7 On approximation of homeomorphisms of 0 0 a Cantor set 2 n Konstantin Medynets a J 4 2 Abstract. We continue to study topological properties of the group ] S Homeo(X)ofallhomeomorphismsofaCantorsetX withrespecttothe D uniform topology τ, which was started in [B-K], [B-D-K 1; 2], [B-D-M], . and[B-M]. Weprovethatthesetofperiodichomeomorphismsisτ-dense h in Homeo(X) and deduce from this result that the topological group t a (Homeo(X),τ) hasthe Rokhlinproperty,i.e., there existsahomeomor- m phismwhoseconjugateclassisτ-denseinHomeo(X). Wealsoshowthat [ for any homeomorphism T the topological full group [[T]] is τ-dense in 2 the full group [T]. v 2 1. Introduction 3 0 Many famous problems in ergodic theory involve the use of topolo- 0 1 gies on the group Aut(X,B,µ) of all measure-preserving transformations 5 of a standard measure space. The first results on group topologies of 0 / Aut(X,B,µ) appeared in the paper [Hal 1]. Halmos introduced two topolo- h gies d and d , which were called later the uniform and weak topolo- t u w a gies, respectively. He defined the uniform topology d by saying that m u two automorphisms T and S are “close” to each other if the quantity : v µ({x ∈ X : Tx 6= Sx}) is small enough. The weak topology is generated i X by the sets of the form N(T;E;ε) = {S ∈ Aut(X,B,µ) : µ(SE△TE) < ε}, r where T ∈ Aut(X,B,µ) and E ∈ B. The use of these topologies turned out a to be very fruitful and led to many outstanding results in ergodic theory (for references, see, for example, [B-K-M] and [C-F-S]). One of the most relevant results in the theory is the Rokhlin lemma [Ro] stating that the set of periodic automorphisms is d -dense in Aut(X,B,µ). u The idea of investigation of transformation groups by means of intro- ducing various topologies into these groups was used in [B-D-K 1] and [B-D-K 2]. In the papers, the authors considered the groups Aut(X,B) of all automorphisms of a standard Borel space and the group Homeo(X) of all homeomorphisms of a Cantor set X with respect to the several topolo- gies analogous to those in ergodic theory. Following [B-D-K 2], we continue studying the group Homeo(X) of all homeomorphisms of a Cantor set X with respect to the topology τ (cf. Definition 1.1), which is obviously a direct analog of the topology d . We u show that the set of all periodic homeomorphisms is τ-dense in Homeo(X) 2000 Mathematics Subject Classification. Primary 37B05;Secondary 54H11. 1 2 KONSTANTINMEDYNETS (Corollary 2.2). This result can be treated as a topological version of the Rokhlin lemma. As a corollary, we prove that the set of topologically free homeomorphisms is τ-dense in Homeo(X) (Theorem 2.8). Recall that a homeomorphism is called topologically free if the set of aperiodic points is dense. In[G-K],aninterestingclassoftopologicalgroupswasdefined: bydefini- tion, a topological group has the Rokhlin property if it has an element whose conjugate class is dense. The authors raised the question: which groups possess this property. At the moment, there is an extensive list of groups that have the Rokhlin property. In particular, the group Homeo(X) with the topology generated by the metric D(T,S) = sup d(Tx,Sx), where x∈X d is a metric on X compatible with the topology [Gl-W], and the group Aut(X,B,µ) with respect to the