n } ∞ n=1 we let ∨ ∞ n=1 P n be the smallest σ-algebra containing ∪ ∞ n=1 P n . We say that a sequence of partitions {P n } ∞ n=1 is a generating sequence if ∨ ∞ n=1 P n = M. A sequence of partitions {P n } n∈N is called decreasing if P n+1 is a refinement of P n for any n ∈ N.
We also have a standard notion of a distance between two ordered partitions: are two ordered partitions with the same number of atoms, then we define where denotes the symmetric difference. Now we are ready to give a concrete description of the weak topology as follows: For T ∈ X , a finite partition P, and ε > 0 we define If {P n } n∈N is a generating sequence of partitions, then generates the weak topology on X .
The following criterion proves useful to check convergence in the weak topology.
Fact 7 ([FW19a], Lemma 5). Let {T n } n∈N be a sequence of MPTs and {P n } n∈N be a generating sequence of partitions. Then the following statements are equivalent: (1) The sequence {T n } n∈N converges to an invertible measure preserving system in the weak topology.
(2) For all measurable sets A and for all > 0 there is an N ∈ N such that for all n, m > N and We will also use the following fact to construct an isomorphism between limits of sequences of MPT's.
Fact 8 ( [FW19a], Lemma 30). Fix a sequence {ε n } n∈N such that ∞ n=1 ε n < ∞. Let (Ω, M, µ) and (Ω , M , µ ) be two standard measure spaces and {T n } n∈N and {T n } n∈N be MPT's of Ω and Ω converging weakly to T and T , respectively. Suppose {P n } n∈N is a decreasing sequence of partitions and {K n } n∈N is a sequence of measure-preserving transformations such that (1) K n : Ω → Ω is an isomorphism between T n and T n , (2) {P n } n∈N and {K n (P n )} n∈N are generating sequences of partitions for Ω and Ω , (3) D µ (K n+1 (P n ), K n (P n )) < ε n . Then the sequence {K n } n∈N converges in the weak topology to a measure-theoretic isomorphism between T and T .
Finally, we also recall some relevant facts on the concept of periodic processes. We refer to [Ka03] for a detailed exposition.
Definition 9. A periodic process is a pair (τ, P) where P is a partition of (Ω, M, µ) and τ is a permutation of P such that each cycle of the permutation has equal length 2 and the atoms in each cycle have the same measure.
We refer to these cycles as towers and their length is called height of the tower. We also choose an atom from each tower arbitrarily and call it the base of the tower. In particular, if t 1 , . . . , t s are the towers (of height h) of this periodic process with B 1 , . . . , B s as their respective bases, then any tower t i can be explicitly written as B i , τ (B i ), . . . , τ h−1 (B i ). We refer to τ k (B i ) as the k-th level of the tower t i . Furthermore, we call τ h−1 (B i ) the top level.
2.2. Symbolic systems. An alphabet is a countable or finite collection of symbols. In the following, let Σ be a finite alphabet endowed with the discrete topology. Then Σ Z with the product topology is a separable, totally disconnected and compact space. A usual base of the product topology is given by the collection of cylinder sets of the form u k = f ∈ Σ Z : f [k, k + n) = u for some k ∈ Z and finite sequence u = σ 0 . . . σ n−1 ∈ Σ n . For k = 0 we abbreviate this by u .
The shift map sh : Σ Z → Σ Z is a homeomorphism. If µ is a shift-invariant Borel measure, then the measure-preserving dynamical system Σ Z , B, µ, sh is called a symbolic system. The closed support of µ is a shift-invariant subset of Σ Z called a symbolic shift or sub-shift. The symbolic shifts that we use are described by specifying a collection of words. A word w in Σ is a finite sequence of elements of Σ, and we denote its length by |w|.
Definition 10. A sequence of collections of words (W n ) n∈N satisfying the following properties is called a construction sequence: (1) for every n ∈ N all words in W n have the same length h n , (2) each w ∈ W n occurs at least once as a subword of each w ∈ W n+1 , (3) there is a summable sequence (ε n ) n∈N of positive numbers such that for every n ∈ N, every word w ∈ W n+1 can be uniquely parsed into segments u 0 w 1 u 1 w 1 . . . w l u l+1 such that each w i ∈ W n , each u i (called spacer or boundary) is a word in Σ of finite length and for this parsing We will often call words in W n n-words or n-blocks, while a general concatenation of symbols from Σ is called a string. We also associate a symbolic shift with a construction sequence: Let K be the collection of x ∈ Σ Z such that every finite contiguous substring of x occurs inside some w ∈ W n . Then K is a closed shiftinvariant subset of Σ Z that is compact since Σ is finite.
In order to be able to unambiguously parse elements of K we will use construction sequences consisting of uniquely readable words.
Definition 11. Let Σ be an alphabet and W be a collection of finite words in Σ. Then W is uniquely readable if and only if whenever u, v, w ∈ W and uv = pws with p and s strings of symbols in Σ, then either p or s is the empty word.
Moreover, our (n + 1)-words will be strongly uniform in the n-words.
Definition 12. We call a construction sequence (W n ) n∈N uniform if there is a summable sequence (ε n ) n∈N of positive numbers and for each n ∈ N a map d n : W n → (0, 1) such that for all words w ∈ W n+1 and w ∈ W n we have where r(w, w ) is the number of occurrences of w in w . Moreover, the construction sequence is called strongly uniform if for each n ∈ N there is a constant c > 0 such that for all words w ∈ W n+1 and w ∈ W n we have r(w, w ) = c.

Remark.
A particular type of symbolic systems are the ones that have odometer systems as their timing mechanism to parse typical elements: Let (k n ) n∈N be a sequence of natural numbers k n ≥ 2 and (W n ) n∈N be a uniquely readable construction sequence with W 0 = Σ and W n+1 ⊆ (W n ) kn for every n ∈ N. The associated symbolic shift is called an odometer-based system.
In the following we will identify K with the symbolic system (K, sh). We introduce the following natural set S which will be of measure one for measures that we consider.
Definition 13. Suppose that (W n ) n∈N is a construction sequence for a symbolic system K with each W n uniquely readable. Let S be the collection of x ∈ K such that there are sequences of natural numbers (a n ) n∈N , (b n ) n∈N going to infinity such that for all m ∈ N there is n ∈ N such that x [−a m , b m ) ∈ W n .
We note that S is a dense shift-invariant G δ subset of K and we recall the following properties from [FW19a, Lemma 11] and [FW19b, Lemma 12].
Fact 14. Fix a construction sequence (W n ) n∈N for a symbolic system K in a finite alphabet Σ. Then: (1) K is the smallest shift-invariant closed subset of Σ Z such that for all n ∈ N and w ∈ W n , K has non-empty intersection with the basic open interval w ⊂ Σ Z .
(2) Suppose that (W n ) n∈N is a uniform construction sequence. Then there is a unique non-atomic shift-invariant measure ν on K concentrating on S and ν is ergodic. (3) If ν is a shift-invariant measure on K concentrating on S, then for ν-almost every s ∈ S there is N ∈ N such that for all n > N there are a n ≤ 0 < b n such that s [a n , b n ) ∈ W n .
Since our symbolic systems K ∼ = Σ Z , B, ν, sh will be built from a uniquely readable uniform construction sequence, they will automatically be ergodic. To each symbolic system we will also consider its inverse K −1 which stands for K, sh −1 . Since it will often be convenient to have the shifts going in the same direction, we also introduce another convention.
Definition 15. If w is a finite or infinite string, we write rev(w) for the reverse string of w. In particular, if x is in K we define rev(x) by setting rev(x)(k) = x(−k).
If we explicitly view a finite word w positioned at a location interval [a, b), then we take rev(w) to be positioned at the same interval [a, b) and we set rev(w)(k) = w(a + b − (k + 1)). For a collection W of words rev(W ) is the collection of reverses of words in W .
Then we introduce the symbolic system (rev(K), sh) as the one built from the construction sequence (rev(W n )) n∈N . Clearly, the map sending x to rev(x) is a canonical isomorphism between K, sh −1 and (rev(K), sh). We often abbreviate the symbolic system (rev(K), sh) as rev(K).
Definition 16. Let Σ be an alphabet. For a word w ∈ Σ k and x ∈ Σ we write r(x, w) for the number of times that x occurs in w and freq(x, w) = r(x,w) k for the frequency of occurrences of x in w. Similarly, for (w, w ) ∈ Σ k × Σ k and (x, y) ∈ Σ × Σ we write r(x, y, w, w ) for the number of i < k such that x is the i-th member of w and y is the i-th member of w . We also introduce freq(x, y, w, w ) = r(x,y,w,w ) k .
2.3. The f metric. In the study of Kakutani equivalence Feldman [Fe76] introduced a notion of distance, now called f , as a substitute for the Hamming distance d in Ornstein's isomorphism theory.
