aa r X i v : . [ m a t h . OA ] O c t LOCAL HOMEOMORPHISMS THAT ∗ -COMMUTEWITH THE SHIFT PAULETTE N. WILLIS
Abstract.
Exel and Renault proved that a sliding block code ona one-sided shift space coming from a progressive block map is alocal homeomorphism. We provide a counterexample showing thatthe converse does not hold. We use this example to generalize thenotion of progressive to a property of block maps we call weaklyprogressive , and we prove that a sliding block code coming from aweakly progressive block map is a local homeomorphism. We alsointroduce the notion of a regressive block map and prove that asliding block code ∗ -commutes with the shift map if and only if itcomes from a regressive block map. We also prove that a slidingblock code is a local homeomorphism and ∗ -commutes with theshift map if and only if it is a k -fold covering map defined from aregressive block map. Introduction
In symbolic dynamics one considers spaces of sequences with entriesfrom a finite alphabet A together with a shift map on the space. Thereare two versions of this theory, one that considers the space of two-sidedinfinite sequences A Z with a shift map σ that is a homeomorphism, andone that considers the space of one-sided infinite sequences A N with ashift map σ that is a local homeomorphism.Morphisms between shift spaces are called sliding block codes, andone can prove that any such morphism τ d comes from a block map d : A n → A (see [3, 6, 5]). In [3, Theorem 3.4] Hedlund proved that themorphisms on two-sided infinite sequences are precisely functions of theform σ k τ d for some k ∈ Z . In Section 2 of this paper we characterizemorphisms on one-sided infinite sequences as functions of the form τ d (Theorem 2.10). More specifically, the morphisms on one-sided infinite Date : April 27, 2018.2010
Mathematics Subject Classification.
Key words and phrases. shift space, ∗ -commuting, local homeomorphism.This research was partially supported by the University of Iowa Graduate CollegeFellowship as part of the Sloan Foundation Graduate Scholarship Program and theUniversity of Iowa Department of Mathematics NSF VIGRE grant DMS-0602242. sequences are functions τ d : A N → A N defined from a block map d : A n → A , for some n ∈ N , by ( τ d ( x )) i = d ( x i · · · x i + n − ). While thisresult has been stated on several occasions in the literature we includea proof for completeness.The first main topic of this paper is to consider when a sliding blockcode on the one-sided shift space is a local homeomorphism. To date,there is no known characterization. In [2, Theorem 14.3] Exel and Re-nault proved that if the block map d that defines τ d is a progressive function (Definition 3.2), then τ d is a local homeomorphism. At thetime, it was not known if the converse is true. In Section 3 we providea counterexample to the converse (Example 3.6). We use this coun-terexample as motivation to generalize the idea of a progressive blockmap and introduce what we call a weakly progressive block map (Def-inition 3.7). We prove that if the block map d is weakly progressivethen τ d is a local homeomorphism (Theorem 3.11). This gives weakerhypothesis under which we can conclude that τ d is a local homeomor-phism. We do not yet know if the converse is true; that is, we do notknow if a sliding block code that is a local homeomorphism must comefrom a weakly progressive block map.The second main topic of this paper is to examine sliding blockcodes that ∗ -commute with the shift. The concept of two functions ∗ -commuting was introduced in [1] and further examined in [2]. Let X be a topological space. Then two functions S, T : X → X ∗ -commute if they commute and for all y, z ∈ X with S ( y ) = T ( z ) there existsa unique x ∈ X such that T ( x ) = y and S ( x ) = z . In Section 4 weintroduce the concept of a regressive block map (Definition 4.5) andcharacterize the sliding block codes that ∗ -commute with the shift asthose for which the block map d is regressive.Local homeomorphisms that ∗ -commute with the shift have interest-ing properties that we discuss in Section 5. In particular, we prove thata sliding block code is a local homeomorphism and ∗ -commutes withthe shift if and only if φ : A N → A N is a k -fold covering map comingfrom a regressive block map (Theorem 5.14).The author thanks Ruy Exel for many enlightening discussions onthe material. Notation and conventions:
Throughtout this paper we let A bea finite alphabet. Give A the discrete topology, then A is a com-pact Hausdorff space. Let A n denote the words of length n , let A ∗ := S n ≥ A n , and let A N denote the one-sided infinite sequence space of ele-ments in A . Since A is a compact Hausdorff space, A N with the producttopology is also a compact Hausdorff space by Tychonoff’s Theorem. OCAL HOMEOMORPHISMS THAT ∗ -COMMUTE WITH THE SHIFT 3 For µ ∈ A ∗ we define a cylinder set Z ( µ ) := { x ∈ A N : x · · · x | µ | = µ } .Note that the family { Z ( µ ) : µ ∈ A ∗ } is a basis for A N and each Z ( µ ) is clopen (and therefore compact). Let σ : A N → A N defined by σ ( x x x · · · ) = x x · · · be the shift map.2. Continuous functions that commute with the shift
In this section we provide a proof of the fact that a function on theone-sided shift space that is continuous and commutes with the shift isa sliding block code. Our proof is constructive in the sense that givenan arbitrary continuous function φ : A N → A N that commutes withthe shift map σ , Lemma 2.5 and Proposition 2.6 demonstrate how toconstruct a block map that defines φ . Definition 2.1. A block map is a function d : A n → A for some n ∈ N . For any block map d we define a function τ d : A N → A N by τ d ( x ) i = d ( x i · · · x i + n − ). We call τ d a sliding block code . Remark 2.2.
