Structure of transition classes for factor codes on shifts of finite type
aa r X i v : . [ m a t h . D S ] A p r STRUCTURE OF TRANSITION CLASSES FOR FACTOR CODESON SHIFTS OF FINITE TYPE
MAHSA ALLAHBAKHSHI, SOONJO HONG, AND UIJIN JUNG
Abstract.
Given a factor code π from a shift of finite type X onto a sofic shift Y ,the class degree of π is defined to be the minimal number of transition classes overthe points of Y . In this paper we investigate the structure of transition classes andpresent several dynamical properties analogous to the properties of fibers of finite-to-one factor codes. As a corollary, we show that for an irreducible factor triple therecannot be a transition between two distinct transition classes over a right transitivepoint, answering a question raised by Quas. Introduction
Given a finite-to-one factor code π from a shift of finite type X onto an irreduciblesofic shift Y , the degree of π is defined to be the minimal number of preimages of thepoints in Y . The notion of degree was first introduced in [12] for endomorphisms offull shifts, and then was extended to those of irreducible shifts of finite type and soficshifts [10]. It is widely studied and is useful in the study of finite-to-one factor codes[2, 5, 14, 19]. If d is the degree of a one-block factor code π , there are well-knownfundamental properties of fibers of points in Y as follows:(1) Every doubly transitive point in Y has exactly d preimages.(2) π ( x ) is doubly transitive if and only if so is x .(3) Any two distinct preimages of a doubly transitive point in Y are mutually sep-arated ; i.e., they do not share a common symbol at the same time.In this work we show that a natural generalization of the degree for general factorcodes is the class degree introduced in [1]. The principal motivation of defining theclass degree was finding a conjugacy-invariant upper bound on the number of ergodicmeasures of relative maximal entropy. Measures of relative maximal entropy appearedin many different topics in symbolic dynamics due to their connections with, for exam-ple, functions of Markov chains [3, 8] or their application in computing the Hausdorffdimension of certain sets [11]. The class degree is in fact a conjugacy-invariant upperbound on the number of such measures over a fully supported ergodic measure [1, 16].The class degree is defined by using a certain equivalence relation on the fiber of eachpoint in Y . Roughly speaking, two preimages x and ¯ x of a point y in Y are equivalent Mathematics Subject Classification.
Primary 37B10; Secondary 37B40.
Key words and phrases. class degree, transition class, shift of finite type, factor code. if we can find a preimage z of y which is equal to x up to an arbitrarily large givenpositive coordinate and right asymptotic to ¯ x and vice versa (see Definition 2.3). The class degree is defined to be the minimal number of equivalence classes (called transitionclasses ) over the points in Y . It is shown in [1] that the class degree is equal to thedegree when π is finite-to-one and moreover, if d is the class degree of π then everyright transitive point in Y has exactly d transition classes (analogous to (1) above).This suggests that the class degree is a candidate of a generalization of the degree.The idea of considering transitions between preimages of a point also appeared inde-pendently in [21]. To find a condition which is invariant under conjugacy and weakerthan the condition which appeared in [9], Yoo defined the notion of fiber-mixing andshowed that a fiber-mixing code between two mixing shifts of finite type sends everyfully supported Markov measure on X to a Gibbs measure on Y . Fiber-mixing codeswere investigated in further studies, e.g., [13, 17]. In terms of our terminology, a fiber-mixing code from a shift of finite type X onto a sofic Y is just a code in which everypoint in Y has only one transition class, thus it is natural to ask what kind of proper-ties a code can have when the code is not fiber-mixing, for example, when the code hasclass degree one but there exist some points with more than one transition class. More-over, since the definition of transition classes is motivated by communicating classes inMarkov chains, Quas asked whether there could be a transition between two distincttransition classes over a right transitive point.To answer such questions one need to have a structural theory on transition classes.In fact unlike the finite-to-one case where the fibers have been well studied, previousresearch on infinite-to-one codes usually had concentrated on their thermodynamicformalism [6, 20], or on the construction of codes with nice properties [4, 7, 18]. In thispaper, we investigate the fibers and transition classes of such factor codes and provideseveral structural results. By these results it is natural to consider the class degree asa natural generalization of the degree.In particular, we provide dynamical properties analogous to (2) and (3) above (seeTheorem 3.4, Corollary 3.6, and Theorem 4.4):(2 ′ ) y is right (resp. doubly) transitive if and only if each transition class over y contains a right (resp. doubly) transitive point.(3 ′ ) Given two points from two distinct transition classes over a right transitivepoint, they are mutually separated.This analogy shows us that as for a finite-to-one code, fibers over almost all imagesfor infinite-to-one factors are well-behaved, in the sense that among the fiber over atypical point of Y a typical point of X always exists, and that any points chosen fromdistinct classes over a typical point of Y have orbits which neither meet nor approachasymptotically to each other. As a corollary, we also show that there cannot be anytransition among distinct classes (see Corollary 4.7). Property (3 ′ ) above is one typeof separation property between distinct transition classes. In Section 5, we presentanother type of separation property which is stronger than the former one; briefly, we RANSITION CLASSES 3 make a partition on the set of preimages of a magic block of π so that one can determinewhether two preimages of a doubly transitive point are in the same transition class ornot only by reading the symbols occurring in their coordinates over the magic block (seeTheorem 5.4). Other structural properties on transition classes will be also providedwhich would hopefully open new doors on further investigation of infinite-to-one factorcodes. 2. Background