We will refer to f (a 1 a 2 · · · a n , b 1 b 2 · · · b m ) as the "f -distance" between a 1 a 2 · · · a n and b 1 b 2 · · · b m , even though f does not satisfy the triangle inequality unless the strings are all of the same length. A match M is called a best possible match if it realizes the supremum in the definition of f .
Then f (a 1 . . . a n , b 1 . . . b m ) = 1 − max 2|D(π)| n+m : π is a match . We also state the following fact that can be proved easily by considering the fit 1 −f (a, b) between two strings a and b.
Fact 18 ( [GKpp], Fact 10). Suppose a and b are strings of symbols of length n and m, respectively, from an alphabet Σ. Ifã andb are strings of symbols obtained by deleting at most γ(n + m) terms from a and b altogether, where 0 < γ < 1, then 2.4. Feldman patterns. To construct the symbolic systems in [GKpp], the nwords in the construction sequence are built using specific patterns of blocks. These patterns are called Feldman patterns since they originate from Feldman's first example of an ergodic zero-entropy automorphism that is not loosely Bernoulli [Fe76]. In particular, different Feldman patterns cannot be matched well in f even after a finite coding. Let T, N, M ∈ Z + . A (T, N, M )-Feldman pattern in building blocks A 1 , . . . , A N of equal length L is one of the strings B 1 , . . . , B M that are defined by Thus N denotes the number of building blocks, M is the number of constructed patterns, and T N 2 gives the minimum number of consecutive occurrences of a building block. We also note that B k is built with N 2(M +1−k) many so-called cycles: Each cycle winds through all the N building blocks.
Moreover, we collect the following properties of (T, N, M )-Feldman patterns. (1) Each building block A i , 1 ≤ i ≤ N , occurs T N 2M +2 many times in each pattern.
(3) For all i, j ∈ {1, . . . , N } and all k, l ∈ {1, . . . , M }, k = l, we have . Moreover, the collection T rees of trees containing arbitrarily long finite sequences is a dense G δ subset. Hence, T rees is a Polish space.
Since the topology on the space of trees was introduced via basic open sets giving us a finite amount of information about the trees in it, we can characterize continuous maps defined on T rees as follows. During our constructions the following maps will prove useful.
Definition 21. We define a map M : T rees → N N by setting M (T ) (s) = n if and only if n is the least number such that σ n ∈ T and the length of σ n is s. Dually, we also define a map s : T rees → N N by setting s (T ) (n) to be the length of the longest sequence σ m ∈ T with m ≤ n.
Remark. When T is clear from the context we write M (s) and s(n). We also note that s(n) ≤ n and that M as well as s is a continuous function when we endow N with the discrete topology and N N with the product topology.
Any real-analytic diffeomorphism on T 2 homotopic to the identity admits a lift to a map from R 2 to R 2 which has the form where f i : R 2 → R are Z 2 -periodic real-analytic functions. Any real-analytic Z 2periodic function on R 2 can be extended as a holomorphic function defined on some open complex neighborhood of R 2 in C 2 , where we identify R 2 inside C 2 via the natural embedding (x 1 , x 2 ) → (x 1 + i0, x 2 + i0). For a fixed ρ > 0 we define the neighborhood Ω ρ := {(z 1 , z 2 ) ∈ C 2 : |Im(z 1 )| < ρ and |Im(z 2 )| < ρ}, and for a function f defined on this set we let We define C ω ρ (T 2 ) to be the space of all Z 2 -periodic real-analytic functions f on R 2 that extend to a holomorphic function on Ω ρ and satisfy f ρ < ∞.
Hereby, we define Diff ω ρ (T 2 , λ) to be the set of all Lebesgue measure-preserving real-analytic diffeomorphisms of T 2 homotopic to the identity, whose lift F to R 2 satisfies f i ∈ C ω ρ (T 2 ), and we also require that the liftF (x) = ( is a Cauchy sequence in the d ρ metric, then f n converges to some f ∈ Diff ω ρ (T 2 , λ). Thus, this space is Polish.

Twisted symbolic systems
In this subsection we introduce a specific type of symbolic system that will turn out to give symbolic representations of our weakly mixing AbC constructions in Section 4.4. These so-called twisted systems are our counterpart of circular systems used in [FW19a, section 4] to describe a symbolic representation of untwisted AbC transformations. Our definition of twisted systems follows in parallel to the definition of circular systems.
3.1. Definition of twisted systems. To state the definition let (C n , l n ) n∈N be a sequence of pairs of positive integers such that n∈N 1 ln < ∞. We use them to inductively define sequences (k n ) n∈N , (p n ) n∈N , (q n ) n∈N of positive integers as follows: We set p 0 = 0 and q 0 = 1. Then for each n ∈ N we define k n = 2 n+2 · q n · C n , q n+1 = k n l n q 2 n , p n+1 = p n k n l n q n + 1.
Obviously, p n+1 and q n+1 are relatively prime.
Remark. In [FW19a] such a sequence (k n , l n ) n∈N of pairs of positive integers is called a circular coefficient sequence. Accordingly, we refer to (C n , l n ) n∈N as a twisting coefficient sequence Furthermore we introduce numbers j i as follows: If n = 0 we take j 0 = 0, and for n > 0 we let j i ∈ {0, . . . , q n − 1} be such that where the (p n ) −1 is the multiplicative inverse of p n modulo q n . We also note that Using these numbers j i we define Remark. These numbers ψ n (i) enter the symbolic representation of our weakly mixing AbC transformations via the parameters a n (i) in the construction of conjugation map h n+1,1 in Section 4.3.1, namely, ψ n (i) = j an(i) . We refer to Remark 33 for a motivation of the choice of parameters a n (i).
Moreover, let Σ be a non-empty finite alphabet and b, e be two additional symbols.
Definition 22 (Twisting operator). Let w 0 , . . . , w kn−1 be words in Σ∪{b, e}. Using the aforementioned notation we let 3 Remark. Suppose that each w i has length q n . Then the length of C twist n (w 0 , . . . , w kn−1 ) is q n 2 n+2 q n C n l n q n = k n l n q 2 n = q n+1 . Remark. Our twisting operator should be compared with the circular operator C n from [FW19a, section 4] defined by In the symbolic representation of our specific weakly mixing AbC constructions the twisting operator plays the role of the circular operator for the symbolic representation of untwisted AbC transformations in [FW19a]. In parallel to the development of circular systems in [FW19a, section 4] we introduce so-called twisted systems.
Given a twisting coefficient sequence (C n , l n ) n∈N we build collections of words W twist n in the alphabet Σ ∪ {b, e} by induction as follows: • Set W twist 0 = Σ. 3 We use and powers for repeated concatenation of words.
• Having built W twist n , we choose a set P n+1 ⊆ W twist n kn of so-called prewords and build W twist n+1 by taking all words of the form C twist Definition 23. A construction sequence W twist n n∈N will be called twisted if it is built in this manner using the C twist -operators and a twisting coefficient sequence, and each P n+1 is uniquely readable in the alphabet with the words from W twist n as letters. (This last property is called the strong readability assumption.) Remark. Similar to the proof of [FW19b, Lemma 45] for circular construction sequences, one can show that each W twist n in a twisted construction sequence is uniquely readable even if the prewords are not uniquely readable. However, the definition of a twisted construction sequence requires this stronger readability assumption.
Definition 24. A symbolic shift K built from a circular construction sequence is called a twisted system. For emphasis we will often denote it by K twist .
Based on Fact 14 we obtain a characterisation of the set S ⊂ K twist from Definition 13 and a strong unique ergodicity result analogous to [FW19a, Lemma 20] for circular systems.
Lemma 25. Let K twist be a twisted system and let ν be a shift-invariant measure on K twist . Then the following are equivalent: (1) ν has no atoms.
(2) ν concentrates on the collection of s ∈ K twist such that {i | s(i) / ∈ {b, e}} is unbounded in both Z − and Z + .
(3) ν concentrates on S If K twist is a uniform twisted system, then there is a unique invariant measure concentrating on S. Remark. The boundary of w constitutes a small portion of 1/l n of the word w.
Definition 27. If s ∈ S or s ∈ W m with m ≥ n we define ∂ n (s) ⊂ Z to be the collection of i ∈ Z such that sh i (s)(0) is in the boundary portion of an n-subword of s. Furthermore, we introduce

3.2.
An explicit description of rev(K twist ). To describe an explicit construction sequence rev(W twist that is, the role of b and e in the twisting operator has been interchanged. Then we note the following connection between forward and reverse words. Proof. From the definition of the numbers j i in (3.1) and the relation (3.2) we obtain Using these identities we calculate Hence, the collections rev(W twist n+1 ) = C twist n (rev(w kn−1 ), . . . , rev(w 1 ), rev(w 0 )) w 0 w 1 . . . w kn−1 ∈ P n+1 constitute a construction sequence of (K twist ) −1 .
3.3. Subscales for twisted words. We end this section by introducing the following subscales for a word w ∈ W twist n+1 analogous to the terminology for circular words in [FW19b, Subsection 3.3].
