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TRIVIAL CENTRALIZERS FOR AXIOM A DIFFEOMORPHISMS 8 TODD FISHER 0 0 2 Abstract. We show there is a residual set of non-Anosov C∞ n Axiom A diffeomorphisms with the no cycles property whose ele- a ments have trivial centralizer. If M is a surface and 2 ≤ r ≤ ∞, J then we will show there exists an open and dense set of of Cr Ax- 1 iomAdiffeomorphismswiththenocyclespropertywhoseelements 3 have trivial centralizer. Additionally, we examine commuting dif- ] feomorphisms preserving a compact invariant set Λ where Λ is a S hyperbolic chain recurrent class for one of the diffeomorphisms. D . h t a 1. Introduction m [ Inspired by Hilbert’s address in 1900 Smale was asked for a list of 1 problems for the 21st century. Problem 12 deals with the centralizer v of a “typical” diffeomorphism. For f ∈ Diffr(M) (the set of Cr diffeo- 8 morphisms from M to M) the centralizer of f is 4 9 Z(f) = {g ∈ Diffr(M)|fg = gf}. 4 . 1 Letr ≥ 1,M beasmooth,connected, compact, boundarylessmanifold, 0 and 8 0 T = {f ∈ Diffr(M)|Z(f) is trivial}. : v Smale asks the following question. i X Question 1.1. Is T dense in Diffr(M)? r a A number of people have worked on these and related problems in- cluding Kopell [10], Anderson [2], Palis and Yoccoz [13] [12], Katok [8], Burslem [5], Togawa [18], and Bonatti, Crovisier, Vago, and Wilkin- son [3]. Palis and Yoccoz [13] are able to answer Question 1.1 in the affir- mative in the case of C∞ Axiom A diffeomorphisms with the added assumption of strong transversality. In [13] Palis and Yoccoz ask if Date: September 1, 2006. 2000 Mathematics Subject Classification. 37C05, 37C20, 37C29,37D05, 37D20. Key words and phrases. Commuting diffeomorphisms, hyperbolic sets, Axiom A. Supported in part by NSF Grant #DMS0240049. 1 2 TODD FISHER their results extend to the case of Axiom A diffeomorphisms with the no cycles property. The first results of the present work extend the results of Palis and Yoccoz to the case of Axiom A diffeomorphisms with the no cycles property. Let Ar(M) denote the set of Cr Axiom A diffeomorphisms with the no cycles property that are non-Anosov. Let Ar(M) denote 1 the subset of Ar(M) containing a periodic sink or source. Theorem 1.2. There is an open and dense subset of A∞(M) whose 1 elements have a trivial centralizer. Theorem 1.3. Let dim(M) ≥ 3. Then there is a residual set of A∞(M) whose elements have trivial centralizer. We can reduce the requirement of r = ∞ for certain Axiom A diffeomorphisms of surfaces. We note that in the following result we are able to include the Anosov diffeomorphisms Theorem 1.4. If M is a surface and 2 ≤ r ≤ ∞, then there exists an open and dense set of Cr Axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer. A general technique in studying the centralizer is to examine prop- erties of the centralizer on the “basic pieces” of the recurrent points. In the hyperbolic case one often looks at what is called a hyperbolic chain recurrent class. An ǫ-chain from a point x to a point y for a diffeomorphism f is a sequence {x = x ,...,x = y} such that the 0 n d(f(x ),x ) < ǫ for all 1 ≤ j ≤ n. The chain recurrent set of f is j−1 j denoted R(f) and defined by: R(f) = {x ∈ M | there is an ǫ-chain from x to x for all ǫ > 0}. Forapoint x ∈ R(f) thechain recurrent classofx consists of allpoints y ∈ R(f) such that for all ǫ > 0 there is an ǫ-chain from x to y and an ǫ-chain from y to x. IfSmale’squestioncanbeansweredintheaffirmativeonewouldhope that the following is true: for a residual