Define d : A → A by d ( a a ) = a . Then τ d = σ . Notethat τ d is not uniquely determined by the choice of d . For example,define d ′ : A → A by d ′ ( a a a ) = a . Then τ d ′ = σ = τ d . Lemma 2.3. If φ : A N → A N is a sliding block code, then φ is contin-uous and commutes with the shift map σ .Proof. Since φ is a sliding block code there exists n ∈ N and a blockmap d : A n → A such that τ d = φ . Let µ ∈ A ∗ . If x = x x x · · · ∈ τ − d ( Z ( µ )), then one can show that x ∈ Z ( x x · · · x k + n − ) ⊆ τ − d ( Z ( µ ))so τ − d ( Z ( µ )) is open. Therefore τ d is continuous. Let x ∈ A N andobserve τ d σ ( x ) = τ d ( x x x · · · ) = d ( x · · · x n +1 ) d ( x · · · x n +2 ) · · · = σ ( d ( x · · · x n ) d ( x · · · x n +1 ) d ( x · · · x n +2 ) · · · ) = στ d ( x ) . (cid:3) Remark 2.4.
For each µ ∈ A ∗ , Z ( µ ) = F a ∈ A Z ( µa ) . Recall that since A N is compact, any open set U ⊆ A N is a finite union of basis elements.Let S be a finite subset of A ∗ . Then for any open set U = S µ ∈ S Z ( µ ) ,we may extend the lengths of the µ ∈ S so that we may find n ∈ N anda finite set T ⊆ A n such that S µ ∈ S Z ( µ ) = F ν ∈ T Z ( ν ) . Note that forany µ ∈ S , | µ | ≤ n . Therefore for any ν ∈ T there exists µ ∈ S suchthat Z ( ν ) ⊆ Z ( µ ) . Lemma 2.5. If φ : A N → A N is a continuous function, then thereexists n ∈ N such that for each λ ∈ A n there exists a unique a ∈ A such that Z ( λ ) ⊆ φ − ( Z ( a )) . PAULETTE N. WILLIS
Proof.
Consider { Z ( a ) | a ∈ A } . Notice that the Z ( a )’s are disjointclopen sets that cover A N . Define V a = φ − ( Z ( a )). Then the V a ’s arealso disjoint clopen sets that cover A N . Since A N is compact and the V a ’s are closed, the V a ’s are also compact. Therefore each V a is theunion of a finite number of basis elements. That is, V a = S µ ∈ S a Z ( µ ),where S a is a finite subset of A ∗ . By Remark 2.4 there exists m ∈ N and a finite set T a ⊆ A m such that V a = S µ ∈ S a Z ( µ ) = F ν ∈ T a Z ( ν ). Let T = S a ∈ A T a . Then F a ∈ A V a = F ν ∈ T Z ( ν ) where T is a finite subset of A ∗ . Observe that the Z ( ν )’s are disjoint since the V a ’s are disjoint andthat for each ν ∈ T there exists a unique a ∈ A such that Z ( ν ) ⊆ V a .By Remark 2.4 there exists n ∈ N and a finite set R ⊆ A n such that F a ∈ A V a = F ν ∈ T Z ( ν ) = F λ ∈ R Z ( λ ). Note that since | ν | ≤ n for every ν ∈ T and the Z ( ν ) ′ s are disjoint, then for every λ ∈ R there exists aunique ν ∈ T such that Z ( λ ) ⊆ Z ( ν ). Recall that F a ∈ A V a is a coverof A N . Therefore F λ ∈ R Z ( λ ) is also a cover of A N , and hence R = A n .Observe that for each λ ∈ A n , there exists a unique ν ∈ T and a unique a ∈ A such that Z ( λ ) ⊆ Z ( ν ) ⊆ V a = φ − ( Z ( a )). (cid:3) Proposition 2.6. If φ : A N → A N is a continuous function that com-mutes with the shift map σ , then φ is a sliding block code.Proof. Since φ is continuous, by Lemma 2.5 there exists n ∈ N suchthat for each λ ∈ A n there exists a unique a ∈ A such that Z ( λ ) ⊆ φ − ( Z ( a )). Define d : A n → A by d ( λ ) = a where a is the unique ele-ment in A such that Z ( λ ) ⊆ φ − ( Z ( a )). The function d is well definedsince the element a is unique. We will now show that τ d = φ . Let k ∈ N and x ∈ A N . Notice that σ k ( x ) ∈ Z ( x k +1 · · · x k + n ) ⊆ φ − ( Z ( a ))for some a ∈ A . Then φ ( σ k ( x )) = a . Also, we have τ d ( σ k ( x )) = d ( σ k ( x ) · · · σ k ( x ) n ) = d ( x k +1 · · · x k + n ) = a . Therefore φ ( x ) k = σ k ( φ ( x )) = φ ( σ k ( x )) = a = τ d ( σ k ( x )) = σ k ( τ d ( x )) = τ d ( x ) k . (cid:3) Example . Fix a ∈ A and define aaa · · · = a ∈ A N . Consider aconstant function φ : A N → A N defined by φ ( y ) = a for all y ∈ A N .Then φ is a continuous function that commutes with σ . Notice thatfor b ∈ A , φ − ( Z ( b )) = ∅ unless b = a , thus we have φ − ( Z ( a )) = A and n = 1. So by Proposition 2.6 d : A → A is defined by d ( b ) = a forall b ∈ A and τ d = φ . Example . Let φ : A N → A N be defined by φ ( x x x · · · ) = x x · · · .Observe that for b ∈ A , φ − ( Z ( b )) = F a ∈ A Z ( ab ). So for all x x ∈ A we have Z ( x x ) ⊂ φ − ( Z ( x )). So by Proposition 2.6 d : A → A isdefined by d ( x x ) = x . OCAL HOMEOMORPHISMS THAT ∗ -COMMUTE WITH THE SHIFT 5 The following proposition shows us the extent to which the function d of Proposition 2.6 is unique. Proposition 2.9.
Let φ : A N → A N be a continuous function thatcommutes with the shift map σ . Let n be the smallest natural numberwith the property that there exists a block map d : A n → A such that τ d = φ . (By Proposition 2.6 such an n exists.) If m ∈ N and thereexists a function d ′ : A m → A such that τ d ′ = φ , then m ≥ n and d ′ ( x i · · · x m + i − ) = d ( x i · · · x n + i − ) for all i ∈ N . In particular, if m = n , then d ′ = d .Proof. Since τ d = φ = τ d ′ , we have m ≥ n by the minimality of n . Also,for all i ∈ N and x ∈ A N we have d ( x i · · · x n + i − ) = τ d ( x ) i = φ ( x ) i = τ d ′ ( x ) i = d ′ ( x i · · · x m + i − ). (cid:3) Theorem 2.10.
The function φ : A N → A N is continuous and com-mutes with the shift map σ if and only if φ is a sliding block code.Proof. The sufficiency is proven in Proposition 2.6 and the necessity isproven in Lemma 2.3. (cid:3)
The following example illustrates the importance of the function φ being continuous. Example . Let A = { , } and φ : A N → A N be defined as follows: φ (0 ∞ ) = 1 ∞ , φ (1 ∞ ) = 0 ∞ , and φ is the identity on all other points.It is clear that φ commutes with σ , however it is impossible to find afunction d such that τ d = φ . If d : A n → A could be defined for some n ∈ N , then for the points where φ acts as the identity we must havethat d (0 n ) = 0 and d (1 n ) = 1. However, defining d in this mannerwould not work for the points 0 ∞ and 1 ∞ .3. Local homeomorphisms that commute with the shift
In this section we examine properties on the block map that forcethe induced sliding block code to be a local homomorphism. Exeland Renault proved that if the block map is progressive, then the in-duced sliding block code is a local homeomorphism [2, Theorem 14.3].The converse, however, remained an open problem. In this sectionwe prove the converse is false by providing a counterexample in Ex-ample 3.6. Specifically, we describe a sliding block code that is a localhomeomorphism such that there does not exist a progressive block mapthat defines it. We then generalize the idea of a progressive block mapby defining a weakly progressive block map (Definition 3.7). In Theo-rem 3.11 we prove that if the block map is a weakly progressive function,then the induced sliding block code is a local homeomorphism. This
PAULETTE N. WILLIS gives weaker hypothesis under which we can conclude that a slidingblock code is a local homeomorphism.
Definition 3.1.
Let
X, Y be topological spaces. A continuous function f : X → Y is a local homeomorphism if for every point x ∈ X thereexists an open neighborhood U of x such that f ( U ) is open in Y and f : U → f ( U ) is a homeomorphism. Definition 3.2.