In this section we introduce some terminology and basic results on symbolic dynamics.If X is a subshift (or shift space) with the shift map σ , then denote by B n ( X ) the setof all n -blocks occurring in the points of X and let B ( X ) = S ∞ n =0 B n ( X ). The alphabetof a shift space X is denoted by A ( X ) = B ( X ).A code π : X → Y is a continuous σ -commuting map between shift spaces. It iscalled a factor code if it is surjective. Every code can be recoded to be a one-blockcode; i.e., a code for which x determines π ( x ) . Given a one-block code π : X → Y ,it naturally induces a map on B ( X ), which we also denote by π for brevity. We call π finite-to-one if π − ( y ) is a finite set for all y ∈ Y .A triple ( X, Y, π ) is called a factor triple if π : X → Y is a factor code from a shiftof finite type X onto a (sofic) subshift Y . A factor triple is called irreducible when X is irreducible. It is called finite-to-one if π is finite-to-one.A point x in a shift space X is called right transitive if every block in X occursinfinitely many times in x [0 , ∞ ) , or equivalently, if the forward orbit of x is dense. Twopoints x, ¯ x in X are called right asymptotic if x [ N, ∞ ) = ¯ x [ N, ∞ ) for some N ∈ Z . Leftasymptotic points and left transitivity are defined similarly. A point is doubly transitive if it is both left and right transitive.If π is a finite-to-one factor code from a shift of finite type X onto a sofic shift Y ,there is a uniform upper bound on the number of preimages of points in Y [15]. Theminimal number of π -preimages of the points in Y is called the degree of the factorcode π and is denoted by d π . Theorem 2.1. [15, § Let ( X, Y, π ) be a finite-to-one factor triple with Y irreducible.Then every doubly transitive point of Y has exactly d π preimages. Two points x and ¯ x in a shift space are mutually separated if x i and ¯ x i are differentfor each integer i . It is well known that if ( X, Y, π ) is a finite-to-one factor triple with X one-step, π one-block and Y irreducible, then each y ∈ Y has d π mutually separatedpreimages. In particular, if y is doubly transitive, then any two distinct preimages of y are mutually separated.We say two factor triples ( X, Y, π ) and ( ˜ X, ˜ Y , ˜ π ) are conjugate if X is conjugate to˜ X under a conjugacy φ , Y is conjugate to ˜ Y under a conjugacy ψ , and ˜ π ◦ φ = ψ ◦ π .An immediate corollary of Theorem 2.1 is that d π is invariant under conjugacy. M. ALLAHBAKHSHI, S. HONG, AND U. JUNG
Definition 2.2. [15] Let (
X, Y, π ) be a factor triple with X one-step and π one-block.Given a block w ∈ B ( Y ), define d ( w ) = min ≤ k ≤| w | (cid:12)(cid:12) { a ∈ A ( X ) : ∃ u ∈ π − ( w ) with u k = a } (cid:12)(cid:12) If a block w satisfies d ( w ) = min v ∈B ( Y ) d ( v ), then it is called a magic block . In thiscase, a coordinate k where the minimum occurs is called a magic coordinate . If | w | = 1then w is called a magic symbol .Let w be a magic block with a magic coordinate k . Given a point y in Y and i ∈ Z with y [ i,i + | w | ) = w , some block in π − ( w ) may not be extendable to a point in π − ( y ). However, due to the minimality of d ( w ), the two sets { x i + k | x ∈ π − ( y ) } and { u k | u ∈ π − ( w ) } are the same. It is well known that for a one-block finite-to-onefactor code π from a one-step shift of finite type X onto an irreducible sofic shift Y , wehave d π = d ( w ) for any magic block w of π .The class degree defined below is a quantity analogous to the degree when the factorcode π is not only limited to be finite-to-one. Definition 2.3.
Let (
X, Y, π ) be a factor triple and x, ¯ x ∈ X . We say there is a transition from x to ¯ x and denote it by x → ¯ x if for each integer n , there exists a point z in X so that(1) π ( z ) = π ( x ) = π (¯ x ), and(2) z ( −∞ ,n ] = x ( −∞ ,n ] , z [ i, ∞ ) = ¯ x [ i, ∞ ) for some i ≥ n .We write x ∼ ¯ x and say x and ¯ x are in the same transition class if x → ¯ x and ¯ x → x .Then the relation ∼ is an equivalence relation. Denote the set of transition classes in X over y ∈ Y by C ( y ). We say there is a transition from a class [ x ] to another class[¯ x ] and denote it by [ x ] → [¯ x ] if x → ¯ x . Note that if [ x ] → [¯ x ] then for each z ∼ x and¯ z ∼ ¯ x we have z → ¯ z . Fact 2.4. [1] Let π : X → Y be a one-block factor code from a one-step shift of finitetype X onto an irreducible sofic shift Y . Then(1) | C ( y ) | < ∞ for each y in Y .(2) Let x, x ′ ∈ π − ( y ) for some y ∈ Y . Given x a i = x ′ a i where ( a i ) i ∈ N is a strictlyincreasing sequence in Z , we have x ∼ x ′ . Definition 2.5.
Let (
X, Y, π ) be a factor triple. The minimal number of transitionclasses over points of Y is called the class degree of π and is denoted by c π .It is clear that c π is invariant under conjugacy. It was shown in [1] that for a finite-to-one factor triple ( X, Y, π ) with Y irreducible, we have c π = d π . Theorem 2.6. [1]
Let ( X, Y, π ) be a factor triple with Y irreducible. Then every righttransitive point of Y has exactly c π transition classes. Theorem 2.9 states the class degree of a factor code in terms of another quantitywhich is defined concretely in terms of blocks.
RANSITION CLASSES 5
Definition 2.7.
Let (
X, Y, π ) be a factor triple with X one-step and π one-block. Let w = w [0 ,p ] ∈ B p +1 ( Y ). Also let n be an integer in (0 , p ) and M be a subset of π − ( w n ).We say a block u ∈ π − ( w ) is routable through a ∈ M at time n if there is a block¯ u ∈ π − ( w ) with ¯ u = u , ¯ u p = u p and ¯ u n = a . A triple ( w, n, M ) is called a transitionblock of π if every block in π − ( w ) is routable through a symbol of M at time n . Thecardinality of the set M is called the depth of the transition block ( w, n, M ). Whenthere is no confusion, for example when y ∈ Y and w = y [ i,i + p ] are fixed, we say thepoints x, ¯ x ∈ π − ( y ) are routable through a ∈ M at time i + n if x [ i,i + p ] and ¯ x [ i,i + p ] areroutable through a at time i + n . Definition 2.8.
Let c ∗ π = min {| M | : ( w, n, M ) is a transition block of π } . A minimal transition block of π is a transition block of depth c ∗ π . Theorem 2.9. [1]
Let ( X, Y, π ) be a factor triple with X one-step, π one-block and Y irreducible. Then c π = c ∗ π . For more details on symbolic dynamics, see [15]. For a perspective on the class degreeand its relation to the degree, see [1].3.
Each transition class over a right transitive pointcontains a right transitive point
In this section, we prove that given an irreducible factor triple (
X, Y, π ), each tran-sition class over a right transitive point contains a right transitive point. This resultcan be seen as an analogue of the well-known fact that for a finite-to-one irreduciblefactor triple (
X, Y, π ), every preimage of a right (resp. doubly) transitive point is aright (resp. doubly) transitive point. We begin with the following definition.
Definition 3.1.