Remark 29. Let w = C twist n (w 0 , . . . , w kn−1 ) ∈ W twist n+1 . • Subscale 0 is the scale of the individual powers of w j ∈ W twist n of the form w ln−1 j and each such occurrence of a w ln−1 j is called a 0-subsection.
• Subscale 1 is the scale of each term in the product that has the form b qn−ψn(i)−jm mod qn (w iCn+c ) ln−1 e ψn(i)+jm mod qn and these terms are called 1-subsections.

Weakly mixing AbC constructions
We start by presenting the general scheme of the abstract Approximation by Conjugation method for the construction of measure-preserving transformations. In this framework we provide a criterion for weak mixing in Section 4.2. We proceed by constructing specific twisted conjugation maps in Section 4.3 so that the resulting AbC transformations satisfy our criterion for weak mixing. In Section 4.4 we find symbolic representations for our specific constructions of weakly mixing AbC maps. In this symbolic representation we use the twisting operator introduced in Section 3. Finally, we show that our specific weakly mixing AbC maps allow realization as smooth or even real-analytic diffeomorphisms. 4.1. Abstract AbC constructions. We give an overview of the abstract Approximation by Conjugation method that will allow us to construct weakly mixing measure-preserving transformations. Here, our constructions can be viewed as taking place on We use M as a proxy for these spaces equipped with Lebesgue measure λ and circle actions {R t } t∈S 1 defined by Furthermore, we introduce the following notation with r, s ∈ Z + : We collect the above sets to form the following partition ξ r,s := {∆ i,j r,s : 0 ≤ i < r, 0 ≤ j < s}. Each transformation is obtained as the limit of an inductive construction process of conjugates with conjugation maps H n = H n−1 • h n and α n = pn qn ∈ Q, where p n and q n are relatively prime. For a start, we choose some arbitrary α 0 ∈ Q and set H 0 = id. In step n + 1 of the construction we build an additional conjugation map h n+1 satisfying In the measure-theoretic AbC construction this map h n+1 will be a permutation of partition elements with some k n , s n+1 ∈ Z + , where we make the following requirement on the sequence (s n ) n∈N : (R1) s n+1 is a multiple of s n and s n → ∞ as n → ∞.
Finally, we complete stage n + 1 of the construction process by setting In case of smooth (or even real-analytic) AbC constructions in Section 4.5, the numbers l n will have to grow sufficiently fast to allow convergence of the sequence (T n ) n∈N to a limit diffeomorphism. In the so-called abstract AbC method of this section we obtain a measure-preserving transformation as a limit of periodic processes.
Lemma 30. Let (T n ) n∈N be a sequence of measure-preserving transformations constructed by the abstract AbC method with (s n ) n∈N satisfying requirement (R1) and with any sequence (l n ) n∈N satisfying (4.7). Then (T n ) n∈N converges in the weak topology to a measure-preserving transformation T . Furthermore, the sequence of partitions is decreasing and generating. We have (4.9) Proof. We recall that h n acts as a permutation of the atoms of ξ kn−1qn−1,sn . Since q n = k n−1 l n−1 q 2 n−1 by (4.6), ξ qn,sn refines ξ kn−1qn−1,sn . Accordingly, we can view h n as permuting the atoms of ξ qn,sn . In this sense, each h m is a permutation of ξ qn,sn for m ≤ n. Hence, ζ n = H n (ξ qn,sn ) is decreasing and generating.
by our assumption on (l n ) n∈N . In particular, this shows convergence of (T n ) n∈N to a measure-preserving transformation T in the weak topology by Fact 7. Moreover, we have shown by triangle inequality that that is, (4.9) holds.
Remark 31. Since R αn gives a periodic process with partition ξ n := ξ qn,sn , the map T n = H n • R αn • H −1 n induces a periodic process with partition ζ n = H n (ξ n ), which we denote by τ n . When we want to view τ n as a collection of towers, we take the bases of τ n to be the sets H n (∆ 0,s qn,sn ) 0 ≤ s < s n . 4.2. Criterion for weak mixing. In this subsection we prove the following criterion for weak mixing in our setting of abstract AbC constructions. Our criterion bases upon the original construction of weakly mixing diffeomorphisms in [AK70, section 5].
Proposition 32 (Criterion for weak mixing). Let (T n ) n∈N be a sequence of measurepreserving transformations constructed by the abstract AbC method with (s n ) n∈N satisfying requirement (R1) and with any sequence (l n ) n∈N such that n∈N 1 ln < ∞. Furthermore, we suppose that there is an increasing sequence (m n ) n∈N of positive integers m n ≤ q n+1 such that for every n ∈ N we have for all 0 ≤ i, j < q n and 0 ≤ t, u < s n . Then (T n ) n∈N converges in the weak topology to a weakly mixing transformation T .
Proof. Since n∈N 1 ln < ∞, Lemma 30 implies the convergence of our sequence (T n ) n∈N of AbC transformations to a measure-preserving transformation T . By [AK70, Theorem 5.1] a measure-preserving transformation T is weakly mixing if and only if there exists a sequence of finite partitions η n converging to the decomposition into points (that is, for every measurable set A and for every n ∈ N there exists a set A n , which is a union of elements of η n , such that lim n→∞ λ(A A n ) = 0) and an increasing sequence of positive integers m n such that We take the partitions η n = H n (ξ qn,sn ) as in (4.8). By Lemma 30, (η n ) n∈N converges to the decomposition into points. Since n∈N 1 ln < ∞ and m n ≤ q n+1 , Lemma 30 also implies that d(η n , T mn , T mn n+1 ) → 0 as n → ∞. Hence, in order to check (4.11) it suffices to show that where we used assumption (4.10) in the last step. Hence, equation (4.12) is satisfied and we conclude that T is weakly mixing.
4.3. Construction of weakly mixing AbC transformations. In our construction of weakly mixing AbC transformations we will take (4.13) k n = 2 n+2 q n C n with some C n ∈ Z + that is a multiple of s 2 n . We also define (4.14) m n := C n l n q n = q n+1 2 n+2 q 2 n as sequence of mixing times in Proposition 32. This yields (4.15) m n α n+1 ≡ 1 2 n+2 q 2 n mod 1.
Furthermore, the conjugation map h n+1 will be a composition of two measure-preserving and 1/q n -equivariant transformations h n+1,1 and h n+1,2 .
Here, h n+1,1 acts as varying horizontal translations by multiples of 1/q n on vertical stripes of full length. In particular, some of these vertical stripes ∆ i,0 knqn,1 are mapped into different fundamental domains. We sometimes refer to h n+1,1 as the twist map in differentiation to the untwisted AbC constructions in [FW19a] and [FW22]. The definitions of h n+1,1 and m n will ensure that h n+1,1 • R mn αn+1 • h −1 n+1,1 distributes each fundamental domain ∆ i,0 qn,1 almost uniformly in the horizontal direction over all the fundamental domains (see Lemma 34).
In contrast, the map h n+1,2 leaves the fundamental domains invariant. It will allow us to also obtain almost uniform distribution in the vertical direction under h n+1 • R mn αn+1 • h −1 n+1 . Altogether, we will be able to verify assumption (4.10) from our criterion for weak mixing in Proposition 32. 4.3.1. Construction of the conjugation map h n+1,1 . Each 0 ≤ i < k n q n can be written in a unique way as In particular, we notice that (4.18) for our number m n from (4.14).
We also use the decomposition from (4.17) to define the conjugation map h n+1,1 by setting As required, we have Remark 33. The choice of a n allows us to deduce the following Lemma 34 on almost uniform distribution in the horizontal direction. The underlying mechanism is illustrated in Figure 4.1. This distribution result in turn is used in the proof of weak mixing in Proposition 35. Our mechanism to produce weak mixing is inspired by the original construction of weakly mixing AbC transformations in [AK70, section 5]. We use the different definitions of a n (i 2 ) for indices 0 ≤ i 2 < 2 n+1 q n and 2 n+1 q n ≤ i 2 < 2 n+2 q n in order to achieve that spacer symbols in our symbolic representation will occur at the same positions in forward and reverse words (see Lemma 28).
Lemma 34. For all pairs of 0 ≤ j, k < q n we have Here, n = 1 and q = q 1 = 4 (for illustration purposes; actual values will be much larger).
Finally, the definition of h n+1,2 is extended to the whole space by the commutativity relation 4.3.3. Verification of the weak mixing property. In order to prove the weak mixing property for our AbC transformation we have to strengthen the uniformity assumption (R2) to the following requirement.
In other words, the requirement (R4) says that all pairs (t, u) occur uniformly in the adjacent tuples b n (i, s) and b n (i + 1 mod 2 n+2 q n , s).
Then we can verify the assumptions of our criterion for weak mixing in Proposition 32 for the AbC constructions described above.