weak topology [Hal 2] have the Rokhlin property. See also the paper [Ke-Ros] for a general approach to the study of groups with dense conjugate classes. Motivated by this, we present a unique approach allowing us to show that the topological groups (Aut(X,B),τ) and (Homeo(X),τ) have the Rokhlin property (Theorem 2.5). The other part of the paper is devoted to the study of full groups [T] of homeomorphisms T ∈ Homeo(X). Our motivation comes from the paper [G-P-S], where full groups were indispensable in the study of orbit equiva- lence of Cantor minimal systems. It is worthwhile to investigate full groups and their dense subsets for any homeomorphism. In this context, we show that for any T ∈ Homeo(X), the topological full group [[T]] is τ-dense in the full group [T] (Theorem 2.1). In the last section, we give a description of homeomorphisms from the topological full group [[T]] of aperiodic T (Theorem 3.3). We consider a subgroup Γ of [[T]], which is an increasing union of permutation groups, Y and find a criterion when Γ is τ-dense in [[T]] (Theorem 3.4). Y Background. Throughout thepaper, X denotes aCantor setandB stands for the σ-algebra of Borel sets of X. A one-to-one Borel map T of X onto itself is called an automorphism of (X,B). Denote by Aut(X,B) the group of all automorphisms of (X,B) and by Homeo(X) the group of all homeomorphisms of X. Following[B-K],recallthedefinitionoftheuniformtopologyonAut(X,B). Let M (X) denote the set of all Borel probability measures on X. For 1 T,S ∈ Aut(X,B), denote E(T,S) = {x ∈ X : Tx 6= Sx}. Definition 1.1. The uniform topology τ on Aut(X,B) is defined by the base of neighborhoods U = {U(T;µ ,...,µ ;ε)}, where 1 n U(T;µ ,µ ,...,µ ;ε) = {S ∈ Aut(X,B) | µ (E(S,T)) < ε, i = 1,...,n}. 1 2 n i Here T ∈ Aut(X,B), µ ,...,µ ∈ M (X), and ε > 0. 1 n 1 As Homeo(X) is a subgroup of Aut(X,B), we also denote by τ the topology on Homeo(X) induced from (Aut(X,B),τ). ObservethatAut(X,B)andHomeo(X)areHausdorfftopologicalgroups with respect to the uniform topology τ. More results related to topological properties of Aut(X,B) and Homeo(X) with respect to τ can be found in [B-D-K 1, B-D-K 2, B-D-M, B-K-M, B-M]. ON APPROXIMATION OF HOMEOMORPHISMS OF A CANTOR SET 3 Let T ∈ Aut(X,B). A point x ∈ X is called periodic of period n > 0 if Tnx = x and Tix 6= x for i = 1,...,n − 1. If Tnx 6= x for n 6= 0, the point x is called aperiodic. We say that T is aperiodic if it has no periodic points. Note that for any T ∈ Aut(X,B) the set X can be decomposed into a disjoint union of Borel sets X = X ∪ X , where X consists of all ∞ n≥1 n n points of period n and X is formed by all aperiodic points. Notice that ∞ S some X ’s can be empty. Moreover, for every X , n < ∞, there exists a n n Borel set X0 ⊂ X such that X = n−1TiX0 is a disjoint union. We call n n n i=0 n {X ,X ,X ,...} the canonical partition of X associated to T. ∞ 1 2 Recall that a finite family of disjoSint Borel sets ξ = {A,TA,...,Tn−1A} is called a T-tower with the base B(ξ) = A and the height h(ξ) = n. A partition Ξ = {ξ ,ξ ,...