set of diffeomorphisms G if Λ is a hyperbolic chain recurrent class for f ∈ G and g ∈ Z(f) that there is a dichotomy, either g| is the identity or g| is a hyperbolic chain Λ Λ recurrent class. In regards to this dichotomy we have the following result. Theorem 1.5. Suppose f ∈ Diffr(M) for any r ≥ 1, g ∈ Z(f), Λ is a mixing hyperbolic chain recurrent class of f and a hyperbolic set for g. Then Λ is a locally maximal hyperbolic set for g. COMMUTING DIFFEOMORPHISMS 3 We briefly remark that the above statement provides some context foranopenproblem inthetheoryofhigher ranksymbolic actions: ifan expansive homeomorphism f commutes with a transitive shift of finite type, must f be topologically conjugate to a shift of finite type? [11] Specifically, Theorem 1.5 is a smooth analog to this question. 2. Background We now review some basic definitions and facts about hyperbolic sets and commuting diffeomorphisms. We assume that all of our maps are diffeomorphisms of a manifold to itself. A compact set Λ invariant under the action of f is hyperbolic if there exists a splitting of the tangent space T f = Eu ⊕ Es and positive Λ constants C and λ < 1 such that, for any point x ∈ Λ and any n ∈ N, kDfnvk ≤ Cλnkvk, for v ∈ Es, and x x kDf−nvk ≤ Cλnkvk, for v ∈ Eu. x x For ǫ > 0 sufficiently small and x ∈ Λ the local stable and unstable manifolds are respectively: Ws(x,f) = {y ∈ M | for all n ∈ N,d(fn(x),fn(y)) ≤ ǫ}, and ǫ Wu(x,f) = {y ∈ M | for all n ∈ N,d(f−n(x),f−n(y)) ≤ ǫ}. ǫ The stable and unstable manifolds are respectively: Ws(x,f) = f−n(Ws(fn(x),f)), and n≥0 ǫ Wu(x,f) =S fn(Wu(f−n(x),f)). n≥0 ǫ For a Cr diffeomorphism tShe stable and unstable manifolds of a hyperbolic set are Cr injectively immersed submanifolds. A point x is non-wandering for a diffeomorphism f if for any neighborhood U of x there exists an n ∈ N such that fn(U)∩U 6= ∅. The set of non-wandering points is denoted NW(f). A diffeomorphism f is Axiom A if NW(f) is hyperbolic and is the closure of the periodic points. A hyperbolic set is locally maximal if there exists a neighborhood U of Λ such that Λ = fn(U). Locally maximal hyperbolic sets have n∈Z some special properTties. First, we have the standard result called the Shadowing Theorem, see [14, p. 415]. Let {x }j2 be an ǫ-chain for f. j j=j1 A point y δ-shadows {x }j2 provided d(fj(y),x ) < δ for j ≤ j ≤ j . j j=j1 j 1 2 Theorem 2.1. (Shadowing Theorem) If Λ is a locally maximal hyperbolic set, then given any δ > 0 there exists an ǫ > 0 and η > 0 such that if {x }j2 is an ǫ-chain for f with d(x ,Λ) < η, then there is a y j j=j1 j which δ-shadows {x }j2 . If the ǫ-chain is periodic, then y is periodic. j j=j1 If j = −j = ∞, then y is unique and y ∈ Λ. 2 1 4 TODD FISHER The Shadowing Theorem also implies the following: Corollary 2.2. If Λ is a locally maximal hyperbolic set of a diffeomorphism f, then cl(Per(f| )) = NW(f| ) = R(f| ). Λ Λ Λ An additional consequence of the Shadowing Theorem is the structural stability of hyperbolic sets. The following is a classical result, see [9, p. 571-572]. Theorem 2.3. (Structural stability of hyperbolic sets) Let f ∈ Diff(M) and Λ be a hyperbolic set for f. Then for any neighborhood V of Λ and every δ > 0 there exists a neighborhood U of f in Diff(M) such that for any g ∈ U there is a hyperbolic set Λ ⊂ V and a homeomorphism g h : Λ → Λ with d (id,h) + d (id,h−1) < δ and h◦g| = f| ◦h. g C0 C0 Λg Λ Moreover, h is unique when δ is sufficiently small. If X is a compact set of a smooth manifold M and f is a continuous map from M to