A block map d : A n → A is progressive if for eachfixed x · · · x n − ∈ A n − , the function p x ··· x n − d : A → A defined by p x ··· x n − d ( a ) = d ( x · · · x n − a ) is bijective . Example . Define d : A → A . by d ( a a ) = a . Then τ d = σ . Fix a ∈ A , let a , a ∈ A and suppose p ad ( a ) = p ad ( a ). We have a = d ( aa ) = p ad ( a ) = p ad ( a ) = d ( aa ) = a , so p ad is injective. Note that b = p ad ( b ) for all b ∈ A so p ad is surjective.Hence d is progressive. Remark 3.4.
Recall from Remark 2.2 that for d ′ : A → A definedby d ( a a a ) = a we have τ d ′ = σ = τ d for the block map d fromExample 3.3. Notice that d ′ is not progressive, therefore it is importantthat we consider the smallest natural number n such that the function d : A n → A defines τ d .Given an arbitrary sliding block code φ we wish to determine if thereis a progressive block map that defines it. Let n be the smallest naturalnumber such that a block map d : A n → A defines φ . Then for any m > n and block map d ′ : A m → A that defines φ the function d ′ isnot progressive. We observe this by recalling from Proposition 2.9 that d ′ ( x i · · · x m + i − ) = d ( x i · · · x n + i − ) . Therefore d ′ can not be bijective.In this section, when we consider a block map d : A n → A thatdefines τ d we assume that n is the smallest natural number such thatthere exists a function d : A n → A that defines τ d . Proposition 2.9allows us to do this.Example . The constant function d ( b ) = a for all b ∈ A from Exam-ple 2.7 is not progressive.The following is an example of a sliding block code that is a localhomeomorphism and can not be defined from a progressive block map. OCAL HOMEOMORPHISMS THAT ∗ -COMMUTE WITH THE SHIFT 7 Example . Let A = { , , , } and define d : A → A by d (00) = 0 d (01) = 0 d (02) = 1 d (03) = 1 d (10) = 3 d (11) = 3 d (12) = 2 d (13) = 2 d (20) = 2 d (21) = 2 d (22) = 3 d (23) = 3 d (30) = 1 d (31) = 1 d (32) = 0 d (33) = 0 . Observe that it is not possible to define a block map d ′ : A → A suchthat τ d = τ d ′ . Therefore if τ d may be defined from a progressive blockmap d is the only possibility. Since d (00) = 0 = d (01) d is not pro-gressive. With a little work one can check that τ d is a homeomorphismon Z (0) ∪ Z (1) and Z (2) ∪ Z (3) such that τ d ( Z (0) ∪ Z (1)) = A N = τ d ( Z (2) ∪ Z (3)). Therefore τ d is a local homeomorphism.Now we generalize the idea of a progressive block map by defining aweakly progressive block map. We prove that if d is weakly progressive,then τ d is a local homeomorphism. Definition 3.7.
Fix n, m ∈ N and let a block map d : A n → A have the property that for every µ ∈ A n and every ν ∈ A m suchthat d ( µ ) = ν there exists a unique a ∈ A such that p µ ··· µ n − d,m ( aα ) = d ( µ · · · µ n − a ) d ( µ · · · µ n − aα ) · · · = ν has a solution α ∈ A m − . Thenwe say that d is weakly progressive of order m .Observe that d is progressive if and only if d is weakly progressive oforder 1. Example . Let A = { , , , } and define d : A → A by: d (00) = 0 d (01) = 0 d (02) = 1 d (03) = 1 d (10) = 2 d (11) = 2 d (12) = 3 d (13) = 3 d (20) = 0 d (21) = 0 d (22) = 1 d (23) = 1 d (30) = 2 d (31) = 2 d (32) = 3 d (33) = 3 . The function d is weakly progressive of order 2. Remark 3.9.
The block map from Example 3.6 is weakly progressiveof order . Proposition 3.10.
Let d : A n → A be a block map and fix x · · · x n − ∈ A n − . If d is weakly progressive, then τ d : Z ( x · · · x n − ) → [ a ∈ A Z ( d ( x · · · x n − a )) is bijective. PAULETTE N. WILLIS
Proof.