Proposition 35. Suppose that (T n ) n∈N is a sequence of AbC transformations with parameters (k n ) n∈N as in (4.13), (s n ) n∈N satisfying requirement (R1), and (l n ) n∈N satisfying n∈N 1 ln < ∞. Furthermore, we assume conjugation maps of the form h n+1 = h n+1,2 • h n+1,1 with maps h n+1,1 as in Subsection 4.3.1 and h n+1,2 as in Subsection 4.3.2 satisfying requirement (R4). Then (T n ) n∈N converges in the weak topology to a weakly mixing transformation T .
Then we observe that Furthermore, for every 0 ≤ s < s n+1 , 0 ≤ i 2 < 2 n+2 q n , and all pairs (t, u) with 0 ≤ t, u < s n there is a set of indices Continuing the calculation from above, we obtain that .
Since this holds true for every 0 ≤ s < s n+1 , we conclude that Then we use Lemma 34 to estimate for any 0 ≤ i 1 , j 1 < 2 n+2 q n and 0 ≤ t, u < s n that Altogether we have Hence, assumption (4.10) is satisfied and Proposition 32 yields that T is weakly mixing.

Symbolic representation of our weakly mixing AbC transformations.
We follow the approach in [FW19a, section 7] to find a symbolic representation for the abstract AbC transformations. While the AbC transformations in [FW19a] are untwisted, we consider our specific twisted constructions of weakly mixing AbC transformations from the previous subsection.
4.4.1. The dynamical and geometric orderings. We start by recalling the dynamical and geometric orderings of intervals from [FW19a, section 7.2]. For q ∈ Z + we let I q := I i q 0 ≤ i < q be the partition of [0, 1) and S 1 = R/Z, respectively, with atoms Definition 36. The geometric ordering of the intervals in I q is given by if and only if i < j, that is, we order these intervals from left to right according to their left endpoints.
To define the dynamical ordering, we fix a rational number α = p/q with p, q relatively prime. Set where the p −1 is the multiplicative inverse of p modulo q. We also note that The rotation by α defined by gives us another ordering of the intervals in I q .
Definition 37. The dynamical ordering of the intervals in I q is given by iff there are n < m < q such that np = i mod q and mp = j mod q. In other words, the list gives the dynamical ordering of I q .
Remark. With j i = p −1 i mod q from equation (4.24) the i-th interval in the geometric ordering, I i q , is the j i -th interval in the dynamical ordering. 4.4.2. An analysis on the circle. To find a symbolic representation of our AbC transformations we start with a simplified analysis of the projection to the horizontal S 1 -coordinate. We explore how the dynamical ordering determined by α n+1 = p n+1 q n+1 = α n + 1 k n l n q 2 n =: α n + β n interacts with the dynamical ordering determined by α n = p n /q n and the varying horizontal translation by a n (·) caused by the conjugation map h n+1,1 . For that purpose, we introduce the notation h n+1,1 for the projectivized action of h n+1,1 on S 1 . Furthermore, we divide the atoms of I knqn into k n many ordered sets ω 0 , . . . , ω kn−1 defined by Each ω j can be viewed as a word of length q n in the alphabet I knqn . We now want to determine a I knqn -name for the trajectory of J : where we used the commutativity relation h n+1,1 • R αn = R αn • h n+1,1 from (4.20). We recall that the twist map h n+1,1 caused a varying horizontal translation by a n (·), where a n (·) was defined in (4.19). Based upon these horizontal translations we define (4.26) ψ n (i) = j an(i) for all 0 ≤ i < 2 n+2 q n , with the numbers j an(i) as defined in equation (4.24), that is, We now follow our original interval J = [0, 1/q n+1 ) through the w j 's under the iterates Φ m n+1 . For 0 ≤ m < C n l n q n we have R m βn (J) ⊂ I (m mod qn)·2 n+2 qn 2 n+2 q 2 n . Since a n (0) = 0, we could write the I knqn -name of any x ∈ J in the first C n l n q n iterates as ω ln 0 ω ln 1 . . . ω ln Cn−1 . Applying R Cnlnqn βn on J makes it the geometrically first 1/q n+1 -subinterval of I 1 2 n+2 q 2 n . On I 1 2 n+2 q 2 n the map h n+1,1 causes a horizontal translation by 1/q n because of a n (1) = 1. Thus, h n+1,1 •R Cnlnqn βn (J) is a subinterval of I Cn+kn knqn , that is, the j 1 -th element of ω Cn . Thus, we must wait q n −j 1 further iterates to have Φ Cnlnqn+qn−j1 n+1 (J) ⊂ I Cn knqn . Then we can follow l n −1 copies of ω Cn . With the remaining j 1 many iterates we can write the I knqn -name of any x ∈ J in the iterates C n l n q n ≤ m < (C n +1)l n q n as b qn−j1 ω ln−1 Cn e j1 = b qn−ψn(1) ω ln−1 Cn e ψn(1) . Since h n+1,1 • R (Cn+1)lnqn βn (J) is a subinterval of I Cn+1+kn knqn (i.e., the j 1 -th element of ω Cn+1 ), we must wait q n − j 1 further iterates before can follow l n − 1 copies of ω Cn+1 . Altogether, we can write the I knqn -name of any x ∈ J in the iterates C n l n q n ≤ m < 2C n l n q n as b qn−ψn(1) ω ln−1 Cn e ψn(1) b qn−ψn(1) ω ln−1 Cn+1 e ψn(1) . . . b qn−ψn(1) ω ln−1 2Cn−1 e ψn(1) . Continuing like this, we deduce the I knqn -name of any x ∈ J in the iterates 0 ≤ m < k n l n q n as (4.27) Applying R knlnqn βn on J makes it the geometrically first 1/q n+1 -subinterval of I 1 qn and I kn knqn , respectively. The pattern in (4.27) would repeat itself up to the fact that we start in the j 1 -th element of ω 0 . By the same reasoning as above, we can write the I knqn -name of any x ∈ J in the iterates k n l n q n ≤ m < 2k n l n q n as 2 n+2 qn−1 i=0 Cn−1 c=0 b qn−ψn(i)−j1 mod qn (ω iCn+c ) ln−1 e ψn(i)+j1 mod qn .
In this way, we obtain the I knqn -name of any x ∈ J in the iterates 0 ≤ m < k n l n q 2 n = q n+1 as (4.28) Cn−1 c=0 b qn−ψn(i)−jm mod qn (ω iCn+c ) ln−1 e ψn(i)+jm mod qn , that is, its coding is given by C twist n (ω 0 , . . . , ω kn−1 ) using our twisting operator from Definition 22. This finding motivated the definition of the twisting operator.
For our goal to find a symbolic representation of our weakly mixing AbC transformations, we note that the action of R α on M ∈ {T 2 , D 2 , A} exactly mimics the action of R α on the circle in the first coordinate. We use this to label all atoms ∆ i,s qn+1,sn+1 of ξ qn+1,sn+1 by b (respectively e) whose projection I i qn+1 ∈ I qn+1 on the first coordinate is labelled with b (respectively e) in (4.28).
Then we inductively define sequences of subsets B n and E n of M as follows: Definition 38. Put B 0 = E 0 = ∅. If B n (respectively E n ) has been defined, let B n+1 (resp. E n+1 ) be the union of B n (resp. E n ) with the set of all x ∈ M whose projection onto the horizontal axis is contained in an atom of I qn+1 labelled with b (resp. e) in (4.28). Furthermore, we define sets B n+1 = B n+1 \ B n , E n+1 = E n+1 \ E n , and For every n ∈ N the measure of B n+1 ∪ E n+1 is 1/l n because the occurences of b and e comprise a proportion of 1/l n of the symbols in (4.28). Since Γ n ⊆ Γ n+1 and n∈N 1/l n < ∞ by (4.7), the Borel-Cantelli Lemma then implies that for almost every x ∈ M there is m ∈ N such that x ∈ Γ n for all n > m.
We describe how to associate a construction sequence to an AbC transformation T obtained as the limit of periodic transformations T n := H n • R αn • H −1 n . Let H n+1 (∆ 0,s * qn+1,sn+1 ) for some s * < s n+1 be the base of a tower of τ n+1 , where τ n is the periodic process given by T n on the partition (4.30) ζ n := H n (ξ n ) with ξ n := ξ qn,sn .
We say that {W n } n∈N is the construction sequence associated with the AbC construction. Under some conditions we can then show that our abstract weakly mixing constructions from Section 4.3 are isomorphic to symbolic twist systems. This ist the content of the subsequent proposition. It should be compared with [FW19a, Theorem 58] for an analogous result that some untwisted AbC transformations are isomorphic to circular symbolic systems.
Proposition 39. Suppose that the measure-preserving system (M, B, λ, T ) is built by the AbC method with parameters (k n ) n∈N as in (4.13), (s n ) n∈N satisfying requirement (R1), and (l n ) n∈N satisfying n∈N 1 ln < ∞. Furthermore, we assume conjugation maps of the form h n+1 = h n+1,2 • h n+1,1 with maps h n+1,1 as in Subsection 4.3.1 and h n+1,2 as in Subsection 4.3.2 satisfying requirements (R2) and (R3). Let Q be the partition defined in (4.29). Then the Q-names describe a strongly uniform twisting construction sequence {W n } n∈N . Let K be the associated twisted system and φ : M → K be the map sending each x ∈ M to its Q-name. Then φ is one-to-one on a set of λ-measure one. Moreover, there is a unique nonatomic shift-invariant measure ν concentrating on the range of φ. In particular, (M, B, λ, T ) is isomorphic to (K, B, ν, sh).