Let (
X, Y, π ) be a factor triple and let ¯ X be a proper subshift of X with π ( ¯ X ) = Y . Let ¯ v be in B ( X ) \ B ( ¯ X ). We say two blocks u and v in B ( X ) forman ( ¯ X, ¯ v ) -diamond if the following hold:(1) π ( u ) = π ( v ),(2) ¯ v is a subblock of v ,(3) u occurs in ¯ X , and(4) u and v share the same initial symbol and the same terminal symbol.The following lemma is a slightly stronger version of [22, Proposition 3.1]. We includea different proof here, which is also more direct than the original one. Later in § Lemma 3.2.
Let ( X, Y, π ) be an irreducible factor triple with X one-step and π one-block. Let ¯ X be a proper subshift of X with π ( ¯ X ) = Y . Then for each block ¯ v in B ( X ) \ B ( ¯ X ) there is an ( ¯ X, ¯ v ) -diamond. M. ALLAHBAKHSHI, S. HONG, AND U. JUNG
Proof.
For w in B ( Y ) define n ( w ) to be the maximal number of mutually separatedpreimages of w in B ( ¯ X ) and let n be the infimum of n ( w ) where w runs over all thenonempty words in B ( Y ). Clearly n is a positive integer. Let w be a block in B ( Y )with n ( w ) = n and let u be a preimage of w in B ( ¯ X ).Let ¯ v be in B ( X ) \ B ( ¯ X ). Since X is irreducible, there is a cycle α in B ( X ) such that α = uγ ¯ vη , for some blocks γ, η . Denote the length of α by l , and consider α n +1 . Thereare at least n mutually separated blocks β (1) , · · · , β ( n ) in B ( ¯ X ) all projecting to π ( α n +1 ).Note that for all 0 ≤ j < n +1 and 1 ≤ m ≤ n we have π ( α n +1[ jl,jl + | w | ) ) = π ( β ( m )[ jl,jl + | w | ) ) = w .Moreover, for all 1 ≤ m, m ′ ≤ n where m = m ′ , the blocks β ( m )[ jl,jl + | w | ) and β ( m ′ )[ jl,jl + | w | ) aremutually separated.Since n ( w ) = n , for each 0 ≤ j < n + 1 there is 1 ≤ m j ≤ n such that β ( m j )[ jl,jl + | w | ) meets u ; i.e, there is 0 < i < | w | such that β ( m j ) jl + i = u i . Thus by Pigeonhole principlethere is 1 ≤ m ≤ n such that β ( m ) meets u twice; say at positions jl + i and j ′ l + i ′ forsome 0 ≤ j < j ′ < n + 1 and 0 ≤ i, i ′ < | w | . It is clear that the blocks β ( m )[ jl + i,j ′ l + i ′ ] and α n +1[ jl + i,j ′ l + i ′ ] form an ( ¯ X, ¯ v )-diamond. (cid:3) Remark . Note that Lemma 3.2 is not necessarily true when X is reducible. Forexample let X be the orbit closure of the point a ∞ .b ∞ and Y = { ∞ } and consider thetrivial map π : X → Y . Let ¯ X = { a ∞ } and ¯ v = b . Theorem 3.4.
Let ( X, Y, π ) be an irreducible factor triple and let y in Y be righttransitive. Then each transition class over y contains a right transitive point.Proof. We may assume that X is one-step and π is one-block. Let C be a transitionclass over y and let x be in C . If x is right transitive, we are done. So suppose that x is not right transitive. Let ¯ X be the ω -limit set of x ; i.e.,¯ X = ω ( x ) = { z ∈ X : ∃ n i ր ∞ with σ n i ( x ) → z } . Then we have ¯ X ( X . Since π ( x ) = y and y is right transitive, it follows that π ( ¯ X ) = Y . Now consider an enumeration ¯ v , ¯ v , · · · of B ( X ). For each i ∈ N with¯ v i ∈ B ( X ) \ B ( ¯ X ), by Lemma 3.2 there is an ( ¯ X, ¯ v i )-diamond ( u i , v i ). Note that forthis i , ¯ v i is a subblock of v i , v i ∈ B ( X ) \ B ( ¯ X ) and u i ∈ B ( ¯ X ).For each i ∈ N , define a block w i ∈ B ( ¯ X ) by w i = ( ¯ v i if ¯ v i ∈ B ( ¯ X ) u i otherwise.Then since each w i is in B ( ¯ X ), we can find an increasing sequence { n i } ∞ i =1 such that x [ n i ,n i + | w i | ) = w i and n i +1 > n i + | w i | for all i ∈ N . Finally, define a new point z ∈ X obtained from x by substituting each occurrence of u i at the coordinates x [ n i ,n i + | w i | ) forall i ∈ N with v i ∈ B ( X ) \ B ( ¯ X ). Since each ( u i , v i ) forms a diamond, z is indeed apoint in X and we have π ( z ) = y . RANSITION CLASSES 7
Since there are infinitely many positive coordinates j for which x j = z j , we have z ∼ x and therefore z ∈ C . Also, since each block in X occurs infinitely many timesas a subblock in the enumeration ¯ v , ¯ v , · · · , it follows that z contains all the ¯ v i ’s andtherefore is a right transitive point, as desired. (cid:3) Remark . Let (
X, Y, π ) be an irreducible factor triple and let y in Y be left transitive.Then each transition class over y contains a left transitive point.The proof of this remark is similar to Theorem 3.4, but much simpler. Let ¯ X bethe α -limit set of x ; i.e., ¯ X = { z ∈ X : ∃ n i ց −∞ with σ n i ( x ) → z } . If x is not lefttransitive, then we may construct a point z ∈ X similarly as in the proof of Theorem 3.4:First find a decreasing subsequence { n i } such that x [ n i ,n i + | w i | ) = w i (same w i defined inTheorem 3.4) and n i +1 < n i − | w i +1 | for all i ∈ N , and then define z by substitutingeach occurrence of u i at the coordinates x [ n i ,n i + | w i | ) with v i for all i ∈ Z . Then z is lefttransitive. Since z and x are right asymptotic, we have z ∼ x .The following corollary is an immediate result of Theorem 3.4 and Remark 3.5. Corollary 3.6.
Let ( X, Y, π ) be an irreducible factor triple and let y in Y be doublytransitive. Then each transition class over y contains a doubly transitive point. With the following corollary, we see that for an irreducible factor triple, the cardinal-ities of the transition classes over right transitive points fall into two categories: Theyare all finite (if a factor code is finite-to-one) or all uncountable (if it is infinite-to-one).
Corollary 3.7.