Fix m such that d is weakly progressive of order m . Notice thatfor x ∈ Z ( x · · · x n − ), τ d ( x ) ∈ Z ( d ( x · · · x n )) ⊆ S a ∈ A Z ( d ( x · · · x n − a )).Therefore we have τ d ( Z ( x · · · x n − )) ⊆ S a ∈ A Z ( d ( x · · · x n − a )). Let y ∈ Z ( d ( x · · · x n − a )) for some a ∈ A . We want to show that thereexists a unique x ∈ Z ( x · · · x n − ) such that τ d ( x ) = y . Notice that x · · · x n − a ∈ A n and y · · · y m ∈ A m satisfy d ( x · · · x n − a ) = y . Sosince d is weakly progressive there exists a unique a ∈ A such that p x ··· x n − d,m ( a α ) = d ( x · · · x n − a ) d ( x · · · x n − a α ) · · · = y · · · y m for some α ∈ A m − . Now consider x · · · x n − a α ∈ A n and y · · · y m +1 ∈ A m such that d ( x · · · x n − a α ) = y . Since d is weakly progressivethere exists a unique a ∈ A such that p x ··· x n − d,m ( a β ) = d ( x · · · x n − a a ) d ( x · · · x n − a a β ) = y · · · y m +1 for some β ∈ A m − . We may continue in this manner to construct x = x · · · x n − a a · · · such that τ d ( x ) = y , hence the function is surjective.Since each a i was unique we have τ d | ( Z ( x · · · x n − )) injective. (cid:3) Observe that if d is progressive, then S a ∈ A Z ( d ( x · · · x n − a )) = A N .Thus τ d is | A n − | to 1. Theorem 3.11. If d : A n → A is weakly progressive block map thenthe induced sliding block code τ d : A N → A N is a local homeomorphism.Proof. By Proposition 3.10 τ d ( Z ( x · · · x n − )) = S a ∈ A Z ( d ( x · · · x n − a ))so τ d ( Z ( x · · · x n − )) is open. Since τ d is continuous, τ d | Z ( x ··· x n − ) isalso continuous. The set Z ( x · · · x n − ) is compact since it is a cylin-der set. Recall A N is Hausdorff, hence S a ∈ A Z ( d ( x · · · x n − a )) ⊆ A N is Hausdorff. So we have τ d | Z ( x ··· x n − ) is a continuous bijective func-tion from the compact space Z ( x · · · x n − ) to the Hausdorff space S a ∈ A Z ( d ( x · · · x n − a )). Therefore by [4, Theorem 5.8], τ d | Z ( x ··· x n − ) is a homeomorphism. Hence τ d is a local homeomorphism. (cid:3) Continuous functions that ∗ -commute with the Shift The concept of ∗ -commuting for functions was introduced in [1] andfurther examined in [2]. In this section we introduce the concept of aregressive block map (Definition 4.5) and prove that sliding block codesthat ∗ -commute with the shift map σ are exactly those defined fromregressive block maps. Definition 4.1.
Let X be a set. Two functions S, T : X → X ∗ -commute if they commute and given ( y, z ) ∈ X × X such that S ( y ) = T ( z ) there exists a unique x ∈ X such that T ( x ) = y and S ( x ) = z . OCAL HOMEOMORPHISMS THAT ∗ -COMMUTE WITH THE SHIFT 9 . x . y . S ( y ) = T ( z ) . zTS ST Remark 4.2.
Definition 4.1 is the same as the definition of “star-commuting” in [2, Definition 10.1] . This is equivalent to the definitionof “ ∗ -commuting” given in [1, Definition 5.6] .Example . Let A = { , } and define the function d : A → A by d (0) = 1 and d (1) = 0. For x ∈ A N denote τ d ( x ) = x . By Lemma2.3 we know that τ d commutes with σ . Let y, z ∈ A N be such that σ ( y ) = τ d ( z ). Since τ d is bijective, observe that y is the unique elementin A N such that τ d ( y ) = y . We also have that τ d ( σ ( y )) = σ ( τ d ( y )) = σ ( y ) = τ d ( z ) and since τ d is bijective σ ( y ) = z . So τ d ∗ -commutes with σ . Example . Recall Example 3.3 where we define d : A → A by d ( a a ) = a so that τ d = σ . Let a , a ∈ A such that a = a and w ∈ A N . Observe that σ ( a w ) = w = σ ( a w ) and a w = a w .Therefore σ does not ∗ -commute with itself.In the two previous examples proving whether or not the function τ d ∗ -commutes with σ using the definition was not terribly difficult.However consider the following example: Let A = { , , · · · n − } and define d : A n → A by d ( a · · · a n ) = ( a + · · · + a n ) (mod n ).Determining whether or not the associated τ d ∗ -commutes with σ isextremely unpleasant. We would like to determine easily verifiableconditions on the block map d that determine when τ d ∗ -commuteswith the shift. Definition 4.5.