Proof. We note that our requirements (R1) and (R3) correspond to Requirements 1 and 3, respectively, in [FW19a]. Moreover, our requirement (R2) implies that for each ∆ 0,u qn,sn ∈ ξ qn,sn and every 0 ≤ s < s n+1 we have that is, the construction sequence {W n } n∈N is strongly uniform (which corresponds to Requirement 2 in [FW19a]). Then the proof follows along the lines of the proof of [FW19a, Theorem 58] using the twisting operator instead of the circular operator.

Smooth realization.
We now show how the AbC method can be used to construct C ∞ diffeomorphisms isomorphic to the abstract AbC transformations described in Sections 4.1 and 4.3.
4.5.1. Approximating partition permutations by smooth diffeomorphisms. In our abstract construction of weakly mixing AbC transformations we used specific partition permutations h n+1 introduced in Section 4.3. We will show that on the manifold M ∈ {T 2 , D, A} we can find for each of these partition permutations h n+1 an areapreserving C ∞ diffeomorphism h (s) n+1 that closely approximates h n+1 . The diffeomorphism h (s) n+1 will coincide with the identity in a neighborhood of the boundary of M and with the action of h n+1 on the "inner kernels" (4.33)∆ i,j knqn,sn+1,εn := of all the partition elements ∆ i,j knqn,sn+1 ∈ ξ knqn,sn+1 . To be more precise, we show the following realization result in this section.
Proposition 40. Let h n+1 = h n+1,2 • h n+1,1 be a partition permutation as defined in Section 4.3 satisfying requirement (R2). Then for any ε > 0 there is a diffeomorphism h Lemma 41. Let σ be a permutation of the rectangles ∆ i,j kq,s ∈ ξ kq,s which commutes with R 1/q and is untwisted, that is, σ(∆ 0,0 q,1 ) = ∆ 0,0 q,1 . Then for any ε > 0 there is a diffeomorphism φ ∈ Diff ∞ λ (M ) such that • φ is the identity in a neighborhood of the boundary of M , • for all 0 ≤ i < kq and 0 ≤ j < s we have kq,s,ε . We say that the partition permutation σ is ε-approximated by φ.
In particular, this lemma allows us to find smooth approximations to the untwisted conjugation map h n+1,2 .
Lemma 42. Let h n+1,2 be a partition permutation as defined in Subsection 4.3.2 with tuples b n (i, s) satisfying requirement (R2), that is, for every 0 ≤ s < s n+1 , 0 ≤ u < s n we have Then for every ε > 0 there is an 1/q n -equivariant area-preserving C ∞ diffeomorphism h (s) n+1,2 that is equal to the identity in a neighborhood of the boundary and ε-approximates h n+1,2 .
To construct smooth approximations to the twisting map h n+1,1 we use the following "pseudo-rotations" introduced in [FS05].
In the construction of smooth approximations to h n+1,1 we use these pseudorotations to map horizontal stripes into vertical ones. The construction is visualised in Figure 4.2.
Lemma 44. Let h n+1,1 be a partition permutation as defined in Subsection 4.3.1. For every ε > 0 there is an 1/q n -equivariant area-preserving C ∞ diffeomorphism h (s) n+1,1 , which is equal to the identity in a neighborhood of the boundary and εapproximates h n+1,1 . Since φ qn,δ coincides with the identity in a neighborhood of the boundary, we can extend it to a diffeomorphism φ qn,δ ∈ Diff ∞ λ (M ) commuting with R 1/qn . Furthermore, we note that for all 0 ≤ i < k n , 0 ≤ j < s n+1 .
In the next step, we build a smooth map that will introduce some horizontal translation depending on the height value i of∆ sn+1−j−1,i sn+1qn,kn,ε . For that purpose, let ρ : R → R be a smooth increasing function that equals 0 for x ≤ 1 2 and 1 for x ≥ 1. Then we define the mapψ an,qn,ε : [0, 1] → R bỹ ψ an,qn,ε (y) = kn−1 i=0 a n i Cn where a n (·) are the numbers from (4.19) defined in the construction of h n+1,1 . Note thatψ an,qn,ε coincides with the identity in a neighborhood of the boundary and for every 0 ≤ i < k n we have Using this mapψ an,qn,ε we define the area-preserving diffeomorphism ψ an,qn,ε : M → M by ψ an,qn,ε (x, y) = x +ψ an,qn,ε (y) , y .
By (4.36), we have for every 0 ≤ i < k n that (4.37) ψ an,qn,ε ∆ j,i sn+1qn,kn,ε =∆ an( i Cn )sn+1+j,i sn+1qn,kn,ε for all 0 ≤ j < s n+1 q n . Finally, we define the area-preserving smooth diffeomorphism h It is 1/q n -equivariant since all composed maps commute with R 1/qn . Using equations (4.35) and (4.37) we also conclude that h (s) n+1,1 |∆i,j knqn,s n+1 ,ε = h n+1,1 |∆i,j kn qn ,s n+1 ,ε for all 0 ≤ i < k n q n , 0 ≤ j < s n+1 , that is, h  that is, the union of the "inner kernels"∆ i,j kn−1qn−1,sn,εn−1 defined in (4.33). We put Exploiting the commutation relation we make the following observation regarding proximity for any n ∈ N: Since the number l n is chosen last in the induction step, we can choose l n ∈ Z + large enough to obtain

This yields
that is, (4.39) holds. Furthermore, this implies for all n, m ∈ Z + that Since (ε n ) n∈N is a summable sequence, the sequence T (s) n is a Cauchy sequence and, hence, converges to some T (s) ∈ Diff ∞ λ (M ). In the next step, we prove that T (s) is in fact measure-theoretically isomorphic to T . Our plan is to use Fact 8 for the proof. In the terminology of the lemma, we put (Ω, M, µ) = (Ω , M , µ ) = (M, B, λ). We define K n : M → M by From the definition it follows that K n is an isomorphism between T n and T (s) n . We define the two sequences of partitions P n := ζ n = H n (ξ n ) = H n (ξ qn,sn ) and P n := K n (ζ n ) = H (s) n (ξ n ). Using Lemma 30 we observe that {P n } ∞ n=1 is generating. Next we need to show that P n is generating, too.
We recall the definition of the set L n from equation (4.40) and note that (4.43) λ(L n ) ≥ 1 − 4ε n−1 .
Then we consider the following sequence of sets: Note that λ(G n ) 1 by summability of (ε n ) n∈N and Borel-Cantelli. By definition we have for every y ∈ G n and all m > n that Then we conclude for c ∈ ξ m and x ∈ c ∩ (h We pick δ > 0. There exists some n 0 such that λ(G m ) > 1 − δ 2 for all m > n 0 . By the observation in the previous paragraph we get for any m > n 0 that

This implies
Combining the previous estimates, we obtain This shows that {P n } ∞ n=1 is a generating sequence of partitions. To verify the remaining assumption of Fact 8 we have to show that D λ (K n+1 (P n ), K n (P n )) < ε n .
On the one hand, we compute On the other hand, We put Q n := h −1 n+1 (ξ n ) and note that by construction h (s) n+1 (Q n ) approximates h n+1 (Q n ). This finishes the proof.
Remark 46. For each choice of sequences {k n } n m=1 , {l n } n−1 m=1 and {s n } n+1 m=1 of natural numbers, we have finitely many permutations of ξ knqn,sn+1 and hence finitely many choices of h n+1 . As seen in the proof of Proposition 45, for each such choice there exists a natural number l n : ) < ε n /4. To get an uniform estimate, we set With information on how to associate a construction sequence with an AbC transformation, we can now state the main theorem of this subsection (compare with [FW19a, Theorem 60]).
Theorem 47. Consider three sequences of natural numbers (k n ) n∈N , (l n ) n∈N , (s n ) n∈N tending to infinity. Assume that (1) l n grows sufficiently fast (see the previous Remark 46); (2) k n is of the form k n = 2 n+2 q n C n for some C n ∈ Z + ; (3) s n divides both k n and s n+1 .
(5) For each w ∈ W n+1 and w ∈ W n , if w = C twist n (w 0 , . . . , w kn−1 ), then there are k n /s n many j with w = w j .
Furthermore, let K be the associated symbolic shift and ν its unique non-atomic ergodic measure. Then there is T (s) ∈ Diff ∞ λ (M ) such that the system (M, B, λ, T (s) ) is isomorphic to (K, B, ν, sh).