Let ( X, Y, π ) be an irreducible factor triple and let y in Y be righttransitive. If π is infinite-to-one, then the cardinality of each transition class over y isuncountable.Proof. We may assume that X is one-step and π is one-block. Recall that π is infinite-to-one if and only if it has a diamond, say, ( u, v ) [15, Theorem 8.1.16]. If C is atransition class over y , there is a right transitive point x in C by Theorem 3.4. Then u occurs infinitely many times to the right in x . Any point made by replacing someoccurrences of u with v is equivalent to x , which implies that C is uncountable. (cid:3) Mutual separatedness for transition classes
If (
X, Y, π ) is an irreducible finite-to-one factor triple of degree d , X is one-step and π is one-block, then for each doubly transitive point y ∈ Y the set of the preimagesof y consists of d mutually separated points in X . This result is one of the importantproperties of fibers of finite-to-one factor codes, since it is used to prove that the degreeindeed equals the number combinatorially defined using a magic block [10, 12, 15, § M. ALLAHBAKHSHI, S. HONG, AND U. JUNG there is no transition between distinct transition classes over a right transitive point,answering a question raised by Quas.
Lemma 4.1.
Let ( X, Y, π ) be an irreducible factor triple with X one-step and π one-block. Given a minimal transition block ( w, n, M ) , any preimage of w is routable througha unique symbol of M .Proof. Let u be in π − ( w ) and let d = c π be the class degree of π . If d = 1 then theresult is trivial, so suppose d ≥
2. Assume that u is routable through two differentmembers a (1) and a (2) of M = { a (1) , a (2) , · · · , a ( d ) } . Let x be a point of X such that u occurs infinitely many times to the right, say at positions { [ i j , i j + | w | ) } j ∈ N where i j +1 > i j + | w | .From each transition class in C ( π ( x )) \ { [ x ] } choose one point and denote them by x (1) , · · · , x ( d − . Each of these points is routable through at least one member of M attime i j + n for each j ∈ N . If there is a point in { x (1) , · · · , x ( d − } which is routablethrough a (1) or through a (2) at i j + n for infinitely many j ’s, then such a point isequivalent to x which gives a contradiction and we are done.Suppose there is no such points; i.e., each of the points x (1) , · · · , x ( d − is routablethrough a symbol in { a (3) , · · · , a ( d ) } at i j + n for all but finitely many j ’s. It follows,by Pigeonhole principle, that there are at least two points in { x (1) , · · · , x ( d − } whichare routable through the same symbol in { a (3) , · · · , a ( d ) } at i j + n for infinitely many j .This forces such two points to be equivalent, which is again a contradiction. (cid:3) Remark . Note that Lemma 4.1 is not necessarily true when X is reducible. Forexample let X be the orbit closure of the point a ∞ .b ∞ and Y = { ∞ } and consider thetrivial map π : X → Y . The triple (000 , , { a, b } ) is a minimal transition block with apreimage abb which is routable through both a and b . Lemma 4.3.
Let ( X, Y, π ) be an irreducible factor triple with X one-step and π one-block. Suppose there are two points x and ¯ x such that(1) x and ¯ x are in two distinct transition classes over a right transitive point y ∈ Y ,and(2) x and ¯ x are not mutually separated.Then given a right transitive point z in X , there is a transition from [ z ] to a transitionclass over π ( z ) other than [ z ] .Proof. Let ( w, n, M ) be a minimal transition block. Since y is right transitive, there isa sequence { i j } j ∈ N such that y [ i j ,i j + | w | ) = w and i j +1 > i j + | w | . By assumption (2) forsome integer i we have x i = ¯ x i . Since x and ¯ x are in distinct transition classes, there is i j > i and two distinct symbols a, b ∈ M such that x and ¯ x are routable through a and b , respectively, at time i j + n . By taking equivalent points in the transition classes of x and ¯ x , we may assume x i j + n = a and ¯ x i j + n = b .Consider the block y [ i,i j + | w | ) which is an extension of w . Denote this extension by ¯ w ,and the coordinate i j − i + n by ¯ n . Then ( ¯ w, ¯ n, M ) is also a minimal transition block. RANSITION CLASSES 9
Note that by above, block ¯ w has two preimages x [ i,i + | ¯ w | ) and ¯ x [ i,i + | ¯ w | ) which share thesame initial symbol; moreover, x i +¯ n = a and ¯ x i +¯ n = b .Now let z be a right transitive point in X . Note that for infinitely many k ∈ Z wehave z [ k,k + | ¯ w | ) = x [ i,i + | ¯ w | ) . Denote the class degree by d and let z (1) = z . From eachtransition class of C ( π ( z )) \ { [ z ] } choose a point and denote them by z (2) , z (3) , · · · , z ( d ) .Since ( ¯ w, ¯ n, M ) is a minimal transition block, by Lemma 4.1 we may assume that z ( j ) k +¯ n ∈ M for each 1 < j ≤ d . Since z ( j ) ’s are from distinct transition classes, the set { z ( j ) k +¯ n : 1 ≤ j ≤ d } consists of d symbols for all large such k . Thus there is a point ¯ z among z (2) , z (3) , · · · , z ( d ) such that among those k at infinitely many l we have ¯ z l +¯ n = b .Then for any such l the points u ( l ) defined by u ( l )( −∞ ,l ) = z [ −∞ ,l ) , u ( l )[ l,l +¯ n ) = ¯ x [ i,i +¯ n ) and u ( l )[ l +¯ n, ∞ ) = ¯ z [ l +¯ n, ∞ ) give a transition [ z ] → [¯ z ] over the right transitive point π ( z ). (cid:3) Theorem 4.4.
Let ( X, Y, π ) be an irreducible factor triple with X one-step and π one-block. Let y ∈ Y be right transitive. Then any two points from two distinct transitionclasses over y are mutually separated.Proof. Suppose not; i.e., there are two points x and ¯ x in distinct transition classes over y such that x and ¯ x are not mutually separated. Recall that any transition class over y contains a right transitive point. Then by Lemma 4.3, given any class C ∈ C ( y )there is a transition from C to some other transition class over y . However, since thereare only finitely many classes in C ( y ), there must be a transition class over y with notransition to any other class which is a contradiction. (cid:3) One can easily check the following corollary, which is a conjugacy-invariant versionof Theorem 4.4.
Corollary 4.5.
Let ( X, Y, π ) be an irreducible factor triple. Then there is c > suchthat, whenever y ∈ Y is right transitive and x, ¯ x are points from two distinct transitionclasses over y , we have d ( x, ¯ x ) > c . Corollary 4.6.