The block map d : A n → A is regressive if for eachfixed x · · · x n − ∈ A n − the function r x ··· x n − d : A → A defined by r x ··· x n − d ( a ) = d ( ax · · · x n − ) is bijective. Example . Recall the block map from Example 3.6. Notice thatwhen the second coordinate is fixed d is bijective. Therefore d is re-gressive. Example . Recall the block map from Example 3.8. Since d (00) =0 = d (20), d is not regressive. Example . For this example, all addition is modulo n . Let A = { , , · · · , n − } and define d : A n → A by d ( a · · · a n ) = ( a + · · · + a n )(mod n ). Fix x · · · x n − ∈ A n − and let x := x + · · · + x n − . To see that r d is injective, let a , a ∈ A and suppose r x ··· x n − d ( a ) = r x ··· x n − d ( a ).Then a + x = d ( a x · · · x n − ) = r x ··· x n − d ( a )= r x ··· x n − d ( a ) = d ( a x · · · x n − ) = a + x, therefore a = a . Let a ∈ A . Then we have r ··· n − d ( a ) = a . Therefore d is regressive. Theorem 4.9.
The block map d : A n → A is regressive if and only ifthe induced sliding block code τ d : A N → A N ∗ -commutes with the shiftmap σ .Proof. By Lemma 2.3, τ d commutes with σ . Suppose we have y, z ∈ A N such that σ ( y ) = τ d ( z ). Since d is regressive there exists a unique x ∈ A such that r z ··· z n − d ( x ) = d ( x z · · · z n − ) = y . Notice that y i +1 = σ ( y ) i = τ d ( z ) i = τ d ( x z ) i +1 . So we have τ d ( x z ) = d ( x z · · · z n − ) τ d ( x z ) τ d ( x z ) · · · = y y y · · · = y and σ ( x z ) = z . To see that x z is unique suppose there exists w ∈ A N such that τ d ( w ) = y and σ ( w ) = z . Then w = az for some a ∈ A .Notice that d ( az · · · z n − ) = τ d ( az ) = y = d ( x z · · · z n − ). Since d isregressive a = x . Therefore τ d ∗ -commutes with σ .Conversely, fix x · · · x n − ∈ A n − . Suppose for a , a ∈ A we have r x ··· x n − d ( a ) = r x ··· x n − d ( a ). Then let z ∈ Z ( x · · · x n − ) and observethat τ d ( a z ) = d ( a z , · · · z n − ) = r x ··· x n − d ( a )= r x ··· x n − d ( a ) = d ( a z , · · · z n − ) = τ d ( a z ) . For i ≥ τ d ( a z ) i = τ d ( z ) i − = τ d ( a z ) i . So τ d ( a z ) = τ d ( a z )and σ ( a z ) = z = σ ( a z ). Since τ d ∗ -commutes with σ we have a z = a z . Therefore r x ··· x n − d is injective. Now let a ∈ A . Suppose z ∈ Z ( x · · · x n − ) and define w = τ d ( z ). Then aw, z ∈ A N satisfy σ ( aw ) = τ d ( z ). Since τ d and σ ∗ -commute there exists a unique v ∈ A N suchthat σ ( v ) = z and τ d ( v ) = aw . Since σ ( v ) = z , there exists b ∈ A such that v = bz . So we have a = τ d ( v ) = τ d ( bz ) = d ( bz · · · z n − ) = d ( bx · · · x n − ). So b ∈ A such that r x ··· x n − d ( b ) = d ( bx · · · x n − ) = a .Therefore d is regressive. (cid:3) OCAL HOMEOMORPHISMS THAT ∗ -COMMUTE WITH THE SHIFT 11 Local homeomorphisms that ∗ -commute with the shift In this section we examine properties of sliding block codes that arelocal homomorphisms and ∗ -commute with the shift. In Theorem 5.14we show that this class of functions is precisely the k -fold covering mapsdefined from regressive block maps. Definition 5.1.
Let X be a topological space, φ : X → X be afunction and k ∈ N . We define the sets Z φk := { y ∈ X : | φ − ( y ) | = k } and Z φ ≥ k := { y ∈ X : | φ − ( y ) | ≥ k } . Remark 5.2.
Let φ : A N → A N commute with σ and fix y ∈ A N . Ifthere exists x ∈ A N such that φ ( x ) = y , then σ ( y ) = σ ( φ ( x )) = φ ( σ ( x )) .This shows that if φ − ( y ) = ∅ , then φ − ( σ ( y )) = ∅ . Definition 5.3.