Afterwards, we extend this map to an invertible measure-preserving transformation h n+1,2 commuting with R 1/qn . This map is of the form as described in Section 4.3.2. It satisfies requirement (R2) by assumption (5) and requirement (R3) since the pre-words in P n+1 are distinct. Additionally, we take the conjugation map h n+1,1 as in Section 4.3.1 and set h n+1 = h n+1,2 • h n+1,1 . The associated AbC construction (T n ) n∈N also satisfies requirement (R1) by assumption (3). Hence, Proposition 45 guarantees that there is T (s) ∈ Diff ∞ λ (M ) measure-theoretically isomorphic to the abstract AbC map T which is isomorphic to the symbolic shift K by Proposition 39.
We note that the sequence {P n } n∈N of prewords determines the conjugation maps h n in the AbC method which in turn determine h (s) n and a neighborhood in the smooth topology which the resulting AbC diffeomorphism belongs to. Therefore, different choices of P n give distant maps h n and, hence, distant diffeomorphisms h (s) n in the smooth topology.
Lemma 48. Let (ε n ) n∈N be a summable sequence of positive reals satisfying (4.38). Suppose {U n } n∈N and {W n } n∈N are two construction sequences for twisting systems and N ∈ Z + such that U n = W n for all n ≤ N . If S (s) and T (s) are the smooth realizations of the twisting systems using the AbC method given in this paper, then We conclude (4.44) by combining these two estimates together with the triangle inequality.
We say that h (a) n+1 ε-approximates h n+1 . These approximations are the counterparts of the smooth realization results for twisting systems obtained in Proposition 40 and of the real-analytic realization results for circular systems from [BKpp,section 4]. As in [BKpp], we use the concept of block-slide type of maps introduced in [Ba17] and their sufficiently precise approximation by area-preserving real-analytic diffeomorphisms.
For a start, we recall that a step function on the unit interval is a finite linear combination of indicator functions on intervals. We define the following two types of piecewise continuous maps on T 2 , where s 1 and s 2 are step functions on the unit interval. Descriptively, the first map h 1 decomposes T 2 into smaller rectangles using vertical lines and slides those rectangles vertically according to s 1 , while the second map h 2 decomposes T 2 into smaller rectangles using horizontal lines and slides those rectangles horizontally according to s 2 . Any finite composition of maps of the above kind is called a block-slide type of map on T 2 .
By the following lemma, block-slide type of maps on T 2 can be approximated well by real-analytic diffeomorphisms that can be extended to entire maps. This can be achieved because step function can be approximated extremely well by real analytic functions (see e.g. [BK19, Lemma 2.13]).

Lemma 50 ([BK19], Proposition 2.22).
Let h : T 2 → T 2 be a block-slide type of map which commutes with R 1/q for some natural number q. Then for any ε > 0 and δ > 0 there exists a measure-preserving real-analytic diffeomorphim h (a) ∈ Diff ω ∞ (T 2 , λ) such that the following conditions are satisfied: (1) Proximity property: There exists a set E ⊂ T 2 such that λ(E) < δ and (2) Commutative property: In this case we say that the diffeomorphism h (a) is (ε, δ)-close to the block-slide type map h.
To approximate our partition permutations h n+1 from Section 4.3 by real-analytic diffeomorphisms, we exploit that those partition permutations are block-side type maps.
Lemma 51 ([BK19], Theorem E). Let k, q, s ∈ N and Π be a partition permutation of ξ kq,s of the torus T 2 . Assume that Π commutes with R 1/q . Then Π is a block-slide type of map.
As a consequence, we can conclude the proof of Proposition 49.
Proof of Proposition 49. We apply Lemma 51 followed by Lemma 50. 4.6.2. Real-analytic AbC method. As an analogue of Proposition 45, we obtain that we can realize any weakly mixing transformation built by the abstract AbC method from Sections 4.1 and 4.3 as an area-preserving real-analytic diffeomorphism provided that the sequence (l n ) n∈N grows sufficiently fast.
Proposition 52. Fix a number ρ > 0. Let (ε n ) n∈N be a summable sequence of positive reals satisfying m>n ε m < ε n 4 .
Suppose T : T 2 → T 2 is a measure-preserving transformation built by the abstract AbC method from Sections 4.1 and 4.3 using parameter sequences (k n ) n∈N and (l n ) n∈N . If (l n ) n∈N grows fast enough, then there exists a sequence of real-analytic AbC diffeomorphisms (T As counterpart to Theorem 47 in the smooth case, we otain the following realization result for twisted symbolic systems. Theorem 53. Fix ρ > 0. Consider three sequences of natural numbers (k n ) n∈N , (l n ) n∈N , (s n ) n∈N tending to infinity. Assume that (1) l n grows sufficiently fast; (2) k n is of the form k n = 2 n+2 q n C n for some C n ∈ Z + ; (3) s n divides both k n and s n+1 .
(5) For each w ∈ W n+1 and w ∈ W n , if w = C twist n (w 0 , . . . , w kn−1 ), then there are k n /s n many j with w = w j .
Furthermore, let K be the associated symbolic shift and ν its unique non-atomic ergodic measure. Then there is T (a) ∈ Diff ω ρ (T 2 , λ) such that the system (T 2 , B, λ, T (a) ) is isomorphic to (K, B, ν, sh).
We also record the subsequent analogue of Lemma 48.
Lemma 54. Fix ρ > 0. Suppose {U n } n ∈ N and {W n } n ∈ N are two construction sequences for twisting systems and N ∈ Z + is such that U n = W n for all n ≤ N . If S and T are the real-analytic realizations of the twisting systems using the AbC method given in this paper, then (4.48)

Building the reduction
In this section we build the continuous reduction Φ : T rees → Diff ∞ λ (M ) that will satisfy the properties required in Theorem 6. For that purpose, we start by constructing strongly uniform and uniquely readable odometer-based construction sequences (W n (T )) n∈N similarly to the ones in [GKpp]. These constructions also specify and use equivalence relations Q n s (T ) on the collections W n (T ) of n-words and group actions on the equivalence classes in W n (T )/Q n s (T ) as in [FRW11]. Then odometer-based (n + 1)-words are constructed by substituting Feldman patterns of finer equivalence classes of n-words into Feldman patterns of coarser classes. We collect important properties of these systems in Section 5.1 and describe such a substitution step in detail in Section 5.2. Here, we point out small modifications to the substitution step from [GKpp] in order to verify the weak mixing property using our criterion from Proposition 35.
We continue the inductive construction process by applying the twisting operator under some growth condition on the parameter sequence (l n ) n∈N that will allow the smooth realization of the associated twisted systems according to Theorem 47. We present the details of this construction process to get Φ : T rees → Diff ∞ λ (M ) in Section 5.3. In the next Section 6 we finally verify that Φ satisfies the properties stated in Theorem 6. 5.1. Specifications. Slightly modifying the constructions in [GKpp] we will construct for each T ∈ T rees an odometer-based construction sequence {W n (T ) | σ n ∈ T } in the basic alphabet Σ = {1, . . . , 2 12 }, where for each n ∈ N with σ n ∈ T the set of words W n = W n (T ) depends only on T ∩ {σ m : m ≤ n}.
The structure of the tree T ⊂ N N is also used to build a sequence of groups G s (T ) as follows. We define G 0 (T ) to be the trivial group and assign to each level s > 0 a so-called group of involutions We have a well-defined notion of parity for elements in such a group of involutions: an element is called even if it can be written as the sum of an even number of generators. Otherwise, it is called odd.
For levels 0 < s < t of T we have a canonical homomorphism ρ t,s : G t (T ) → G s (T ) that sends a generator τ of G t (T ) to the unique generator σ of G s (T ) that is an initial segment of τ . The map ρ t,0 is the trivial homomorphism ρ t,0 : G t (T ) → G 0 (T ) = {0}. We denote the inverse limit of G s (T ) , ρ t,s : s < t by G ∞ (T ) and we let ρ s : G ∞ (T ) → G s (T ) be the projection map.
Since there is a one-to-one correspondence between the infinite branches of T and infinite sequences (g s ) s∈Z + of generators g s ∈ G s (T ) with ρ t,s (g t ) = g s for t > s > 0, we obtain the following characterization.
Fact 55. Let T ⊂ N N be a tree. Then G ∞ (T ) has a nonidentity element of odd parity if and only if T has an infinite branch.
During the construction one uses the following finite approximations: For every n ∈ N we let G n 0 (T ) be the trivial group and for s > 0 we let G n s (T ) = (Z 2 ) τ where the sum is taken over τ ∈ T ∩{σ m : m ≤ n} , lh(τ ) = s.
We also introduce the finite approximations ρ (n) t,s : G n t (T ) → G n s (T ) to the canonical homomorphisms.
In the following, we simplify notation by enumerating {W n (T ) | σ n ∈ T } and {G n s (T ) | σ n ∈ T } as {W n } n∈N and {G n s } n∈N , respectively. During the course of construction one also defines an increasing sequence of prime numbers (p n ) n∈N satisfying We now collect important properties of the odometer-based construction sequence {W n } n∈N . To start we set W 0 = Σ.