itself, then f| is transitive if for any open sets U and X V of X there exists some n ∈ N such that fn(U)∩V 6= ∅. A set X is mixing if for any open sets U and V in X there exists an N ∈ N such that fn(U)∩V 6= ∅ for all n ≥ N. A standard result for locally maximal hyperbolic sets is the following Spectral Decomposition Theorem [9, p. 575]. Theorem 2.4. (Spectral Decomposition) Let f ∈ Diffr(M) and Λ a locally maximal hyperbolic set for f. Then there exist disjoint closed sets Λ ,...,Λ and a permutation σ of {1,...,m} such that NW(f| ) = 1 m Λ m Λ , f(Λ ) = Λ , and when σk(i) = i then fk| is topologically i=1 i i σ(i) Λi Smixing. Corollary 2.2 implies that Theorem 2.4 can be stated for a decomposition of the chain recurrent set of f restricted to Λ where Λ is a locally maximal hyperbolic set. In the case where f is Axiom A we have the following version of the Spectral Decomposition Theorem. Theorem 2.5. [14, p. 422] Let f ∈ Diff1(M) and assume that f is Axiom A. Then there are a finite number of sets Λ ,...,Λ closed, 1 N N pairwise disjoint, and invariant by f such that NW(f) = Λ . Fur- i=1 i thermore, each Λi is topologically transitive. S In the theorem above the sets Λ are called basic sets. We define a i relation ≪ on the basic sets Λ ,...,Λ given by the Spectral Decom- 1 m position Theorem as follows: Λ ≪ Λ if i j (Wu(Λ )−Λ )∩(Ws(Λ )−Λ ) 6= ∅. i i j j COMMUTING DIFFEOMORPHISMS 5 A k-chain is a sequence Λ ,...,Λ where Λ 6= Λ for i,l ∈ [1,k] and j1 jk ji jl Λ ≪ Λ ≪ ... ≪ Λ . j1 j2 jk A k-cycle is a sequence of basic sets Λ ,...,Λ such that j1 jk Λ ≪ Λ ≪ ... ≪ Λ ≪ Λ . j1 j2 jk j1 A diffeomorphism is Axiom A with the no cycles property if the diffeomorphisms is Axiom A and there are no cycles between the basic sets given by the Spectral Decomposition Theorem. The set R(f) is hyperbolic if and only if f is Axiom A with the no cycles property. For any r ≥ 1 the set of Cr diffeomorphisms with R(f) hyperbolic is open. For a hyperbolic set Λ let Ws(Λ) = {x ∈ M | lim d(fn(x),Λ) = 0}. n→∞ If Λ is a topologically transitive locally maximal hyperbolic set and p ∈ Per(f) ∩ Λ, then Ws(O(p)) is dense in Ws(Λ). If Λ is a mixing locally maximal hyperbolic set and p ∈ Per(f)∩Λ, then Ws(p) is dense in Ws(Λ). The following proposition found in [1] will be used in the proof of Theorem 1.5. Proposition 2.6. If Λ is a hyperbolic chain recurrent class, then there exists a neighborhood U of Λ such that R(f)∩U = Λ. A set X ⊂ M has an attracting neighborhood if there exists a neighborhood V of X such that X = fn(V). A set X ⊂ M has a n∈N repelling neighborhood if there existTs a neighborhood U of X such that X = f−n(U). Aset Λ ⊂ M iscalled a hyperbolic attractor (hyper- n∈N bolic Trepeller) if Λ is a transitive hyperbolic set for a diffeomorphism f with an attracting neighborhood (a repelling neighborhood). A hyperbolic attractor (repeller) is non-trivial if it is not the orbit of a periodic sink (source). Remark 2.7. For Axiom A diffeomorphisms with the no cycles property there is an open and dense set of points of the manifold that are in the basin of a hyperbolic attractor. Wenowreviewsomebasicpropertiesofcommutingdiffeomorphisms. Let f and g be commuting diffeomorphisms. Let Pern(f) be the peri- odicpointsofperiodnforf andPern(f)denotethehyperbolicperiodic h points in Pern(f). If p ∈ Pern(f), then g(p) ∈ Pern(f) so g permutes the points of Pern(f). Furthermore, if p ∈ Pern(f), then T fnT g = T gT fn. g(p) p p p 6 TODD FISHER Hence, the linear maps T fn and T fn are similar. If p ∈ Pern(f), g(p) p h then g(p) ∈ Pern(f). Since #(Pern(f)) < ∞ it follows that if p ∈ h h Pern(f), then g(p) ∈ Per(g). Additionally, If p ∈ Pern(f), then h h g(Wu(p,f)) = Wu(g(p),f) and g(Ws(p,f)) = Ws(g(p),f). 