Let ( X, Y, π ) be an irreducible factor triple. Then each transition classover a right transitive point is a closed set (with respect to the usual topology on X ).Proof. We may assume that X is one-step and π is one-block. Let { x ( i ) } i ∈ N be aconvergent sequence in a transition class C over a right transitive point y . Denote thelimit of this sequence by x . Then π ( x ) = y . Since x ( i ) → x , for large i we have x ( i )0 = x .Then x is not mutually separated from this x ( i ) , so by Theorem 4.4 that x must belongto C . (cid:3) Corollary 4.7.
Let ( X, Y, π ) be an irreducible factor triple and let y be a right transitivepoint in Y . There is no transition between any two distinct transition classes over y . Proof.
We may assume that X is one-step and π is one-block. Suppose, on the contrary,that there is a transition [ x ] → [¯ x ] between two distinct transition classes [ x ] and [¯ x ]over y . Then there is a point z ∈ π − ( y ) such that z ( ∞ , = x ( ∞ , and z [ i, ∞ ) = ¯ x [ i, ∞ ) forsome i >
0. Since z ∈ [¯ x ] and z and x are not mutually separated, by Theorem 4.4 wehave a contradiction. (cid:3) The following examples show that there may be a transition if the domain is notirreducible, or the point in Y is not right transitive. Example 4.8. (1) Let X be the orbit closure of the point a ∞ .b ∞ and Y = { ∞ } andconsider the trivial map π : X → Y . Above the point 0 ∞ there are two transitionclasses: one class, say C , consists of only one point a ∞ and the other class C consistsof all the points of the form a ∞ b ∞ and b ∞ . Even though 0 ∞ is a (right) transitive point,there is a transition from C to C . Note that X is not irreducible.(2) Let X be an irreducible edge shift given by the diagram and Y = { , } Z . Let π : X → Y be a one-block factor code given by π ( a ) = π ( b ) = π ( d ) = 0 and π ( c ) = π ( e ) = 1. Note that c π = 1. I Ja bc de
Let y in Y be any point with y [0 , ∞ ) = 0 ∞ . Since each vertex has incoming edgeslabeled by 0 and by 1 respectively, there are two left infinite paths, say α and β ,mapping to y ( −∞ , and terminating at I and J , respectively. Hence as in the previousexample, y has two transition classes: one containing α.a ∞ and the other containing α.a n bd ∞ for all n ∈ N and β.d ∞ . It follows that there is a transition from one transitionclass over y to the other. Note that one can even take y to be left transitive, so even theleft transitivity of y does not guarantee no transition between transition classes over y .In the definition of a transition x → ¯ x between two points x and ¯ x in X , we requirean infinite number of points which are left asymptotic to x and right asymptotic to ¯ x ,since for each n ∈ N we need a point which equals x until time n . However, if the point y = π ( x ) = π (¯ x ) is right transitive, to satisfy this definition, a single asymptotic pointto x and ¯ x suffices, as condition (3) in the following corollary shows. Corollary 4.9.
Let ( X, Y, π ) be an irreducible factor triple and y ∈ Y be right transitive.Then for x, ¯ x ∈ π − ( y ) the following are equivalent:(1) x ∼ ¯ x (2) x → ¯ x (3) There is a point in π − ( y ) which is left asymptotic to x and right asymptotic to ¯ x . RANSITION CLASSES 11
Proof.
Since all the conditions (1), (2) and (3) are conjugacy-invariant, we may assumethat X is one-step and π is one-block. The equivalence of (1) and (2) follows fromCorollary 4.7. (2) clearly implies (3). Suppose that x and ¯ x are not equivalent. If z is a point satisfying condition (3), we have z ∈ [¯ x ]. Since z and x are left asymptotic,they are not mutually separated, which contradicts Theorem 4.4. (cid:3) The following corollary is immediate.
Corollary 4.10.
Let ( X, Y, π ) be an irreducible finite-to-one factor triple. Let y in Y be right transitive. Then for each x and ¯ x in π − ( y ) , they are either mutually separatedor right asymptotic.Proof. First note that since π is finite-to-one, x and ¯ x must be right transitive. Suppose x and ¯ x are neither mutually separated nor right asymptotic. Then by Corollary 4.9they are equivalent and therefore produce a diamond which implies that π is not finite-to-one (see [15, § (cid:3) Symbol partition properties for a factor triple
In this section, we provide further separation properties for transition classes over aright transitive point. Throughout this section, we assume that X is one-step and π isone-block for a factor triple ( X, Y, π ). Also in what follows, for a transition class C and A ⊂ Z , denote by C | A the set { x A : x ∈ C } . If A = { i } for some integer i then C | A isdenoted simply by C | i for the convenience. With this notation, Theorem 4.4 says thatif y ∈ Y is right transitive, then C | i ∩ ¯ C | i = ∅ for C = ¯ C ∈ C ( y ) and all i ∈ Z .We will first see that if ( w, n, M ) is a minimal transition block and y is right transitive,then | C | i + n ∩ M | = 1 for any C in C ( y ) and i ∈ Z with y [ i,i + | w | ) = w . Lemma 5.1.
Let ( X, Y, π ) be an irreducible factor triple. Let y ∈ Y be right transitivewith y [ i,i + | w | ) = w for some minimal transition block ( w, n, M ) . Given C ∈ C ( y ) , thereis a unique symbol b in M such that every point in C is routable through b at time i + n .Proof. Let M C = M ∩ C | i + n for any C in C ( y ). Then S C ∈ C ( y ) M C = M . As each M C is contained in C | i + n , by Theorem 4.4 the sets M C ’s are mutually disjoint. So P C ∈ C ( y ) | M C | = | M | = c π . Since | C ( y ) | = c π and each M C is nonempty, we have that | M C | = 1 for each C ∈ C ( y ). (cid:3) The following proposition intuitively states that given a right transitive point y in Y , if two preimages of the symbol y appear in two distinct transition classes over y attime 0, then these two symbols cannot ever appear at the same time in a single classover any right transitive point. Proposition 5.2.
Let ( X, Y, π ) be an irreducible factor triple and let y ∈ Y be a righttransitive point with a minimal transition block in y ( −∞ , . Then for any right transitivepoint z ∈ Y with z i = y , i ∈ Z , and any given D ∈ C ( z ) , we have D | i ∩ C | = ∅ for atmost one C ∈ C ( y ) . Proof.
