Random walks associated to beta-shifts
aa r X i v : . [ m a t h . D S ] O c t RANDOM WALKS ASSOCIATED TO BETA-SHIFTS
BING LI, YAO-QIANG LI, AND TUOMAS SAHLSTEN
Abstract.
We study the dynamics of a simple random walk on subshifts defined by thebeta transformation and apply it to find concrete formulae for the Hausdorff dimension ofdigit frequency sets for β > that solves β m +1 − β m − generalising the work of Fanand Zhu. We also give examples of β where this approach fails. Introduction
Let
Σ = { , } N be the full shift and let Σ ∗ be the set all finite words. Then any closedshift invariant subset of Σ is called a subshift . For any subshift of Σ we can always writethem as a set Σ W for some subset W ⊂ Σ ∗ by removing all the sequences from Σ containingsubstrings from W . The set W is called the collection of all forbidden words. If W is finite,then Σ W is called a subshift of finite type.The main example in this paper we consider is the subshift Σ β ⊂ Σ defined by the possible β -expansions w w . . . to x = ∞ X j =1 w j β − j of real numbers x , for β > , where the digits w j ∈ { , } are obtained by the naturalfiltration of [0 , defined by the β -transformation T β ( x ) = βx mod 1 on [0 , . For examplein the case β is the Golden ratio, then Σ β = Σ { } with forbidden word . These expansionswere introduced by Rényi [16] in 1957 and they have since been of wide interest throughoutmetric number theory and fractal geometry, and in analog-to-digital signal conversions inthe study beta-encoders [19].The algebraic properties of the number β link deeply to the dynamical properties of thesubshift Σ β , for example, a classical result of Parry [14] says is that Σ β is a subshift of finitetype if and only if β is a simple number, that is, has a finite β -expansion. In this paperwe will study further dynamical characterisations of Σ β from the point of view of randomwalks on the finite words Σ ∗ β associated to Σ β .Let W be any set of forbidden words of the full shift Σ . Given < p < , there is anatural biased random walk X n = ω ω . . . ω n on Σ ∗ for random variables ω , ω , · · · ∈ { , } defined as follows. If X n − = w ∈ Σ n − W , where w / ∈ W , then the probability of ω n = 0 is p and ω n = 1 by − p respectively. If w ∈ W , then the probability of ω n = 0 is . Therandom walk ( X n ) defines a probability distribution µ p supported on the subshift Σ W bysetting µ p [ w ] := P ( X | w | = w ) for all w ∈ Σ ∗ and cylinder [ w ] . Then µ p [0] = p , µ p [1] = 1 − p , and if w / ∈ W , we have µ p [ w
0] = pµ p [ w ] and µ p [ w
1] = (1 − p ) µ p [ w ] . If w ∈ W , we have µ p [ w
0] = µ p [ w ] . Then µ p Date : October 30, 2019.2010
Mathematics Subject Classification.
Primary 28A12; Secondary 28A75, 28A80.
Key words and phrases. β -expansion, Bernoulli-type measure, digit frequency, Hausdorff dimension. defines a natural probability measure ν p = π β µ p on [0 , under the natural projection π ( w ) = ∞ X j =1 w j β − j . In the case of β -shift Σ β , we notice that the measure µ p could be considered some whatnatural construction of a Bernoulli type measure for Σ β , but in general µ p does fail to be,for example, T β invariant under the β transformation T β . However, what we see that havinga type of quasi-Bernoulli is closely related to the algebraic properties of β : Theorem 1.1.