Let X ⊆ A N . We say that X is shift invariant if σ ( X ) = X . Proposition 5.4. If φ : A N → A N ∗ -commutes with the shift map σ ,then Z φk is shift invariant for all k ∈ N .Proof. Let y ∈ A N and fix k ∈ N such that σ ( y ) ∈ Z φk . Then σ ( y ) has k preimages under φ and we define { z i } ki =1 = φ − ( σ ( y )) (see Figure 1below). Since σ and φ ∗ -commute, for each z i there exists a unique x i such that φ ( x i ) = y and σ ( x i ) = z i . So | φ − ( y ) | ≥ k , but we wantto show that | φ − ( y ) | = k . Suppose x ∈ φ − ( y ). Then φ ( σ ( x )) = σ ( φ ( x )) = σ ( y ). So σ ( x ) ∈ φ − ( σ ( y )) = { z i } ki =1 , and there exists i such that σ ( x ) = z i . Hence φ ( x ) = y , σ ( x ) = z i , but x i is the uniqueelement with those properties thus x = x i . Therefore | φ − ( y ) | = k thatmeans y ∈ Z φk .(1):= . x i . y . σ ( y ) . z i φσ σφ (2):= . x i . y . σ ( y ) . σ ( x i ) φσ σφ Conversely, let y ∈ A N and fix k ∈ N such that y ∈ Z φk . Thendefine { x i } ki =1 = φ − ( y ) (see Figure 2 above). Suppose w ∈ φ − ( σ ( y ))(which exists by Remark 5.2). Since σ and φ ∗ -commute there exists x such that φ ( x ) = y and σ ( x ) = w . However { x i } ki =1 = φ − ( y ) so x = x i for some i . So for each w ∈ φ − ( σ ( y )), w = σ ( x i ) for some i . So | φ − ( σ ( y )) | ≤ k . Suppose | φ − ( σ ( y )) | < k , then there exists x i , x j ∈ φ − ( y ) with i = j such that σ ( x i ) = σ ( x j ) = z , say. Sowe have y, z ∈ A N such that σ ( y ) = φ ( z ) and x i = x j such that φ ( x i ) = y = φ ( x j ) and σ ( x i ) = z = σ ( x j ). Since φ and σ ∗ -commute x i = x j , that is a contradiction. Therefore | φ − ( σ ( y )) | = k whichmeans σ ( y ) ∈ Z φk . (cid:3) Proposition 5.5. If φ : A N → A N is a local homeomorphism, then Z φ ≥ k is open in A N for all k ∈ N .Proof. Let y ∈ A N and fix k ∈ N such that y ∈ Z φ ≥ k . Then there exists l ≥ k such that φ − ( y ) = { x i } li =1 where each x i is distinct. Therefore y ∈ Z φl . Since φ is a local homeomorphism, for each x i there existsa neighborhood U i containing x i such that x j is not an element of U i for i = j , y ∈ φ ( U i ) for each i , and φ ( U i ) is open in A N . Since A N isHausdorff, let x i ∈ V i for each i and V i ∩ V j = ∅ for each i = j . Thendefine W i = V i ∩ U i . Thus the set { W i } li =1 are pairwise disjoint. Nowlet W = T li =1 φ ( W i ). Then W is open in A N and y ∈ W . We wantto show that W ⊆ Z φ ≥ k . Let z ∈ W . Then z ∈ φ ( W i ) for each i , sothere exists w i ∈ W i such that φ ( w i ) = z . Since { W i } li =1 are pairwisedisjoint, each w i is distinct. Therefore z has at least l preimages. Hence z ∈ Z φ ≥ l ⊆ Z φ ≥ k , thus W ⊆ Z φ ≥ k . Therefore Z φ ≥ k is open. (cid:3) Remark 5.6.
It is important to note that the only shift invariant opensets in A N are ∅ and A N . Any non-empty open set U contains a basicopen set Z ( x , · · · , x l ) for some l ∈ N . If σ ( U ) = U , then σ k ( U ) = U for any k ∈ N . So σ l ( Z ( x , · · · , x l )) = A N ⊆ U . Lemma 5.7. If φ : A N → A N is a local homeomorphism, then thereexists an M ∈ N such that { Z ( µ ) : µ ∈ A M } is a finite covering of A N by disjoint sets and φ is a homeomorphism on each Z ( µ ) .Proof. Let { W α } be a covering basic open sets of A N such that φ is ahomeomorphism on each set and let { W i } ni =1 be a finite subcover. ByRemark 2.4 there exists M ∈ N and T ⊆ A M such that S ni =1 W i = F ν ∈ T Z ( ν ). Since { W i } ni =1 is a cover, T = A M . Observe that for each ν ∈ A M , Z ( ν ) ⊆ W i , therefore φ is a homeomorphism on each Z ( ν ). (cid:3) Proposition 5.8.
If the sliding block code φ : A N → A N is a localhomeomorphism that ∗ -commutes with the shift map σ , then φ is sur-jective and there exists k ∈ N such that φ is k -to- . OCAL HOMEOMORPHISMS THAT ∗ -COMMUTE WITH THE SHIFT 13 Proof.