Let y = a . Suppose, on the contrary, that for some right transitive point z ∈ Y with z i = a , i ∈ Z , and D ∈ C ( z ), there are two symbols b ∈ C | and c ∈ ¯ C | , where C = ¯ C ∈ C ( y ), occurring at the time i in D , that is, b, c ∈ D | i . Then there are twopoints x (1) and x (2) in D with x (1) i = b and x (2) i = c . As b ∈ C | and c ∈ ¯ C | , we havetwo points ¯ x (1) ∈ C and ¯ x (2) ∈ ¯ C such that ¯ x (1)0 = b and ¯ x (2)0 = c .Define new points ˆ x ( l ) for l = 1 , x ( l ) = ¯ x ( l )( −∞ , .x ( l )[ i, ∞ ) and consider ˆ z = π (ˆ x (1) ) = π (ˆ x (2) ). Then ˆ z is right transitive since it is right asymptoticto σ i ( z ), and the points ˆ x ( l ) ’s are equivalent since x ( l ) ’s are equivalent.Recall that by assumption, a minimal transition block ( w, n, M ) occurs in y [ j,j + | w | ) =ˆ z [ j,j + | w | ) for some j ≤ −| w | . Since ¯ x (1) and ¯ x (2) are not equivalent then by Theorem 4.4they are routable through different members of M at time j + n . It follows that ˆ x (1) and ˆ x (2) are routable through different members of M at time j + n which contradictsLemma 5.1 since ˆ x (1) and ˆ x (2) are equivalent. (cid:3) A finite-to-one factor code possesses a kind of permutation property, as discussed in[15, § Proposition 5.3.
Let ( X, Y, π ) be an irreducible factor triple of class degree d and let ( w, n, M ) be a minimal transition block. For each block u = wvw ∈ B ( Y ) for some v ,there is a permutation τ u : M → M such that given any right transitive point y in Y with y [0 , | u | ) = u and C ∈ C ( y ) , we have τ u ( C | n ∩ M ) = C | | wv | + n ∩ M .Proof. Let y be a right transitive point in Y with y [0 , | u | ) = u . Given C in C ( y ), byLemma 5.1, we have C | n ∩ M = { a } and C | | wv | + n ∩ M = { b } for some symbols a and b . Define a permutation τ u,y of M by τ u,y ( a ) = b , and let x ∈ C be a point with x n = a and x | wv | + n = b .We show that τ u,y does not depend on y . Let ¯ y be a right transitive point in Y with¯ y [0 , | u | ) = u , and let ¯ x be a point in π − (¯ y ) with ¯ x n = a . Consider a point x ′ ∈ π − (¯ y )with x ′| wv | + n = b . Then the point ¯ x ( −∞ ,n ] x ( n, | wv | + n ) x ′ [ | wv | + n, ∞ ) ∈ π − (¯ y ) is left asymptoticto ¯ x and right asymptotic to x ′ , which implies that ¯ x and x ′ are equivalent. It followsthat τ u, ¯ y ( a ) = b , and thus τ u,y = τ u, ¯ y as C was chosen arbitrarily. (cid:3) Theorem 5.4 informally states that given any irreducible factor triple (
X, Y, π ), inorder to determine whether two points with the same image are equivalent or not, oneonly needs to locally compare the preimages of a magic block which occur within thesepoints. More precisely, there is a partition on the set of preimages of a magic block w of π such that given all right transitive points y in Y and any i ∈ Z with y [ i,i + | w | ) = w ,two preimages x, x ′ of y are equivalent if and only if x [ i,i + | w | ) and x ′ [ i,i + | w | ) belong to thesame class of the partition on π − ( w ). RANSITION CLASSES 13
Theorem 5.4.
Let ( X, Y, π ) be an irreducible factor triple of class degree d . Let w bea magic block of π . There is a partition of π − ( w ) into d subsets B , · · · , B d such thatfor any doubly transitive z in Y and any i ∈ Z with z [ i,i + | w | ) = w , we have a bijection ρ z,i : C ( z ) → { , . . . , d } with D | [ i,i + | w | ) ⊆ B ρ z,i ( D ) for each D ∈ C ( z ) .Proof. Let k be a magic coordinate of w and A w,k = { u k | u ∈ π − ( w ) } . Let y be a doubly transitive point in Y with y [0 , | w | ) = w . For C ∈ C ( y ) let B C = { u ∈ π − ( w ) | u k ∈ C | k } . List C ( y ) = { C (1) , · · · , C ( d ) } and denote B j = B C ( j ) . We claimthat {B j : 1 ≤ j ≤ d } is a desired partition.Note that since k is a magic coordinate of w , we have S C ∈ C ( y ) C | k = A w,k and hence S C ∈ C ( y ) B C = π − ( w ). Moreover, B C ’s with C ∈ C ( y ) are mutually disjoint, since byTheorem 4.4 the sets C | k ’s with C ∈ C ( y ) , are mutually disjoint.Now we show that such a partition does not depend on y . Let z be a doubly transitivepoint in Y with z [ i,i + | w | ) = w for some i ∈ Z . By considering a point σ i ( z ) we may assumethat i = 0. For each D ∈ C ( z ), let B ( z ) D = { u ∈ π − ( w ) | u k ∈ D | k } . Again since k is amagic coordinate of w , we have S D ∈ C ( z ) D | k = A w,k and thus S D ∈ C ( z ) B ( z ) D = π − ( w ).It follows that for each D ∈ C ( z ) there is a transition class C ∈ C ( y ) such that C | k ∩ D | k = ∅ , which implies that B C ∩ B ( z ) D = ∅ . We show that B C = B ( z ) D . Supposenot. First assume that there is a block u in B C \ B ( z ) D . There must be another class D ′ ∈ C ( z ) such that u k ∈ D ′ | k . This means C | k intersects both D | k and D ′ | k whichcontradicts Proposition 5.2. So B C ⊆ B ( z ) D . The case u ∈ B ( z ) D \ B C is also impossible bythe symmetric argument, and hence we have B C = B ( z ) D .Define ρ z,i : C ( z ) → { , · · · , d } to send each D ∈ C ( z ) to the unique 1 ≤ ρ z,i ( D ) ≤ d with B ( σ i ( z )) D = B ρ z,i ( D ) . By definition, D | [ i,i + | w | ) is contained in B ρ z,i ( D ) . (cid:3) Note that in general, the partition in Theorem 5.4 need not be unique. However, insome special cases, for example when π has a magic symbol, we obtain the uniquenessof the partition. The following remark which follows directly from Proposition 5.2explains such cases. Remark . Let (
X, Y, π ) be an irreducible factor triple of class degree d and let a be in A ( Y ). If for some doubly transitive point y in Y we have y = a and π − ( y ) | = π − ( a )then there is a unique partition of π − ( a ) into d subsets B , · · · , B d such that for anyright transitive z ∈ Y with z i = a , we have a bijection ρ z,i : C ( z ) → { , · · · , d } with C | i ⊆ B ρ z,i ( C ) for each class C in C ( z ).The following example shows that the partition property stated in Remark 5.5 neednot hold for every symbol. Example 5.6.