Let β > and Σ β the associated subshift. Then the measure µ p is quasi-shift-invariant, that is, the shift action preserve the µ p null sets. Moreover, the followingare equivalent(1) β is simple number, that is, the β -expansion of is finite;(2) µ p is quasi-Bernoulli, that is, there is a constant C > such that C − µ p [ w ] µ p [ v ] ≤ µ p [ wv ] ≤ Cµ p [ w ] µ p [ v ] for all admissible w, v ∈ Σ ∗ with wv admissible.(3) µ p is strongly quasi-invariant with respect to the shift.When the β is simple, by the strong quasi-invariance, there exists a unique ergodic probabilitymeasure on Σ β equivalent to µ p . This could be considered as an analogue of Parry’s characterisation [14] of subshift offinite type with β being simple, and indeed we will use this as an ingredient of the proof.This work was initiated from the question to establish concrete formulae for the Hausdorffdimensions of the sets of real numbers with specified digit frequencies associated to β -expansions, and for this purpose Theorem 1.1 becomes useful. Here we define the levelsets F p := n x ∈ [0 ,
1) : lim n →∞ ♯ { ≤ k ≤ n : ε k ( x, β ) = 0 } n = p o ,F p := n x ∈ [0 ,
1) : lim n →∞ ♯ { ≤ k ≤ n : ε k ( x, β ) = 0 } n = p o ,F p := n x ∈ [0 ,
1) : lim n →∞ ♯ { ≤ k ≤ n : ε k ( x, β ) = 0 } n = p o where ε ( x, β ) ε ( x, β ) · · · ε k ( x, β ) · · · is the β -expansion of x . A well-known result associatedto the digit frequencies is the result of Fan and Zhu [9], who prove that dim H F p = p log p − (2 p −
1) log(2 p − − (1 − p ) log(1 − p )log β where β = √ is the golden ratio and ≤ p ≤ .We employ the random walks on Σ ∗ β above to extend the work [9] to more general numbersand obtain the following extension: Theorem 1.2.
For < β < such that ε (1 , β ) = 10 m ∞ with some m ∈ { , , , , · · · } ,the following exact formulas of the Hausdorff dimension of F p , F p and F p hold:(1) If ≤ p < m +1 m +2 , then F p = F p = F p = ∅ and dim H F p = dim H F p = dim H F p = 0 .(2) If m +1 m +2 ≤ p ≤ , then dim H F p = dim H F p = dim H F p = ( mp − m + p ) log( mp − m + p ) − ( mp − m + 2 p −
1) log( mp − m + 2 p − − (1 − p ) log(1 − p )log β . ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 3
For calculating the Hausdorff dimension of the level set F p , there is a variation formulain [15] says that we only need to calculate the measure-theoretic entropy of T β with respectto the invariant probability Borel measure with maximal entropy taking value p on [0 , β ) (see also [11, Proposition 4.2]). The following two examples show that if we assume that β has the form assumed in Theorem 1.2, then m p , the T β -ergodic invariant probability Borelmeasure we study in Section 4, is a measure with maximal entropy: Example . Let β ∈ (1 , such that ε (1 , β ) = 10 m ∞ with some m ∈ { , , , , · · · } .Then for any p ∈ (0 , , we have h m p ( T β ) = sup n h ν ( T β ) : ν is a T β -invariant [0 , and ν [0 , β ) = m p [0 , β ) o . However, if we do not assume that β has the form assumed in Theorem 1.2, then thereexists β ∈ (1 , such that m p will never be the measure with maximal entropy: Example . Let β ∈ (1 , such that ε (1 , β ) = 1110 ∞ . Then for any p ∈ (0 , , we have h m p ( T β ) < sup n h ν ( T β ) : ν is a T β -invariant on [0 , and ν [0 , β ) = m p [0 , β ) o . See Section 7 for proofs of these examples. As a future problem it would be interesting tosee how the random walk we use could be used