} of X is called a Kakutani-Rokhlin (K-R) partition 1 2 if every ξ is a T-tower. For a K-R partition Ξ, we denote B(ξ ) by i n≥1 i B(Ξ) and call it the base of the K-R partition. Notice that for a K-R S partition Ξ, one has that T−1B(Ξ) = Th(ξ)−1B(ξ). ξ∈Ξ For T ∈ Aut(X,B), let Orb (x) = {Tnx : n ∈ Z} denote the T-orbit T S of x. With any homeomorphism T ∈ Homeo(X), we can assign two full groups [T] and [T] , where C B [T] = {S ∈ Homeo(X) : Orb (x) ⊆ Orb (x), x ∈ X} C S T [T] = {S ∈ Aut(X,B) : Orb (x) ⊆ Orb (x), x ∈ X}. B S T Here the subindeces C and B stand for the cases of Cantor and Borel dy- namics, respectively. Clearly, [T] is a subgroup of [T] . Observe that if C B S ∈ [T] , then there is a Borel function n : X → Z such that Sx = TnS(x)x B S for all x ∈ X. The subgroup [[T]] = {S ∈ [T] : n is continuous} is called C S the topological full group of T. One of the main results in the approximation theory of Borel automor- phisms is a Borel version of the Rokhlin lemma. The following τ-version of the Rokhlin lemma was proved in [B-D-K 1, Proposition 3.6]. We also refer the reader to the works [N, Section 7] and [W, Section 4] for measure free versions of the result. Theorem 1.2. Let T be an aperiodic automorphism of X. Then there exists a sequence of periodic automorphisms (P ) ∈ Aut(X,B) such that n τ P −→ T, n → ∞. Moreover, the automorphisms P can be taken from n n [T] . B Denote by Per the set of all homeomorphisms P such that Pn = I for 0 some n ∈ N; and for T ∈ Homeo(X), set Per (T) = Per ∩[[T]]. 0 0 2. Rokhlin lemma In the section, we prove a topological version of the Rokhlin lemma, namely, we show that the set of periodic homeomorphisms is τ-dense in Homeo(X). Then, we deduce several corollariesofthisresult. Inparticular, we prove that the topological group (Homeo(X),τ) possesses the Rokhlin property and the topological full group [[T]] is τ-dense in [T] for any B T ∈ Homeo(X). Theorem 2.1. (1) The set Per is τ-dense in Homeo(X). 0 (2) Let T ∈ Homeo(X), then for any automorphism S ∈ [T] and any B 4 KONSTANTINMEDYNETS τ-neighborhood U = U(S;µ ,...,µ ;ε) of S there exists a periodic homeo- 1 p morphism P ∈ [[T]] such that P ∈ U. Proof. Notice that statement (1) is an immediate corollary of (2). By Theorem 1.2, it is enough to prove (2) for a periodic automorphism S. Let us sketch the main stages of the proof. (i) We find a finite number of disjoint S-towers consisting of closed sets and covering “almost” the entire space X with respect to the measures µ such that on each level of these S- i towers the automorphism S coincides with a power of T. (ii) We extend the S-towers found in (i) to clopen ones constructed by powers of T. (iii) Using the clopen towers, we define a periodic homeomorphism P which belongs to U(S;µ ,...,µ ;ε). 1 p (i)LetΞ = {X ,X ,...}bethecanonicalBorelpartitionofX associated 1 2 to S. Without loss of generality, we will assume that the sets X are not i empty, i ∈ N. We first find N ∈ N such that ε (2.1) µ (X ∪X ∪...∪X ) > 1− for all j = 1,...,p. j 1 2 N 3 For n ≥ 1, set Z = {x ∈ X : Sx = Tix for some −n ≤ i ≤ n}. For i ≥ 1 n define the sets i−1 X0(n) = S−j SjX0 ∩Z , i i n j=0 \ (cid:0) (cid:1) where X = X0 ∪SX0 ∪...