Consider the irreducible vertex shifts X and Y displayed in Figure 1.In the graphs of X and Y , each pair of leftmost and rightmost vertices with the samesymbol ( m , m and m ) is identified. Let π be the map erasing the subscripts, andsending all numbered vertices of X to the symbol a . m α α + β β + γ γ γ ′ α ′ + β ′ β ′ + γ ′ γ ′ m m X : m α α β γ γ α ′ β ′ β ′ + α ′ + γ ′ β ′ + γ ′ α ′ m m Y : m α + β + γ a α ′ + β ′ + γ ′ mπ Figure 1.
Graph of Example 5.6A point y in Y is of the form ( m ( α + β + γ ) a ( α ′ + β ′ + γ ′ )) ∞ where the notation α + β + γ implies that one can choose freely either α or β or γ as a symbol to appear in the givenposition (this is a standard notion in the theory of formal languages). The choices maydiffer up to positions so that a word like mαaβ ′ mβaγ ′ is allowed. If u = mδaδ ′ m with δ ∈ { α, β, γ } and δ ′ ∈ { α ′ , β ′ , γ ′ } , this u defines a unique permutation τ u of { m , m } :If m and u are given, for any preimage v of u with v = m , the last symbol of v isuniquely determined. Similarly for m . Note that this property is inductively true forgeneral blocks of the form u = mδ (1) aδ ′ (1) m · · · mδ ( k ) aδ ′ ( k ) m . This means that for each y in Y and x in π − ( y ) there are only two choices of putting m i ’s in x . Hence, there areexactly two transition classes over y . It follows that every point in Y has 2 transitionclasses and the class degree of π is 2. Note that m is a magic symbol, and also, it is aminimal transition block.Now consider the symbol a ∈ A ( Y ) and consider the sets C | with C ∈ C ( y ), with y right or doubly transitive.(1) If y = · · · mα.aα ′ m · · · , then we have { } and { } as C | ’s in Proposition 5.2.(2) If y = · · · mβ.aβ ′ m · · · , then we have { , } and { } .(3) If y = · · · mγ.aγ ′ m · · · , then we have { , } and { } . RANSITION CLASSES 15 (4) If y = · · · mδ.aδ ′ m · · · , where ( δ, δ ′ ) is not a pair in the above cases, then wehave each C | is a singleton.Symbols 1 , ,
3; however, symbols 1 , π − ( a ) into 2 sets satisfying theproperty stated in Remark 5.5.6. Applications and examples
The definitions of transition, transition classes, and the class degree are asymmetric.So it is natural to consider reversed transition as follows: x → r ¯ x if for each integer n ,there is a point which is left asymptotic to x and equal to ¯ x in [ n, ∞ ). We say that x ∼ r ¯ x if and only if x → r ¯ x and ¯ x → r x . The reversed transition classes of a point in Y are the equivalence classes made by this new relation ∼ r . Denote by [ x ] r the reversedtransition class containing x . Let x T be the point that ( x T ) i = x − i for every integer i and define X T = { x T : x ∈ X } . Note that the reversed transition classes of a point y can be obtained from the transition classes of a point y T under the transposed code π T : X T → Y T . All the results in the previous sections hold for the reversed transitionclasses if we replace left transitive with right transitive.Note that the set of transition classes and that of the reversed transition classes of apoint y ∈ Y need not coincide, nor do they need to have the same cardinality. However,they do coincide for almost all points (Proposition 6.2). Example 6.1.
We show that given any finite-to-one irreducible factor triple (
X, Y, π )which is not bi-closing, there are uncountably many points in Y for which the numberof transition classes differs from the number of reversed transition classes.Suppose, without loss of generality, that π is not left closing ; i.e., there are twodistinct points x = ¯ x ∈ X which are right asymptotic and y = π ( x ) = π (¯ x ). Since thesubshift X is of finite type and x [ i, ∞ ) = ¯ x [ i, ∞ ) for some i , by changing the common righttail we may assume that x and ¯ x are right transitive. Then y is also right transitiveand hence y has exactly c π transition classes. Since π does not have any diamond, x and ¯ x are not equivalent with respect to the reversed transition relation. Moreover,since any given two points z and ¯ z from distinct transition classes over y are mutuallyseparated, they are also not equivalent with respect to the reversed transition relation.It follows that the number of reversed transition classes of y is at least c π + 1. Sincethe right tail of x can be changed in uncountably many ways in order to produce aright transitive point, we have uncountably many points in Y with a different numberof transition classes than the number of its reversed transition classes. Proposition 6.2.
Let ( X, Y, π ) be an irreducible factor triple. If y is doubly transitiveand x ∈ π − ( y ) , then the transition class of x equals the reversed transition class of x .Proof. Suppose, on the contrary that [ x ] = [ x ] r . We may assume that there is a point¯ x ∈ [ x ] r \ [ x ]. Since ¯ x ∼ r x , there is a point z ∈ π − ( y ) which is left asymptotic to ¯ x and right asymptotic to x . Then by Corollary 4.9, [ x ] = [ z ] = [¯ x ] and therefore ¯ x ∈ [ x ],which is a contradiction. (cid:3) Recall Definition 3.1: Let (
X, Y, π ) be a factor triple and ¯ X a proper subshift of X with π ( ¯ X ) = Y . Let ¯ v be in B ( X ) \ B ( ¯ X ). A block u in B ( ¯ X ) and a block v in B ( X ) form an ( ¯ X, ¯ v ) -diamond if π ( u ) = π ( v ), ¯ v is a subblock of v , and u and v sharethe same initial symbol and the same terminal symbol. Lemma 3.2 states that if X isirreducible, one-step and π is one-block then for each block ¯ v in B ( X ) \ B ( ¯ X ) there isan ( ¯ X, ¯ v )-diamond.As mentioned before, we strengthen Lemma 3.2 in the present section. Proposition6.3 gives an upper bound for the length of ( ¯ X, ¯ v )-diamond of a given word ¯ v in B ( X ) \B ( ¯ X ). Note that the proof of Proposition 6.3 employs Theorem 3.4 which was shownusing Lemma 3.2. Proposition 6.3.