to characterise further arithmetic propertiesof β , and also if one can prove similar results for other β transformations like the intermediate T β,α ( x ) = βx + α mod 1 .The article is organised as follows. In Section 2 we give some notations and preliminar-ies about the beta-shifts and their properties. In Section 3 we define the digit frequencyparameters and establish some key properties of them using the structure of the beta-shift.In Section 4 we prove the dynamical properties of the random walk X n on Σ ∗ β . In Sections5 and 6 we prove local dimension bounds for µ p and Hausdorff dimension bounds for thedigit frequency sets. Finally, in Section 7 we prove the Examples 1.3 and 1.4.2. Notation and preliminaries
Throughout this paper, we use N to denote the positive integer set { , , , , · · · } and N ≥ to denote the non-negative integer set { , , , , · · · } .In this section, we assume β > . We will give some basic notations and recall somenecessary preliminary work.Similar to [4], we consider the β -transformation T β : [0 , → [0 , given by T β ( x ) := βx − ⌊ βx ⌋ for x ∈ [0 , where ⌊ βx ⌋ denotes the integer part of βx . Let A β := (cid:26) { , , · · · , β − } if β ∈ N { , , · · · , ⌊ β ⌋} if β / ∈ N and for any n ∈ N , x ∈ [0 , , we define ε n ( x, β ) := ⌊ βT n − β ( x ) ⌋ ∈ A β . Then we can write x = ∞ X n =1 ε n ( x, β ) β n and call the sequence ε ( x, β ) := ε ( x, β ) ε ( x, β ) · · · ε n ( x, β ) · · · the β - expansion of x . BING LI, YAO-QIANG LI, AND TUOMAS SAHLSTEN
We use ε ε · · · ε n · · · to denote ε (1 , β ) = ε (1 , β ) ε (1 , β ) · · · ε n (1 , β ) · · · for abbreviationin this paper. We say that ε (1 , β ) is infinite if there are infinitely many n ∈ N such that ε n = 0 . Conversely, if there exists M ∈ N such that j > M implies ε j = 0 , we say that ε (1 , β ) is finite and call β a simple beta-number. If additionally ε M = 0 , we say that ε (1 , β ) is finite with length M .The modified β -expansion of is very useful for showing the admissibility of a sequence(see for example Lemma 2.3). It is defined by ε ∗ (1 , β ) := (cid:26) ε (1 , β ) if ε (1 , β ) is infinite ;( ε · · · ε M − ( ε M − ∞ if ε (1 , β ) is finite with length M. No matter whether ε (1 , β ) is finite or not, we denote ε ∗ (1 , β ) = ε ∗ (1 , β ) ε ∗ (1 , β ) · · · ε ∗ n (1 , β ) · · · by ε ∗ ε ∗ · · · ε ∗ n · · · for abbreviation.For a finite word w , we use | w | to denote its length . On the other hand, we write w | k := w w · · · w k to be the prefix of w with length k for w ∈ A N β or w ∈ A nβ where n ≥ k .Let σ : A N β → A N β be the shift σ ( w w · · · ) = w w · · · for w ∈ A N β . We define the usual metric d on A N β by d ( w, v ) := β − inf { k ≥ w k +1 = v k +1 } for any w, v ∈ A N β . Then σ is continuous. Definition 2.1 (Admissibility) . A sequence w ∈ A N β is called admissible if there exists x ∈ [0 , such that ε i ( x, β ) = w i for all i ∈ N . We denote the set of all admissible sequencesby Σ β . A word w ∈ A nβ is called admissible if there exists x ∈ [0 , such that ε i ( x, β ) = w i for i = 1 , · · · , n . We denote the set of all admissible words with length n by Σ nβ and write Σ ∗ β := ∞ [ n =1 Σ nβ . Remark . It is not difficult to check w | n ∈ Σ nβ and w n +1 w n +2 · · · ∈ Σ β for any n ∈ N and w ∈ Σ β by definition. Lemma 2.3 (Parry’s criterion [14]) . Let w ∈ A N β . Then w is admissible (that is, w ∈ Σ β )if and only if σ k ( w ) ≺ ε ∗ (1 , β ) for all k ≥ where ≺ means the lexicographic order smaller in A N β . Noting that σ β (Σ β ) = Σ β , we use σ β : Σ β → Σ β to denote the restriction of σ on Σ β andthen (Σ β , σ β ) is a dynamical system.The continuous projection map π β : Σ β → [0 , defined by π β ( w ) = w β + w β + · · · + w n β n + · · · for w ∈ Σ β is bijective with ε ( · , β ) : [0 , → Σ β as its inverse. Definition 2.4 (Cylinder) . Let w ∈ Σ ∗ β . We call [ w ] := { v ∈ Σ β : v i = w i for all ≤ i ≤ | w |} the cylinder in Σ β generated by w and I ( w ) := π β ([ w ]) ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 5 the cylinder in [0 , generated by w . For any x ∈ [0 , , the cylinder of order n containing x is denoted by I n ( x ) := I ( ε ( x, β ) ε ( x, β ) · · · ε n ( x, β )) . Definition 2.5 (Full words and cylinders) . Let w ∈ Σ nβ . If T nβ I ( w ) = [0 , , we call theword w and the cylinders [ w ] , I ( w ) full . Lemma 2.6 ([1, 8, 13]) . Let w · · · w n ∈ Σ ∗ β with w n = 0 . Then for any ≤ w ′ n < w n , w · · · w n − w ′ n is full. Proposition 2.7 ([12]) . Let w ∈ Σ nβ . Then the following are equivalent. (1) The word w is full, i.e., T nβ I ( w ) = [0 , . (2) | I ( w ) | = β − n . (3) The sequence ww ′ is admissible for any w ′ ∈ Σ β . (4) The word ww ′ is admissible for any w ′ ∈ Σ ∗ β . (5) The word wε ∗ · · · ε ∗ k is admissible for any k ≥ . (6) σ n [ w ] = Σ β . Proposition 2.8 ([12]) . Let w, w ′ ∈ Σ ∗ β be full and | w | = n ∈ N . Then (1) the word ww ′ is full (see also [1] ); (2) the word σ k ( w ) := w k +1 · · · w n is full for any ≤ k < n ; (3) the digit w n < ⌊ β ⌋ if β / ∈ N . In particular, w n = 0 if < β < . Proposition 2.9 ([12]) . (1) Any truncation of ε (1 , β ) is not full (if it is admissible). Thatis, ε (1 , β ) | k is not full for any k ∈ N (if it is admissible).(2) Let k ∈ N . Then ε ∗ (1 , β ) | k is full if and only if ε (1 , β ) is finite with length M whichexactly divides k , i.e., M | k . Proposition 2.10 ([12]) . Let w ∈ Σ nβ . Then w is not full if and only if it ends with a prefixof ε (1 , β ) . That is, when ε (1 , β ) is infinite (finite with length M ), there exists ≤ s ≤ n ( ≤ s ≤ min { M − , n } respectively) such that w = w · · · w n − s ε · · · ε s . For n ∈ N , we use l n ( β ) to denote the number of s following ε ∗ n (1 , β ) as in [13], i.e., l n ( β ) := sup { k ≥ ε ∗ n + j (1 , β ) = 0 for all ≤ j ≤ k } where by convention sup ∅ := 0 . The set of β > such that the length of the strings of sin ε ∗ (1 , β ) is bounded is denoted by A := { β > { l n ( β ) } n ≥ is bounded } . Proposition 2.11 ([13]) . Let β > . Then β ∈ A if and only if there exists a constant c > such that for all x ∈ [0 , and n ≥ , c · β n ≤ | I n ( x ) | ≤ β n Proposition 2.12 ([1] Covering properties) . Let β > . For any x ∈ [0 , and any positiveinteger n , the ball B ( x, β − n ) intersected with [0 , can be covered by at most n +1) cylindersof order n . Definition 2.13 (Absolute continuity and equivalence) . Let µ and ν be measures on ameasurable space ( X, F ) . We say that µ is absolutely continuous with respect to ν anddenote it by µ ≪ ν if ν ( A ) = 0 implies µ ( A ) = 0 for any A ∈ F . Moreover, if µ ≪ ν and ν ≪ µ we say that µ and ν are equivalent and denote it by µ ∼ ν . BING LI, YAO-QIANG LI, AND TUOMAS SAHLSTEN
By the structure of cylinders, the following lemma follows from a similar proof of Lemma1. (i) in [17].