∪Si−1X0 is a disjoint union. Since S ∈ [T] , i i i i B we have that X = X (n), where X (n) = i−1 SjX0(n). Then, find i n≥1 i i j=0 i K ∈ N such that S S N ε (2.2) µ (X \X (K)) < for all j = 1,...,p. j i i 3 ! i=1 [ Denote by S the set of all maps from {0,...,i − 1} to {−K,...,K}. i For σ ∈ S , define the set i i−1 X0(K,σ) = S−j {x ∈ SjX0(K) : Sx = Tσ(j)x} . i i j=0 \ (cid:0) (cid:1) Thus, we get a finite cover X0(K) = X0(K,σ). Applying the stan- i σ∈Si i dard argument, make the X0(K,σ)’s disjoint and denote the obtained sets i S by X0(K,σ) again. Some of the X0(K,σ)’s can be empty, but, without i i loss of generality, we will assume they are not. Observe that S restricted to SjX0(K,σ) is equal to Tσ(j), for i ≥ 1, j = 0,...,i−1, and σ ∈ S . This i i means that S is a homeomorphism on SjX0(K,σ). i ForeveryX (K,σ) = i−1 SjX0(K,σ),findaclosedsetA0(σ) ⊂ X0(K,σ) i j=0 i i i such that S N ε (2.3) µ ( (X (K,σ)\A (σ))) < for j = 1,...,p, j i i 3 i[=1σ[∈Si where A (σ) = i−1 SjA0(σ). i j=0 i (ii) Summing up the above, we get that {A (σ) : 1 ≤ i ≤ N ,σ ∈ S } is a i i S family of disjoint closed S-towers such that the automorphism S restricted ON APPROXIMATION OF HOMEOMORPHISMS OF A CANTOR SET 5 to SjA0(σ) is equal to Tσ(j). Furthermore, it follows from (2.1),(2.2), and i (2.3) that N (2.4) µ ( A (σ)) > 1−ε. j i i[=1σ[∈Si As the closed S-towers A (σ)’s are disjoint, we can find clopen sets i A0(σ) ⊃ A0(σ) so that all the sets A0(σ) and Tσ(0)+...+σ(j)A0(σ) are mu- i i i i tually disjoint for i = 1,...,N j = 0,...,i−2, and σ ∈ S . i (iii) Define the periodic homeomorphism P as follows: Tσ(0)x if x ∈ A0 i Px =  Tσ(j+1)x if x ∈ Tσ(0)+...+σ(j)A0i(σ) σ ∈ Si  T−σ(0)−...−σ(i−2)x if x ∈ Tσ(0)+...+σ(i−2)A0(σ) 0 ≤ j ≤ i−3  i x otherwise.  Clearly, P is well-defined and belongs to [[T]]. By the definition of P,  we have N {x ∈ X | Px = Sx} ⊃ A (σ). i i[=1σ[∈Si Hence, we get by (2.4) that P ∈ U(S;µ ,...,µ ;ε). This completes the 1 p (cid:3) proof. Remark. After this work was submitted, B. Miller showed how using ideas of the proof above one can generalize Statement (2) of Theorem 2.1 to any countable group acting by homeomorphisms on a zero-dimensional Polish space [Mil]. Rokhlin property We give several immediate corollaries of Theorem 2.1, which have the well- known analogs in ergodic theory. Corollary 2.2. Let T ∈ Homeo(X). Then, for every τ-neighborhood U of T, there exists a homeomorphism P ∈ Per (T)∩U whose associated 0 canonical partition is clopen. The next statement generalizes Theorem 4.5 of [B-K] proved originally for minimal homeomorphisms. Corollary 2.3. Let T ∈ Homeo(X). The topological full group [[T]] of T is τ-dense in [T] . C As the group Homeo(X) is not τ-closed in Aut(X,B), in [B-D-K 2] the authors brought up the question: how to describe the closure of [[T]] in (Aut(X,B),τ). They answered it for minimal homeomorphisms (see The- orem 2.8 of [B-D-K 2]) and we generalize it up to an arbitrary homeomor- phism. τ τ Corollary 2.4. Let T ∈ Homeo(X). Then [[T]] = [T] = [T] . C B 6 KONSTANTINMEDYNETS Definition. A topological group G possesses the Rokhlin property if the action of G on itself by conjugation is topologically transitive, i.e. there is an element of G whose conjugate class is dense. Thefollowingpropositionextendsthelistoftopologicalgroupsthathave the Rokhlin property. See also [Gl-W] and [Ke-Ros] for other examples. Theorem 2.5. Thetopologicalgroups(Aut(X,B),τ) and(Homeo(X),τ) possess the Rokhlin property. Proof. We prove this theorem for the group (Homeo(X),τ) only, for the other case the proof is similar. Take a decomposition of the Cantor set X = {x }∪ X such