3. Trivial centralizer for Axiom A diffeomorphisms with no cycles We will show that Theorems 1.2 and 1.3 will follow from extending the results in [13] if one can show the following theorem: Theorem 3.1. There exists and open and dense set V of A∞(M) such that if f ∈ V and g ,g ∈ Z(f) where g = g on a non-empty open set 1 2 1 2 of M, then g = g . 1 2 Before proceeding with the proof of Theorem 3.1 we review a result of Anderson [2]. Let f ∈ Diff∞(Rn) be a contraction. Anderson shows that if g ,g ∈ Z(f) and g = g on an open set of Rn, then g = g on 1 2 1 2 1 2 all of Rn. In the proof of Theorem 3.1 it is sufficient to show that there exists an open set V of A∞(M) such that if f ∈ V, g ∈ Z(f), and there exists an open set U of M where g| = id| , then g = id . U U M Now suppose that f ∈ A∞(M) and g ∈ Z(f) where g is the identity for a non-empty open set U of M. Then U intersects the basin of either a hyperbolic attractor or repeller Λ for f in an open set denoted U . Λ Let p ∈ Pern(Λ). Then there exists a i ∈ N such that Ws(fi(p)) ∩ U 6= ∅. Since g(Ws(fi(p))) = Ws(q) for some periodic point q and g is the identity on U we know that g(fi(p)) = fi(p). The map fn restrictedtoWs(fi(p))isacontractionthatcommuteswithg restricted to Ws(fi(p)). Hence, g is the identity on Ws(fi(p)) from Anderson’s result. The density of Ws(O(p)) in Ws(Λ) implies thatg is the identity on Ws(Λ). A similar argument holds for repellers. To prove Theorem 3.1 we then need a way to connect the basins of adjacent attractors so that if g is the identity in one it will be the identity in the other. To do this we will prove the next proposition. We note that the next proposition is similar to the Lemma in the proof of Theorem 1 from [13, p. 85]. Proposition 3.2. There exists an open and dense set V of Ar(M), 1 ≤ r ≤ ∞, such that if f ∈ V and Λ and Λ′ are attractors for f where Ws(Λ)∩Ws(Λ′) 6= ∅, then there exists a hyperbolic repeller Λ such that r Ws(Λ)∩Wu(Λ ) 6= ∅ and Ws(Λ′)∩Wu(Λ ) 6= ∅. r r COMMUTING DIFFEOMORPHISMS 7 Before proving the above proposition we show how it implies Theo- rem 3.1. ProofofTheorem 3.1. LetV beopenanddenseinA∞(M)satisfying Proposition 3.2 and let f ∈ V. Since f ∈ A∞(M) we know there is an openanddensesetofM containedinthebasinofhyperbolicattractors. Denote the hyperbolic attractors of f as Λ ,...,Λ . Let g ∈ Z(f) 1 k such that g is the identity on a non-empty open set U contained in M. Then there exists some attractor Λ where 1 ≤ i ≤ k such that i Ws(Λ )∩U 6= ∅. Hence, g is the identity on Ws(Λ ). i i For any attractor Λ such that j Ws(Λ )∩Ws(Λ ) 6= ∅ i j there exists a repeller Λ such that r Ws(Λ )∩Wu(Λ ) 6= ∅ and Ws(Λ )∩Wu(Λ ) 6= ∅. i r j r This follows from Proposition 3.2. It then follows that g is the identity on Wu(Λ ) and Ws(Λ ) since the intersection of the basins for an r j attractor and a repeller is an open set. Continuingtheargumentweseethatg istheidentityon k Ws(Λ ). n=1 n Hence, g is the identity on all of M from Remark 2.7. 2 S We now state and prove two lemmas that will be helpful in proving Proposition 3.2. Lemma 3.3. There exists an open and dense set V of Ar(M) for 1 1 ≤ r ≤ ∞ such that if f ∈ V , Λ is a hyperbolic repeller for f, and 1 Λ = Λ ≪ Λ ≪ ... ≪ Λ , then Λ ≪ Λ . 