(E1) All words in W n have the same length h n and the cardinality |W n | is a power of 2. (E2) There are f n , k n ∈ Z + such that every word in W n+1 is built by concatenating k n words in W n and such that every word in W n occurs in each word of W n+1 exactly f n times. The number f n is a product of p 2 n and powers of 2. Clearly, we have k n = f n |W n |. (E3) If w = w 1 . . . w kn ∈ W n+1 and w = w 1 . . . w kn ∈ W n+1 , where w i , w i ∈ W n , then for any k ≥ kn Remark. In particular, these specifications say that {W n } n∈N is a uniquely readable and strongly uniform construction sequence for an odometer-based system.
For each s ≤ s(n), there is an equivalence relation Q n s on W n satisfying the following specifications. To start, we let Q 0 0 be the equivalence relation on W 0 = Σ which has one equivalence class, that is, any two elements of Σ are equivalent.
(Q4) Suppose that n = M (s) for some s ∈ Z + . There is a specific number J s,n ∈ Z + such that 2J s,n p 2 n divides k n−1 and two words w = w 0 . . . w kn−1−1 ∈ W n and w = w 0 . . . w kn−1−1 ∈ W n are in the same Q n s class iff w i = w i for all i with k n−1 2p n J s,n ≤ i mod k n−1 J s,n < k n−1 J s,n − k n−1 2p n J s,n .
(Q5) For n ≥ M (s) + 1 we can consider words in W n as concatenations of words from W M (s) and define Q n s as the product equivalence relation of Q M (s) s . (Q6) Q n s+1 refines Q n s and each Q n s class contains 2 4e(n) many Q n s+1 classes. We write Q n s for the number of equivalence classes in Q n s and enumerate the classes by c (n,s) j : j = 1, . . . , Q n s . Occasionally, we will identify W n /Q n s with an alphabet denoted by (W n /Q n s ) * of Q n s symbols {1, . . . , Q n s }. Each equivalence relation Q n s will induce an equivalence relation on rev(W n ), which we will also call Q n s , as follows: rev(w), rev(w ) ∈ rev(W n ) are equivalent with respect to Q n s if and only if w, w ∈ W n are equivalent with respect to Q n s .
Remark. By (Q5) we can view W n /Q n s as sequences of elements W M (s) /Q M (s) s and similarly for rev(W n )/Q n s . In particular, it follows that Q n 0 is the equivalence relation on W n which has one equivalence class. Moreover, it allows us to regard elements in In addition to the odometer-based construction we will define a twisting coefficient sequence (C n , l n ) n∈N with positive integers l n growing sufficiently fast such that (5.2) m>n 1 l m < 1 l n for every n ∈ N as well as an associated twisted construction sequence (W n ) n∈N and bijections κ n : W n → W n by induction: • Let W 0 = Σ and κ 0 be the identity map.
5.1.1. Propagating equivalence relations and actions. We also propagate our equivalence relations and group actions to the twisted system. We proceed in an analogous manner to propagating equivalence relations and actions to circular systems in [FW22, section 5.10].
In order to prepare the construction of isomorphisms between K twist and (K twist ) −1 in case of T having an infinite branch (see Section 6.2), it proves useful to also give an intrinsic and inductive description of equivalence relations as well as group actions. An inductive definition of (Q n s ) twist is given as follows: • Define (Q n 0 ) twist to have exactly one class in W n . • For s ∈ Z + and w 1 , Let n > m. By specification (E2) we can write w ∈ W n ⊂ (W m ) hn/hm as w = w 0 . . . w hn/hm−1 with w i ∈ W m . Then let C twist m,n denote the operator such that κ n (w) = C twist m,n κ m (w 0 ), κ m (w 1 ), . . . , κ m (w hn/hm−1 ) .
In particular, C twist m,m+1 = C twist . Furthermore, we define the operator C twist m,n by interchanging the role of b and e in C twist m,n .
Remark 58. For every w ∈ W n there is a unique w ∈ W n such that w = κ n (w). Since w can be written as w = w 0 . . . w hn/hm−1 with w i ∈ W m by specification (E2), we have w = C twist m,n κ m (w 0 ), κ m (w 1 ), . . . , κ m (w hn/hm−1 ) .
Lemma 60. Let s ∈ N and n > m ≥ M (s). We write w ∈ W n ⊂ (W m ) hn/hm as w = w 0 . . . w hn/hm−1 with w i ∈ W m . Then for a canonical generator g ∈ G m s we have where we used (5.3) and the induction assumption from (5.5).
Remark 61. In continuation of Remark 57 we see that W n /(Q n s ) twist is closed under the twisted skew diagonal action. π s (x) = π s (sh k (x)) k∈Z . We denote the image of π twist s by K twist s which is a factor of K twist by construction. There is an analogous map from rev(K twist ) to rev(K twist s ) that we also denote by π twist s . Next, we describe a convenient base for the topology on K twist s . For this purpose, we recall from Remark 58 that for n ≥ M (s) any word w ∈ W n can be written We also define the measure ν s := (π twist s ) * ν on K twist s . To be more explicit, with the aid of specifications (E2) and (Q6) for the odometer-based system, the propagation specifications, and the proportion of symbols in the boundary for a typical x ∈ K we can show that for any w ∈ W n and 0 ≤ k < q n .
Finally, we let H twist Let n = M (s) for some s ∈ Z + . We define G n ⊂ K twist \ m≥n ∂ m to be the collection of x ∈ K twist such that there is an n-block at position 0 of x and if this principal n-word is given by C twist n−1 (w 0 , . . . , w kn−1−1 ), then x(0) belongs to an (n − 1)-word w i with k n−1 2p n J s,n ≤ i mod k n−1 J s,n < k n−1 J s,n − k n−1 2p n J s,n .
We note that Let G be the collection of x ∈ K twist such that for all large s, if n = M (s) then x belongs to G n . Then G has measure one by equations (5.1), (5.2) and the Borel-Cantelli Lemma.
In analogy with [FRW11, Proposition 23] we can prove the following statement making use of specification (Q4). Using the set G from above, we collect the following properties as in Propositions 24 and 25 of [FRW11].
(1) For all x = y belonging to G, there is an open set S ∈ s H twist s such that x ∈ S and y / ∈ S.
(2) For all s ≥ 1, H twist s is a strict subalgebra of H twist s+1 . 5.2. A general substitution step. In this subsection we describe a step in our iteration of substitutions that we use in the following subsection. It is very similar to the general substitution step in [GKpp,section 7] with some small modifications to satisfy the requirements in our criterion for weak mixing in Proposition 35. For the reader's convenience we present the substitution step in detail and emphasize the modifications to [GKpp,section 7]. As in [GKpp] the substitution will have the following initial data: • An alphabet Σ and a collection of words X ⊂ Σ h • Equivalence relations P and R on X with R refining P • Groups of involutions G and H with distinguished generators • A homomorphism ρ : H → G that preserves the distinguished generators. We denote the range of ρ by G and its kernel by H 0 with cardinality |H 0 | = 2 t for some t ∈ N.
• A free G action on X/P and a free H action on X/R such that the H action is subordinate to the G action via ρ. • There are N different equivalence classes in X/P denoted by [A i ] P , i = 1, . . . , N , where N = 2 ν+N with N , ν ∈ N. • Each equivalence class [A i ] P contains 2 4e elements of X/R, where e ∈ Z + with e ≥ max(2, t). We subdivide these R classes into tuples as follows. Pick where u ∈ {0, . . . , 2 t − 1}. • For some R ≥ 2 and some α ∈ 0, 1 8 we have (5.6) f (A,Ā) ≥ α for any substantial substrings A andĀ of at least h R consecutive symbols in any representatives of two different P-equivalence classes, that is, representatives of [A i1,j1 ] R and [A i2,j2 ] R for i 1 = i 2 and any j 1 , j 2 ∈ 1, . . . , 2 4e . • For some β ∈ (0, α] we have (5.7) f (A,Ā) ≥ β for any substantial substrings A andĀ of at least h R consecutive symbols in any representatives of two different R-equivalence classes, that is, representatives of [A i,j1 ] R and [A i,j2 ] R , respectively, for j 1 = j 2 .
Let K, T 2 ∈ Z + andR ≥ 2 be given. Moreover, let an even number D ∈ Z + be given.
Remark. This parameter D is introduced in addition to the parameters from [GKpp,section 7]. In the applications of our substitution step in Section 5.3, we choose D = 2 n+2 q n decribing a division of newly constructed pre-words into segments as required for the application of the twisting operator (recall the condition k n = 2 n+2 q n C n from (4.13)). This corresponds to the subdivision of the fundamental domain ∆ 0,0 qn,1 into 2 n+2 q n vertical segments in our weakly mixing constructions, particularly in the definition of the numbers a n (·) in the construction of map h n+1,1 .
Then we construct a collection S ⊂ (X/R) k of substitution instances of Ω as follows.