Let ( X, Y, π ) be an irreducible factor triple with X one-step and π one-block. Let ¯ X be a proper subshift of X with π ( ¯ X ) = Y . Then there is a positiveinteger N such that for each block ¯ v in B ( X ) \ B ( ¯ X ) we have an ( ¯ X, ¯ v ) -diamond oflength less than | ¯ v | + N .Proof. Let k be a positive integer such that for any blocks u, v in B ( X ) there is a block w in B ( X ) with | w | ≤ k and uwv in B ( X ). Such k exists since X is irreducible andof finite type. Let ( w, n, M ) be a minimal transition block of π with | w | = l and u apreimage of w in B ( ¯ X ).Consider a block ¯ v in B ( X ) \ B ( ¯ X ), and let γ = uα ¯ vβu for some α and β with | α | , | β | ≤ k . Denote | γ | = L . Let y be a right transitive point of Y and ¯ x a preimageof y in ¯ X . By Theorem 3.4 there is a right transitive preimage x of y in X whichis equivalent to ¯ x . Note that x = ¯ x . For convenience, let x [0 ,L ) = γ . Note that x and ¯ x are both routable through the same symbol of M , say a , at time n , and at time L − l + n . Let δ be a block of length l in B ( X ) such that δ = u , δ l − = u l − and δ n = a .Also let ¯ δ and ¯ δ ′ be blocks of length l in B ( X ) such that ¯ δ = ¯ x , ¯ δ l − = ¯ x l − , ¯ δ n = a and ¯ δ ′ = ¯ x L − l , ¯ δ ′ l − = ¯ x L − , ¯ δ ′ n = a . Then the two blocks ¯ δ [0 ,n ) δ [ n,l ) x [ l,L − l ) δ [0 ,n ) ¯ δ ′ [ n,l ) and¯ x [0 ,L ] form an ( ¯ X, ¯ v )-diamond of length smaller than or equal to | ¯ v | + 2 l + 2 k . Letting N = 2 l + 2 k completes the proof. (cid:3) In the case of finite-to-one factor codes π = π ◦ π , we have c π = d π = d π · d π = c π · c π . Since class degree is a conjugacy invariant generalization of degree, it isnatural to consider whether this equality holds for the infinite-to-one case. The followingexample shows that it actually does not; however we are still able to get an inequalityas says Proposition 6.5. Example 6.4.
Let X be the full 2-shift and Y = { ∞ } and consider the trivial map π : X → Y . By letting π = π and π : X → X by π ( x ) i = x i + x i +1 mod 2, we have π = π ◦ π . However, 1 = c π < c π · c π = 2. RANSITION CLASSES 17
Proposition 6.5.
Let ( X, Y, π ) and ( Y, Z, π ) be irreducible factor triples. If π = π ◦ π , then c π ≤ c π · c π .Proof. Since class degree is invariant under conjugacy, we may assume X and Y areone-step, and π and π (hence π ) are one-block. For convenience, rename c = c π and c = c π . Fix a doubly transitive point z in Z and let C be a transition class over z withrespect to π . By Corollary 3.6, C contains a doubly transitive point y . Moreover, bythe same corollary there are c doubly transitive points x (1) , · · · , x ( c ) in π − ( y ) whichare not equivalent to each other with respect to π .We claim that any doubly transitive point x ′ in π − ( C ) is equivalent to some x ( i ) , ≤ i ≤ c . To show the claim, observe that y ′ = π ( x ′ ) lies in C and is equivalent to y with respect to π . It follows that there is a point ~y in Y such that π ( ~y ) = z and ~y ( −∞ , = y ( −∞ , , ~y [ j, ∞ ) = y ′ [ j, ∞ ) for some j > y ′ is doubly transitive, a minimal transition block ( w, n, M ) of π occurs in y ′ [ k,k + | w | ) = ~y [ k,k + | w | ) = w for some k ≥ j . Let the point x ′ be routable through a symbol a ∈ M at time k + n . There is a point ~x in π − ( ~y ) which is also routable through a attime k + n . Reset ~x [ k, ∞ ) to have ~x k + n = a and ~x t = x ′ t for all t ≥ k + | w | .Since y is doubly transitive, the minimal transition block ( w, n, M ) also occurs in y [ − l, − l + w | ) = ~y [ − l, − l + | w | ) = w for some l ≥ | w | . Let the point ~x be routable through asymbol b ∈ M at time − l + n . By Lemma 4.1 and Theorem 4.4 there is exactly one x ( i ) among x (1) , · · · , x ( c ) which is routable through b at time − l + n . Reset ~x ( −∞ , − l + | w | ) tohave ~x − l + n = b and ~x t = x ( i ) t for all t ≤ − l . Therefore we have ~x left asymptotic to x ( i ) ,right asymptotic to x ′ , and π ( ~x ) = π ( ~y ) = z . Corollary 4.9 implies that x ′ ∼ x ( i ) withrespect to π . It follows that π − ( C ) contains at most c doubly transitive π -preimagesof z which are not equivalent to each other with respect to π .Now let C , C , · · · , C c be all the transition classes over z with respect to π . Notethat by Corollary 3.6 the class degree of π is the maximal number of doubly transitivepoints in π − ( z ) = S c j =1 π − ( C j ) which are not equivalent to each other with respectto π . By the above argument, each π − ( C j ) contains at most c doubly transitivepoints which are not equivalent to each other with respect to π . Therefore we have c π ≤ c c . (cid:3) We finish with the following question, which can be regarded as a measure-theoreticalversion of Theorem 3.4.
Question 6.6.
Let (
X, Y, π ) be an irreducible factor triple and let ν be an ergodicmeasure on Y . Given a right transitive point y ∈ Y which is ν -generic, does eachtransition class over y contain a generic point of a measure of relative maximal entropyover ν ?Note that the class C ∈ C ( y ) may not contain generic points for different measuresof relative maximal entropy over ν . For example consider the factor code π on the full 2-shift in Example 6.4. Then c π = 2 and π maps the (1 / , /
3) and (2 / , / X . However each transition class over apoint in X is a singleton. Acknowledgment.
The first author was supported by Fondecyt project 3120137, thesecond author was supported by Fondecyt project 3130718, and the third author wassupported by Basic Science Research Program through the National Research Founda-tion of Korea(NRF) funded by the Ministry of Education (2012R1A1A2006874). Theauthors would like to thank Michael Schraudner and the referee for helpful comments.
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Centro de Modelamiento Matem´atico, Universidad de Chile, Av. Blanco Encalada2120, Piso 7, Santiago de Chile, Chile
E-mail address : [email protected] Centro de Modelamiento Matem´atico, Universidad de Chile, Av. Blanco Encalada2120, Piso 7, Santiago de Chile, Chile
E-mail address : [email protected] Department of Mathematics, Ajou University, Suwon 443-749, South Korea
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