Lemma 2.14.
Any cylinder (in Σ β or [0 , ) can be written as a countable disjoint unionof full cylinders. In order to extend some properties from a small family to a larger one in some proofs inSection 4, we recall the following two well-known theorems as basic knowledge of measuretheory. For more details, see for examples [2] and [3].
Theorem 2.15 (Monotone class theorem) . Let A be an algebra and M ( A ) be the smallestmonotone class containing A . Then M ( A ) is precisely the σ -algebra generated by A , i.e., σ ( A ) = M ( A ) . Theorem 2.16 (Dynkin’s π - λ theorem) . Let C be a π -system and G be a λ -system with C ⊂ G . Then the σ -algebra generated by C is contained in G , i.e., σ ( C ) ⊂ G . The following approximation lemma follows from Theorem 0.1 and Theorem 0.7 in [18].
Lemma 2.17.
Let ( X, B , µ ) be a probability space, C be a semi-algebra which generates the σ -algebra B and A be the algebra generated by C . Then(1) A = C Σ f := { S ni =1 C i : C , · · · , C n ∈ C are disjoint, n ∈ N } ;(2) for each ε > and each B ∈ B , there is some A ∈ A with µ ( A △ B ) < ε . Digit frequency parameters
Let < β ≤ . Write N ( w ) := { k ≥ w k +1 = 0 and w w . . . w k is admissible } for any w ∈ Σ β , N ( w ) := { ≤ k < | w | : w k +1 = 0 and w w . . . w k is admissible } for any w ∈ Σ ∗ β , N ( w ) := { k ≥ w k = 1 } for any w ∈ Σ β , N ( w ) := { ≤ k ≤ | w | : w k = 1 } for any w ∈ Σ ∗ β and let N ( w ) := ♯ N ( w ) , N ( w ) := ♯ N ( w ) for any w ∈ Σ ∗ β or Σ β ,N ( x, n ) := N ( ε ( x, β ) | n ) , N ( x, n ) := N ( ε ( x, β ) | n ) for any x ∈ [0 , where ♯ N means the cardinality of the set N . Remark . Noting that N ( w ) is just the number of the digit appearing in w , it isimmediate from the definition that if w, w ′ ∈ Σ ∗ β such that ww ′ ∈ Σ ∗ β , then N ( ww ′ ) = N ( w ) + N ( w ′ ) . Denote the first position where w and ε ∗ (1 , β ) are different by m ( w ) := min { k ≥ w k < ε ∗ k } for w ∈ Σ β and m ( w ) := m ( w ∞ ) for w ∈ Σ ∗ β . For any w ∈ Σ β , combing the facts w ≺ ε ∗ (1 , β ) , ε ∗ (1 , β ) | n ∈ Σ ∗ β , ∀ n ∈ N and Lemma 2.6,we know that there exists k ∈ N such that w | k is full. Therefore we can write τ ( w ) := min { k ≥ w | k is full } for any w ∈ Σ β , and τ ( w ) := τ ( w ∞ ) for any w ∈ Σ ∗ β . For any w ∈ Σ ∗ β , regarding w | as the empty word which is full, we write τ ′ ( w ) := max { ≤ k ≤ | w | : w | k is full } . ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 7
Lemma 3.2.
Let β > . For any w ∈ Σ β ∪ Σ ∗ β , we have τ ( w ) = (cid:26) m ( w ) if ε (1 , β ) is infinite ;min { m ( w ) , M } if ε (1 , β ) is finite with length M. Proof.