0 1 k k Proof. Let U be a connected component of Ar(M). Let Λ ,...,Λ 0 M be basic sets such that Λ ,...,Λ are the hyperbolic repellers for each 0 j f ∈ U. We will prove the lemma inductively on k. For k = 1 the statement is trivially true. Assume for k ≥ 1 that there is an open and dense set U of U such that if Λ = Λ ≪ ... ≪ Λ where 0 ≤ n ≤ j, then k n0 nk 0 Λ ≪ Λ . nk Fix α = (α ,...,α ) ∈ {0,...,j}×{j +1,...,M}k+1. 0 k+1 LetI bethesetofallsuchαandletf ∈ U suchthatΛ ≪ ... ≪ Λ k α0 αk+1 for f. Since f ∈ U we know that Λ ≪ Λ ≪ Λ . k α0 αk αk+1 ThenextclaimwillshowthereisanarbitrarilysmallCr perturbation of f such that Λ ≪ Λ . α0 αk+1 Claim 3.4. If f ∈ Ar(M), Λ ≪ Λ ≪ Λ , Λ is a repeller, y ∈ 1 2 Wu(Λ )∩Ws(Λ ), and U is a sufficiently small neighborhood of y, then 1 2 8 TODD FISHER there exists an arbitrarily small Cr perturbation fõf f with support in U such that f(y) ∈ Wu(Λ)∩Ws(Λ ) for f˜. 2 Proof of Claim 3.4. Let p be a periodic point of Λ . Since Ws(O(p)) 1 is dense in Ws(Λ ) and Wu(Λ) is open we know that there exists some 1 m such that Wu(Λ)∩Ws(fm(p)) 6= ∅. Let x ∈ Wu(Λ)∩Ws(fm(p)). If we take a transversal to Ws(fm(p)) at x such that the transversal is contained in Wu(Λ), then the Inclination Lemma (or λ-lemma) [4, p. 122] implies that the transversal accumulates on Wu(fm(p)). By the invariance of Wu(Λ) the same holds for any power of p. Let y ∈ Wu(Λ ) ∩ Ws(Λ ). Then there exists an n such that y ∈ 1 2 Wu(fn(p)) and hence Wu(Λ) accumulates on y. Since y is a wandering point there exists a sufficiently small neighborhood U of y such that fn(U)∩U = ∅ for all n ∈ Z−{0}, and U is disjoint from a neighborhood of Λ∪Λ ∪Λ . 1 2 Let y be a sequence in Wu(Λ) converging to y. Then there exists k an arbitrarily small Cr perturbation f˜, with support in U, of f such that y gets mapped to f(y) for some k sufficiently large. We know k that f˜−n(y ) = f−n(y ) and fñ(y) = fn(y) for all for all n ∈ N. Hence k k f(y) ∈ Wu(Λ)∩Ws(Λ ) for f˜. 2 2 We now return to the proof of the lemma. The previous claim shows ˜ thatbyanarbitrarilysmallperturbationf wehave Λ ≪ Λ . Since α0 αk+1 U is open we may assume f˜∈ U . Since Wu(Λ ) is open and varies k k α0 continuously with f˜ as does Ws(Λ ) we know that it is an open αk+1 condition that Λ ≪ ... ≪ Λ and Λ ≪ Λ . α0 αk+1 α0 αk+1 Let U0 ⊂ U such thatforallf ∈ U0 thereexists someǫ > 0where k,α k k,α Λ ≪ ... ≪ Λ is not a chain for all g ∈ B (f). Let U1 ⊂ U α0 αk+1 ǫ k,α k such that Λ ≪ ... ≪ Λ and Λ ≪ Λ . From the previous α0 αk+1 α0 αk+1 argument we know that U1 is open. We want to show that U1 is k,α k,α dense in U −U0 . k k,α Let f ∈ U −U0 . Then there exists a sequence of f converging to k k,α n f such that for each f we have Λ ≪ ... ≪ Λ . Hence, there exists n α0 αk+1 a sequence f˜ converging to f such that each f˜ ∈ U1 . n n k,α Define U = U0 ∪U1 . The set U is then open and dense in U . k,α k,α k,α k,α k Define U = U . k+1 k,α α\∈I Since the set I is finite we know that U is open and dense in U . The k+1 k no cycles property implies there are no chains of length M +1. Define V = U this will be open and dense in Ar(M) and by construction if 1 M COMMUTING DIFFEOMORPHISMS 9 Λ is a hyperbolic repeller for f ∈ V , and Λ = Λ ≪ Λ ≪ ... ≪ Λ , 1 0 1 k then Λ ≪ Λ for f. 2 k Lemma 3.5. There exists an open and dense set V of Ar(M) for 2 1 ≤ r ≤ ∞ such that if f ∈ V , Λ is a hyperbolic attractor for f, Λ′ is 2 a basic set for f, Λ is a hyperbolic repeller for f with Λ ≪ Λ′, and r r Ws(Λ)∩Wu(Λ′) 6= ∅, then Λ ≪ Λ. r Proof. Let C be a connected component of Ar(M) ∩ V where V is 1 1 an open and dense set of diffeomorphisms satisfying Lemma 3.3 and let Λ ,...,Λ be the basic