(1) We start by choosing a set Υ ⊂ Ω that intersects each orbit of the action by the group G exactly once. (2) We construct a collection of M 2 many different T 2 , 2 4e−t , M 2 -Feldman patterns, where the tuple of building blocks is to be determined in step (6).
Note that each such pattern is constructed as a concatenation of T 2 2 (4e−t)·(2M2+3) many building blocks in total which motivates the definition of the number T 1 from equation (5.10).
(3) By assumption on Ω we can subdivide each element r ∈ Υ as a concatenation of U 1 2 ν·(2M1+3) = U 2 strings of the form [A i ] T1 P and each i ∈ {1, . . . , N } occurs exactly 1 N U 2 many times in this decomposition.
Using this collection S ⊂ (X/R) k we define Ω = HS.
Remark. Here, steps (5) and (6) are modifications from the corresponding steps in [GKpp]. The two different sequences ψ and φ are used to satisfy assumption (B3) for a next substitution step. We refer to the proof of part (3) of the subsequent Proposition 65. In Remark 66 we use this part (3) to verify requirement (R4) in our weakly mixing constructions.
As in [GKpp, Proposition 42] we collect some properties of the collection Ω . In addition to strong uniformity of R-classes in elements of Ω , part (3) implies that assumption (B3) is satisfied for a next substitution step.
Proposition 65. This collection Ω ⊂ (X/R) k satisfies the following properties.
(1) Ω is closed under the skew diagonal action by H.
We end this subsection by pointing out that our modifications do not affect the estimates in [GKpp,Proposition 43] on the f distance of elements in Ω that are equivalent with respect to the P product equivalence relation but are not Requivalent. 5.3. The Construction Process. In this section, we describe the construction of our continuous reduction Φ : T rees → Diff ∞ λ (M ) for the proof of Theorem 6. For each T ∈ T rees we build a construction sequence {W n (T ) | σ n ∈ T } based on collections of odometer-based words {W n (T ) | σ n ∈ T } satisfying our specifications from Section 5.1 and bijections κ n : W n (T ) → W n (T ). To prove continuity of our map Φ : T rees → Diff ∞ λ (M ) in Lemma 69, this has to be done in such a way that Therefore, we follow [FRW11] and [GKpp, section 8] to organize our construction. To simplify notation we enumerate {W n (T ) | σ n ∈ T } as {W n } n∈N , that is, (n+1)words are built by concatenating n-words. Compared to the construction process in [GKpp, section 8], we use a slightly different order of choices of parameters p n , R n , e(n), and l n . We carefully describe their interdependencies and show that there are no circular dependencies. As in [GKpp,section 10.4] we will also use a sequence (R c n ) ∞ n=1 , where R c 1 = R 1 (with R 1 from the sequence (R n ) n∈N above) and R c n = l n−2 · k n−2 · q 2 n−2 for n ≥ 2. We note that for n ≥ 2, (5.15) q n R c n = k n−1 · l n−1 · k n−2 · l n−2 · q 2 n−2 · q n−1 l n−2 · k n−2 · q 2 n−2 ≥ l n−2 · k n−1 · l n−1 · q n−1 .
Hence, for n ≥ 2 a substring of at least q n /R c n consecutive symbols in a twisted n-block contains at least l n−2 − 1 complete 2-subsections, which have length k n−1 l n−1 q n−1 (recall the notion of a 2-subsection from Remark 29). When conducting f estimates on the twisting system, this allows us to ignore incomplete 2-subsections at the ends of the substring.
We describe how to construct W n+1 (T ), W n+1 (T ), Q n+1 s (T ), and the action of G n+1 s (T ), which we abbreviate by W n+1 , W n+1 , Q n+1 s , and G n+1 s , respectively. We also define a bijection κ n+1 : W n+1 → W n+1 . If n = 0 we have W 0 = W 0 = Σ = 1, . . . , 2 12 and κ 0 is the identity map. We also take a prime number p 1 > 2 and an integer R 1 ≥ 40p 1 . If n ≥ 1 our induction assumption says that we have W n , W n , Q n s , and G n s satisfying our specifications. In particular, the odometer-based words in W n have length h n and the words in W n have length q n . We also assume that there is a bijection κ n : W n → W n and that there are 1 8 > α (t) 1,n > · · · > α (t) s(n),n > β (t) n such that the following estimates on f distances hold: • For every s ∈ {1, . . . , s(n)} we assume that if w, w ∈ W n with [w] s = [w] s , then s,n on any substrings A, A of at least q n /R c n consecutive symbols in w and w, respectively.
• For w, w ∈ W n with w = w, we have (5.17) f A, A > β (t) n on any substrings A, A of at least q n /R c n consecutive symbols in w and w, respectively. .
We now distinguish between the two possible cases s(n+1) = s(n) and s(n+1) = s(n) for the construction of (n + 1)-words as well as the extension of G n s actions to G n+1 s actions.
Remark 66. As in [GKpp] these constructions satisfy the specifications stated in Section 5.1. In both cases of construction we iteratively apply the general substitution step with D = 2 n+2 q n . Hence, k n is of the form k n = 2 n+2 q n C n for some C n ∈ Z + as a multiple of s 2 n as required in (4.13) for our twisted AbC constructions. Since the odometer-based words in W n+1 from above determine the combinatorics of the abstract conjugation map h n+1 , we can verify requirements (R1), (R3), and (R4) from properties of odometer-based words in W n+1 : • Since s n+1 = |W n+1 | = |W n+1 | is a multiple of s n , assumption (R1) holds.
• At each application of the substitution step we can apply part (3) of Proposition 65 with D = 2 n+2 q n . By induction we deduce that (R4) holds. • Clearly, (R3) holds because different (n + 1)-blocks in W n+1 are obtained by different concatenations of n-blocks from W n .
We recall that the odometer-based words W n+1 from above determine the combinatorics of the abstract conjugation map h n+1 . Finally, we choose the parameter l n ∈ Z + sufficiently large to allow the smooth or real-analytic realization of the twisting system with odometer-based words W n+1 by Theorem 47 or Theorem 53, respectively. Additionally, we can choose l n large enough to satisfy (5.24) l n ≥ max 4R 2 n+2 , 9l 2 n−1 . Then we also know q n+1 = k n l n q 2 n from the relation (4.6).
In particular, the prewords are Proof. Since the isomorphisms ζ s cohere, their inverse limit defines a measurepreserving isomorphism between the subalgebra of B K twist generated by s H twist s and the subalgebra of B rev(K twist ) generated by s rev(H twist s ). By Lemma 63 this extends uniquely to a measure-preserving isomorphismζ between B K twist and B rev(K twist ) . Then by part (1) of Lemma 64 we can find sets D ⊂ K twist , D ⊂ rev(K twist ) of measure zero such thatζ determines a shift-equivariant isomorphism ζ between K twist \ D and rev(K twist ) \ D . Now we are in the position to prove the first half of Theorem 6.
Proof of part (1) in Theorem 6. Suppose that T ∈ T rees has an infinite branch. Then G ∞ (T ) has an element g of odd parity according to Fact 55. By Lemma 70 we obtain a coherent sequence of isomorphisms ζ s := η ρs(g) between K twist s (T ) and rev(K twist s (T )). Hence, Lemma 71 yields an isomorphism between K twist (T ) and rev(K twist (T )). Since rev(K twist (T )) is isomorphic to (K twist (T )) −1 , we conclude that K twist (T ) ∼ = (K twist (T )) −1 in case that the tree T has an infinite branch. Since our smooth realization of twisted systems in Theorem 47 preserves isomorphism, we conclude that Φ(T ) ∼ = Φ(T ) −1 in case that the tree T has an infinite branch.
6.3. Proof of Non-Kakutani Equivalence. In [GKpp, sections 10.4 and 10.5] it is shown that for a tree T ∈ T rees without an infinite branch the circular system T c = F(Ψ(T )) and T −1 c = F(Ψ(T )) −1 are not Kakutani equivalent. Within the required f estimates, the newly introduced spacers b and e are always ignored. As already observed in Section 5.3, the explicit form of the circular operator does not matter as long as each 0-subsection w ln−1 j comes with q n many newly introduced spacers, that is, we have a string of the form b qn−i w ln−1 j e i for some 0 ≤ i < q n as an 1-subsection. Since our twisting operator is an operator of such type, we can apply the same analysis to conclude that K twist (T ) and (K twist (T )) −1 are not Kakutani equivalent if T ∈ T rees does not have an infinite branch. Since our Theorem 47 produces diffeomorphisms isomorphic to the the symbolic systems, we conclude that Φ(T ) and Φ(T ) −1 are not Kakutani equivalent if T ∈ T rees does not have an infinite branch. This proves part (2) of Theorem 6.
Altogether we completed the proof of Theorem 6.
Remark 72. In order to prove Theorem C we proceed in an analogous manner to the proof of Theorem B by substituting Theorem 53 and Lemma 54 for Theorem 47 and Lemma 48, respectively.