Since φ is a local homeomorphism, by Lemma 5.7 there exists an M ∈ N such that φ is a homeomorphism on each Z ( µ ) for all ν ∈ A M .Observe that for any l ∈ N such that l > | A M | we have Z φl = ∅ . Let k := max { l : Z φl = ∅} and notice that Z φk = Z φ ≥ k . Since φ ∗ -commuteswith σ , by Proposition 5.4 we have σ ( Z φk ) = Z φk . By Proposition 5.5 Z φk = Z φ ≥ k is open and by Remark 5.6 Z φk = A N . Thus every elementof A N has exactly k preimages under φ .Suppose φ is not surjective. Then there exists y ∈ A N such that y ∈ Z . We have just proven that every element of A N must have thesame number of preimages under φ . Then k = 0 which means φ is notdefined for any element in A N . Thus φ must be surjective. (cid:3) Definition 5.9.
Let p : E → B be a continuous surjective function.The open set U of B is said to be evenly covered by p if the inverseimage p − ( U ) can be written as the union of disjoint open sets V α in E such that for each α , the restriction of p to V α is a homeomorphismof V α onto U . Definition 5.10.
Let p : E → B be a continuous surjective function.If every point b ∈ B has a neighborhood U that is evenly covered by p ,then p is called a covering map and E is said to be a covering space of B . If p − ( b ) has k elements for every b ∈ B , then E is called a k -foldcovering of B . The condition that p be a local homeomorphism doesnot suffice to ensure that p is a covering map (see [7, Chapter 9, page338, Example 2]). Example . The shift σ is a | A | -fold covering map. Example . Define V := Z (0) ∪ Z (1) and W := Z (2) ∪ Z (3). Observethat for both the τ d functions from Example 3.6 and Example 3.8 wehave τ d ( V ) = A N = τ d ( W ). Therefore the τ d functions are 2-foldcovering maps. Proposition 5.13.
If the sliding block code φ : A N → A N is a localhomeomorphism that ∗ -commutes with the shift map σ , then φ is a k -fold covering map.Proof. The function φ is continuous by definition and surjective byProposition 5.8. Let y ∈ A N , then φ − ( y ) = { x i } ki =1 for some k ∈ N byProposition 5.8. Since φ is a local homeomorphism there exists an openneighborhood W i of x i such that φ : W i → φ ( W i ) is a homeomorphismand φ ( W i ) for 1 ≤ i ≤ k . Since A N is Hausdorff we may define opensets W ′ i such that x i ∈ W ′ i ⊆ W i and { W ′ i } ki =1 are pointwise disjoint.Notice that φ | W ′ i is a homeomorphism onto its image and φ ( W ′ i ) is open. Let U := ∩ ki =1 φ ( W ′ i ). Then U is an open set such that y ∈ U .Define V i := φ − ( U ) ∩ W ′ i . Then x i ∈ V i and the V i ’s are open andpairwise disjoint. Notice that φ − ( U ) = F ki =1 V i . Observe that φ | V i isa homeomorphism and φ ( V i ) = φ ( φ − ( U ) ∩ W ′ i ) = U ∩ φ ( W ′ i ) = U .Hence φ ( V i ) is onto U for each i . Therefore A N is evenly covered by φ . By Theorem 5.8 φ is k -to-1 for some k ∈ N , therefore φ is a k-foldcovering map. (cid:3) Theorem 5.14.
A sliding block code φ : A N → A N is a local homeo-morphism and ∗ -commutes with the shift map σ if and only if φ is a k -fold covering map defined from a regressive block map.Proof. Since φ is a sliding block code that ∗ -commutes with the shift,by Theorem 4.9 there exists a regressive block map d such that τ d = φ .Since φ is a local homeomorphism that ∗ -commutes with the shift, byProposition 5.13 φ is a k -fold covering map.Conversely, if φ is a k -fold covering map, then φ is a local homeomor-phism by definition. Since φ is defined from a regressive block map, byTheorem 4.9 φ ∗ -commutes with the shift. (cid:3) References [1] V. Arzumanian & J. Renault,
Examples of pseudogroups and their C ∗ -algebras ,Operator Algebras and Quantum Field Theory (Rome, 1996), 93–104, Int.Press, Cambridge, MA, 1997.[2] R. Exel & J. Renault, Semigroups of local homeomorphisms and interactiongroups , Ergodic Theory Dynam. Systems (2007), no. 6, 1737–1771.[3] G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamicalsystems , Math. Systems Theory (1969), 320–375.[4] J. L. Kelley, General topology , Reprint of the 1955 edition [Van Nostrand,Toronto, Ont.]. Graduate Texts in Mathematics, No. 27. Springer-Verlag, NewYork-Berlin, 1975.[5] B. Kitchens,
Symbolic dynamics. One-sided, two-sided, and countable stateMarkov shifts , Springer-Verlag, Berlin, 1998.[6] D. Lind & B. Marcus,
An introduction to symbolic dynamics and coding , Cam-bridge University Press, Cambridge, 1995.[7] J. R. Munkres,
Topology: a first course , Prentice Hall, Inc., Englewood Cliffs,N.J., 1975.
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