sets for f ∈ C where Λ ,...,Λ are the hy- 0 M 0 j perbolic attractors and Λ ,Λ ...,Λ are hyperbolic repellers. Before J J+1 M proceeding with the proof of the lemma we prove a claim. Claim 3.6. If f ∈ C satisfies the following: • Λ is a hyperbolic attractor for f, a • Λ is a basic set for f, b • Λ is a repeller for f, r • Ws(Λ)∩Wu(Λ′) 6= ∅, and • Λ ≪ Λ , r b then there exists an arbitrarily small perturbation of f such that Λ ≪ r Λ . a Proof of claim. Lemmas 2.4 and 2.5 in [16] show that Ws(Λ )∩Λ 6= ∅ and Ws(Λ )∩(Wu(Λ )−Λ ) 6= ∅. a b a b b Let x ∈ Ws(Λ )∩(Wu(Λ )−Λ ). Since x is wandering there exists a a b b neighborhood V of x such that fn(V)∩V = ∅ for all n ∈ Z−{0}. Since Λ ≪ Λ we know that Wu(x) ⊂ Wu(Λ ). As in the proof of r b r Claim 3.4 there exists a Cr small perturbation f˜ with support in V such that f(x) ∈ Wu(Λ ) for f˜. r Since fn(V)∩V = ∅ for all n ∈ N, the perturbation had support in V, and f(x) ∈ Wu(Λ )∩Ws(Λ ) for f˜ we know that Λ ≪ Λ for f˜. r a r a 2 We now return to the proof of the lemma. Let α = (α ,α ,α ) ∈ {0,...,j}×{j +1,...,M}×{J,...,M} 1 2 3 and I be the set of all such α. Let C0 be the set of all f ∈ C such that α if Ws(Λ )∩Wu(Λ ) 6= ∅ and Λ ≪ Λ , then Λ ≪ Λ . We want α1 α2 α3 α2 α3 α1 to show that C = intC0 is dense in C. α α Let f ∈ C − C . Then for all neighborhoods U of f there exists a α function g ∈ U satisfying the following: • Ws(Λ )∩Wu(Λ ) 6= ∅, α1 α2 10 TODD FISHER • Λ ≪ Λ , and α3 α2 • Wu(Λ )∩Ws(Λ ) = ∅. α3 α1 Then from Claim 3.6 there exists an arbitrarily small perturbation g˜ of g such that Λ ≪ Λ for g˜. Since the intersection of the basins α3 α1 for attractors and repellers is an open condition among the diffeomorphisms we know that g˜ ∈ C . Hence, f ∈ C and C is open and dense α α α in C. Let V = C . 2 α α\∈I Since I is a finite set this will be an open and dense set in C. 2 Proof of Proposition 3.2. Let V = V ∩ V where V and V are 1 2 1 2 open and dense sets in Ar(M) satisfying Lemma 3.3 and Lemma 3.5, respectively. Let f ∈ V and Λ and Λ′ be attractors such that Ws(Λ)∩ Ws(Λ′) 6= ∅. Fix x ∈ Ws(Λ) ∩ Ws(Λ′). Then x ∈ Wu(Λ˜) for some basic set Λ˜. Then from Lemma 3.5 there exists a hyperbolic repeller Λ such that Λ ≪ Λ and Λ ≪ Λ′. 2 r r r To extend the proofs of Theorems 2 and 3 from [13] to Theorems 1.2 and 1.3 we now need to show that the lack of strong transversality is not essential in the arguments. Let U(M) be the set of C∞ Axiom A diffeomorphisms with the strong transversality condition. Let U (M) consist of all elements of 1 U(M) that have a periodic sink or source. To prove Theorem 2 in [13] it is shown there is an open and dense set C (M) of U (M) such that if f ∈ C (M), then there is a periodic sink 1 1 1 (or source) p such that if g ∈ Z(f), then g = fk in Ws(p) (Wu(p)). Theorem 1 in [13] (that is similar to Theorem 3.1 in the present work) is then used to connect the regions to show that g is a power of f for all of M. Similarly, to prove Theorem 3 in [13] it is shown there is a set C(M) that is residual in U(M) if dim(M) ≥ 3 such that for any f ∈ C(M) there is a hyperbolic attractor (or repeller) Λ for f such that if g ∈ Z(f), then g = fk in Ws(Λ) (Wu(Λ)). Theorem 1 in [13] is then used to connect the regions to show that g is a power of f for all of M. Let p be a periodic point of period k for a diffeomorphism f ∈ A∞(M) of an n-dimensional manifold. The periodic point p is non- resonant if the eigenvalues of Dfkp are distinct and for all (j ,...,j ) ∈ 1 n Nn such that j ≥ 2, we have k P λ 6= λj1...λjn for all 1 ≤ i ≤ n. i 1 n