that 0 i≥1 i theX ’sarenon-emptyclopensetswithdiam(X ∪{x }) → 0asi → ∞. Let i i 0 S S be a homeomorphism such that Sx = x and Six = x, Sjx 6= x for any 0 0 x ∈ X , j = 1,...,i−1. Our goal is to show that we can approximate any i T ∈ Homeo(X) by elements from the conjugate class of S. By Corollary 2.2, it suffices to approximate periodic homeomorphisms whose canonical k partitions are clopen. Thus, suppose T has a clopen partition X = Y , i=1 i where Y is the set of all points having T-periodn for some n ≥ 0. Observe i i i that there exists a clopen set Y0 such that Y = ni−1TjY0 is a Sdisjoint i i j=0 i union (see Lemma 3.2 of [B-D-K 2]). Analogously, there exists a clopen set S X0 with X = ni−1SjX0 a disjoint union. ni ni j=0 ni LetU = U(T;µ ,...,µ ;ε)beaτ-neighborhoodofT. Takeanon-empty 1 p S clopen T-invariant set Z with µ (Z) < ε for i = 1,...,p. Without loss of i generality, we may assume that Y0\Z is not empty for i = 1,...,k. Let R i i be any homeomorphism from X0 onto Y0\Z. Define a homeomorphism R ni i as follows: let R be equal to TjR S−j whenever x ∈ SjX0 for i = 1,...,k, i ni j = 0,...,n −1 and let R map the rest of the space X onto Z. It is not i hard to check that RSR−1 ∈ U. In the setting of Borel dynamics, we need to produce a periodic trans- formation that has uncountably many orbits of any finite length. Then, the (cid:3) application of the Rokhlin lemma shows that its conjugate class is dense. Remark. Let p be the topology on Homeo(X) generated by the metric D(T,S) = sup d(Tx,Sx), where d is a metric on X compatible with x∈X the topology. In [Gl-W], it is shown that (Homeo(X),p) has the Rokhlin property. Moreover, the elements whose conjugate classes are dense form a residual set with respect to p. Topologically free homeomorphisms. It is interesting to compare the topological properties of the set Ap of all aperiodic homeomorphisms with respect to the both topologies τ and p. The following statement is proved in [B-D-K 2, Theorem 2.1]. Theorem 2.6. The set Ap is dense in (Homeo(X),p). However, the situation in (Homeo(X),τ) is completely different. The set Ap is nowhere dense with respect to the topology τ. To see this, one can check that the set Ap is τ-closed in Homeo(X). Then, the application of Theorem 2.1 implies the result. ON APPROXIMATION OF HOMEOMORPHISMS OF A CANTOR SET 7 The question we investigate in this section is “How can we extend the class of aperiodic homeomorphisms to produce a τ-dense class?”. Appar- ently, the most natural extension of aperiodic homeomorphisms is the class of topologically free homeomorphisms. Definition. It is said that a homeomorphism is topologically free if the set of all aperiodic points is dense. In Theorem 2.8, we prove that the set of topologicallyfree homeomorphisms is τ-dense. To begin with, we need the following lemma on homeomorphism extensions proved in [Kn-R]. We will need the arguments used in its proof. Thus, we give a sketch of the proof, but without going into the details. Lemma 2.7. Let A and B be closed nowhere dense subsets of Cantor sets X and Y, respectively. Supposethere is a homeomorphismh : A → B. Then h can be extended to a homeomorphism h∗ : X → Y such that h∗| = h. A Sketch of the proof. Find clopen sets {U } and {V } such that X \A = i j U , Y \B = V , and their diameters tend to zero. Find the points i≥1 i j≥1 j a ∈ A such that dist(U ,A) = dist(U ,a ) and b ∈ B with dist(V ,B) = i i i i j j F F dist(V ,b ). j j Set I = J = N. There exist injective functions f : I → J and g : J → I such that dist(U ,a ) > dist(V ,h(a )) for i ∈ I i i f(i) i dist(V ,b ) > dist(U ,h−1(b )) for j ∈ J. j j g(j) j Applying the usual Schr¨oder-Bernshtein argument to f and g, find disjoint sets I = I′ ⊔I′′ and J = J′ ⊔J′′ such that f(I′) = J′ and g(J′′) = I′′. Let φ be an arbitrary homeomorphism of U′ = U onto V′ = i∈I′ i V such that φ(U ) = V . Analogously, let ψ be a homeomorphism j∈J′ i i f(i) S of V′′ = V onto U′′ = U such that ψ(V ) = U . S j∈J′′ j i∈I′′ i j g(j) Define S S φ(x) x ∈ U′ h∗(x) = ψ−1(x) x ∈ U′′  h(x) x ∈ A.  For the verification of continuity of h∗, we refer the reader to [Kn-R]. (cid:3)  Theorem 2.8. The set of topologically free homeomorphisms is τ-dense in Homeo(X). Proof. By Corollary 2.2, it suffices to approximate only homeomor- phisms from Per that have clopen canonical partitions. Assume that R 0 belongs to Per and its canonical partition X = X ∪...∪X is clopen. 0 n1 nm Recall that the set X consists of all points with the period n . Consider ni i a τ-neighborhood U = U(R;µ ,...,µ ;ε). Since the X ’s are R-invariant 1 k ni and clopen, we will prove the theorem under the assumption that X = X ni for some i and leave the generalization to the reader. Suppose X = p−1RiF is a clopen partition and Rpx = x for all i=0 x ∈ X. Using the standard Cantor argument, find a closed nowhere dense set P ⊂ Rp−1F suchSthatµ (P) > 1−ε fori = 1,...,k. Repeating theproof i of Lemma 2.7, we extend the homeomorphism R : P → RP to a homeo- morphism T : Rp−1F → F so that the homeomorphism T∗ ∈ Homeo(X) 8 KONSTANTINMEDYNETS defined as T∗|Rp−1F = T|Rp−1F and T∗ = P elsewhere is topologically free. To do this, it suffices to choose the functions ψ and φ so that φ(x) 6= Rx and ψ−1(x) 6= Rx for x ∈ Rp−1F \P. Since E(T∗,R) = Rp−1F \P, we get that T∗ ∈ U. (cid:3) 3. Structure of homeomorphisms from topological full group In this section, we discus the structure of homeomorphisms from the topological full group [[T]] for arbitrary aperiodic T ∈ Homeo(X). Consider a Cantor aperiodic system (X,T). A Borel set Y ⊂ X is called wandering if TnY ∩Y = ∅ for all n ≥ 1. Definition. Wsay that a closed wandering set Y is basicif every clopen neighborhood of Y meets every T-orbit. Theorem 3.1. Every Cantor aperiodic system has a basic set. Sketch of the proof. Applying the argument developed in [B-D-M, The- orem 2], we can find a decreasing sequence of clopen sets {U } such that: n U ⊂ U ; TiU ∩U = ∅ for i = 1,...,n−1; and U meets every T-orbit. n+1 n n n n Then Y = U is a basic set. (cid:3) n n Remark For more results related to basic sets and their interaction with T Bratteli diagrams, see [M]. Fix a triple (X,T,Y), where (X,T) is a Cantor aperiodic system and Y is a basic set. Consider a clopen neighborhood U of Y. It is not hard to check that for every x ∈ U, there is n = n(x) > 0 such that Tnx ∈ U. Therefore, it follows from the definition of a basic set that, by the first return function, we can construct a clopen K-R partition Ξ of X with the base B(Ξ) = U. Take a decreasing sequence of clopen sets {U } such that Y = U . n n n Constructing clopenK-Rpartitions forthe U ’s andrefining them, we prove n T the following: Theorem 3.2. Let (X,T,Y) be a Cantor aperiodic system with a basic set Y. There exists a sequence of clopen K-R partitions {P } of X such n that for all n ≥ 1 (i) P refines P ; (ii) h > h , where h is the height n+1 n n+1 n n of the lowest T-tower in P ; (iii) B(P ) ⊃ B(P ); (iv) the sequence {P } n n n+1 n generates the clopen topology of X; (v) B(P ) = Y. n n We will follow here the method deveTloped in [B-K] for minimal homeo- morphisms (see also [K-W]). Let P be a clopen K-R partition with towers P(i), i = 1,...,k. Define two partitions α = α(P) and α′ = α′(P) of {1,2,...,k}. We say that J is an atom of α if there exists a subset J′ of α′ such that (3.1) T( Th(i)−1Di) = Di′ i∈J i′∈J′ [ [ and for every proper subset J of J, the T-image of Th(i)−1D is not a 0 i∈J0 i union of atoms from P. It follows from (3.1) that J′ is uniquely defined by S J and T. Let S ∈ [[T]]. Then, there are a finite set K ⊂ Z and clopen partition E = {E : k ∈ K} of X such that Sx = Tkx for x ∈ E and k ∈ K. Denote k k by E(K) the clopen partition {SkE : k ∈ K}. By Theorem 3.2, find a K-R k ON APPROXIMATION OF HOMEOMORPHISMS OF A CANTOR SET 9 partition P = {P(i) : i = 1,...,k} with P(i) = {D ,...,D } and 0,i h(i)−1,i D = TD that refines E and E(K) and so that K ⊂ (−h,h), where h j+1,i j,i is the height of the lowest T-tower in P. Let F = {(j,i)|i = 1,...,k, j = 0,...,h(i)−1}. Observe that for every pair (j,i) ∈ F there is a unique l = l(j,i) ∈ K such that (3.2) S(D ) = TlD . j,i j,i Divide F = F(P) into three disjoint sets F ,F and F as follows: in top bot (a) (j,i) ∈ F if S(D ) ⊂ P(i), i.e. 0 ≤ l+j ≤ h(i)−1; in j,i (b) (j,i) ∈ F if S(D ) goes through the topof P(i), i.e. l+j ≥ h(i); top j,i (c) (j,i) ∈ F ifS(D )goesthroughthebottomofP(i),i.e. l+j < 0, bot j,i here l is taken from (3.2). Let α and α′ be the partitions of {1,...,k} defined by T and P. For J ⊂ {1,...,k}, set h = min{h(i)|i ∈ J}. For J ∈ α and J′ ∈ α′, let J F1(r,J) = Dh(i)−hJ+r,i F2(r′,J′) = Dr,i′ i∈J i′∈J′ [ [ where r = 0,...,hJ −1 and r′ = 0,...,hJ′ −1. Definition. Wesay thatS ∈ [[T]]belongs toΓ(P) iffor eachpair (j,i) ∈ F the following conditions hold: (a) if (j,i) ∈ F and D ⊂ E , then F (h −h(i)+j,J) ⊂ E , where top j,i l 1 J l J is an atom of α containing i; (b) if (j,i) ∈ F and D ⊂ E , then F (j,J′) ⊂ E , where J′ is an bot j,i l 2 l atom of α′ containing i. Condition (a) means that whenever the set D goes through the top j,i of P(i) under the action of S, then the entire level F (r,J) containing D 1 j,i also goes through the top of P. Similarly, one can clarify condition (b) by taking the D ’s and levels F (j,J′) containing them that go through the j,i 2 bottom of P. Observe that if (j,i) ∈ F , then the entire levels F (r,J) and in 1 F (j,J′) containing D remain “within” P. 2 j,i Clearly, Γ(P) is a finite set. The following theorem reveals the structure ofhomeomorphismsfrom[[T]]foranarbitraryaperiodichomeomorphismT. Notice that this structure was found earlier for minimal homeomorphisms (see Theorem 2.2 in [B-K]). Since our proof is similar to that in [B-K, Theorem 2.2], we omit it. Theorem 3.3. Let (X,T,Y) be a Cantor aperiodic system with a ba- sic set Y and a sequence of K-R partitions {P } satisfy the conditions of n Theorem 3.2. Then [[T]] = Γ(P ) with Γ(P ) ⊂ Γ(P ). n n n n+1 The subgroup ΓY S Let (X,T) be a Cantor aperiodic system with a basic set Y. Define the subgroup Γ of[[T]]asfollows: S ∈ Γ if S ∈ Γ(P ) (andhence S ∈ Γ(P ), Y Y n m for m > n) implies that F(P ) = F . In other words, S ∈ Γ if no level n in Y from P goes over the top as well as through the bottom under the action n of S. This means that S acts as a permutation on each T-tower from P . n Therefore, the group Γ is an increasing union of permutation groups. Y Our object is to find a criterion when Γ is dense in [T]. Y Remark. (1) Denote by [[T]] the subgroup of [[T]] consisting of homeo- Y 10 KONSTANTINMEDYNETS morphisms that preserve the forward T-orbit of every y ∈ Y, i.e., S ∈ [[T]] Y if S({Tny : n ≥ 0}) = {Tny : n ≥ 0}. Observe that Γ ⊂ [[T]] . Y Y (2) The subgroup [[T]] is not τ-dense in [T]. Therefore, so is Γ . In- Y Y deed, take any z ∈ T−1Y and the Dirac measure δ supported by {z}. z Consider S ∈ U := U(T;δ ;1/2). As z ∈/ Y and Sz = Tz ∈ Y, S does not z preserve the forward T-orbit of Tz. Therefore, U contains no elements from [[T]] . Y The fact that Γ is not τ-dense in [[T]] is mainly caused by the presence Y of discrete measures. We can partly overcome this obstacle by considering only continuous measures in the definition of the topology τ. Denote by τ the topology defined by continuous measures as in Definition 1.1. One 0 can check that τ is a Hausdorff group topology on Homeo(X). The next 0 theorem answers the question when Γ is τ -dense in [T]. Y 0 Theorem 3.4. Suppose we have a Cantor aperiodic system (X,T) with a basic set Y. Then the subgroup Γ is τ -dense in [T] if and only if the Y 0 basic set Y is at most countable. Proof. (1) Assume that Y is uncountable. Take any continuous measure µsupportedbyT−1Y. ThenforeveryS ∈ U := U(T;µ;1/2)thereisatleast onez ∈ T−1Y suchthatSz = Tz. Thisimplies that{Tn(Tz) : n ≥ 0}isnot S-invariant. Therefore, by (1) of the remark above we get that Γ ∩U = ∅. Y (2) Now, assume that Y is countable. Observe that by Corollary 2.3, it is enough to approximate homeomorphisms from [[T]] with elements of Γ . Y Consider R ∈ [[T]] and a τ -neighborhood U = U(R;µ ,...,µ ;ε) of R. By 0 1 p definition of R, the sets E = {x ∈ X : Rx = Tkx}, k ∈ K, |K| < ∞, form k a clopen partition of X. Let k = sup{|k| : k ∈ K}. As Y is countable, 0 µ(TnY) = 0 for any continuous measure µ and integer n. Therefore, by Theorem 3.2 we can find a K-R partition P such that k < 2h , where h n 0 n n is the height of the lowest T-tower in P , and n k0 (3.3) µ ( TiB(P )) < ε, for j = 1,...,p. j n i=[−k0 Define a homeomorphism S ∈ Γ ∩U as follows: Y Take a T-tower (say λ = {D,...,Th(λ)D}) from P . Consider an atom n TlD of λ. We have two possibilities: (i) The R-orbit of the set TlD does not leave the T-tower λ. In this case, we define the homeomorphism S to be equal to R on the R-orbit of TlD. (ii)ThesetTlD leaves λundertheactionofR. Thenthereexist integers q < 0 < d such that Rd+1TlD and Rq−1TlD do not lie in λ entirely, whereas the sets RjTlD, j = q,...,d are contained in λ. In this case, we set S = R on RiTlD, i = q,...,d−1, and S = R−d+q on RdTlD. Observe that the choice of P guarantees that n k0 h(λ)−1 RdTlD ⊂ TiD ∪ TiD. i[=0 i=h(λ[)−1−k0