For any w ∈ Σ β ∪ Σ ∗ β . Let k = m ( w ) . Then w | k = ε ∗ · · · ε ∗ k − w k and w k < ε ∗ k . (When w ∈ Σ ∗ β and k > | w | , we regard w | k = w · · · w k as w · · · w | w | k −| w | ). By ε ∗ · · · ε ∗ k − ε ∗ k ∈ Σ ∗ β and Lemma 2.6, w | k is full.(1) When ε (1 , β ) is infinite, for any ≤ i ≤ k − , we have w | i = ε ∗ (1 , β ) | i = ε (1 , β ) | i whichis not full by Proposition 2.9. Therefore τ ( w ) = k = m ( w ) .(2) when ε (1 , β ) = ε · · · ε M ∞ with ε M = 0 : (cid:13) If k ≤ M , then for any ≤ i ≤ k − < M , we have w | i = ε ∗ (1 , β ) | i which is not full byProposition 2.9. Therefore τ ( w ) = k = m ( w ) . (cid:13) If k > M , then w | M = ε ∗ (1 , β ) | M is full by Proposition 2.9. For any ≤ i ≤ M − , wehave w | i = ε ∗ (1 , β ) | i which is not full by Proposition 2.9. Therefore τ ( w ) = M . (cid:3) Lemma 3.3.
Let β > and w ∈ Σ β . Then (1) there exists a strictly increasing sequence ( n j ) j ≥ such that w | n j is full for any j ∈ N ; (2) N ( w ) = + ∞ if < β ≤ .Proof. (1) Let k := m ( w ) , n := k , k j := m ( σ n j − w ) and n j := n j − + k j for any j ≥ . Then n j is strictly increasing. By ε ∗ · · · ε ∗ k − ε ∗ k ∈ Σ ∗ β , w n < ε ∗ k and Lemma 2.6, we know that w · · · w n − w n = ε ∗ · · · ε ∗ k − w n is full. Similarly for any j ≥ , by ε ∗ · · · ε ∗ k j − ε ∗ k j ∈ Σ ∗ β , w n j < ε ∗ k j and Lemma 2.6, we know that w n j − +1 · · · w n j − w n j = ε ∗ · · · ε ∗ k j − w n j is full.Therefore, by Proposition 2.8 (1), w | n j is full for any j ∈ N .(2) Noting that < β ≤ , by w n j < ε ∗ k j , we get w n j = 0 , ε ∗ k j = 1 for any j ∈ N . Thus w · · · w n j − ε ∗ · · · ε ∗ k − w n · · · · · · ε ∗ · · · ε ∗ k j − − w n j − ε ∗ · · · ε ∗ k j − ε ∗ k j ∈ Σ ∗ β for any j ∈ N by Proposition 2.8 (1) and Proposition 2.7 (5). Therefore N ( w ) = + ∞ . (cid:3) Lemma 3.4.
Let < β ≤ , w, w ′ ∈ Σ ∗ β with ww ′ ∈ Σ ∗ β . Then (1) N ( w ) ≤ N ( ww ′ ) ≤ N ( w ) + N ( w ′ ) ; (2) when w is full, we have N ( ww ′ ) = N ( w ) + N ( w ′ ) ; (3) when ε (1 , β ) = ε · · · ε M ∞ with ε M = 0 , we have N ( ww ′ ) ≥ N ( w ) + N ( w ′ ) − M .Proof. Let a = | w | , b = | w ′ | and then ww ′ = w · · · w a w ′ · · · w ′ b .(1) (cid:13) N ( w ) ≤ N ( ww ′ ) follows from N ( w ) ⊂ N ( ww ′ ) . (cid:13) Prove N ( ww ′ ) ≤ N ( w ) + N ( w ′ ) .i) We prove N ( ww ′ ) ⊂ N ( w ) ∪ ( N ( w ′ ) + a ) first. Let k ∈ N ( ww ′ ) .If ≤ k < a , then w k +1 = 0 and w · · · w k ∈ Σ ∗ β . We get k ∈ N ( w ) .If a ≤ k < a + b , then w ′ k − a +1 = 0 and w · · · w a w ′ · · · w ′ k − a ∈ Σ ∗ β . It follows from w ′ · · · w ′ k − a ∈ Σ ∗ β that k − a ∈ N ( w ′ ) and k ∈ N ( w ′ ) + a .ii) Combining N ( w ) ∩ ( N ( w ′ ) + a ) = ∅ , ♯ ( N ( w ′ ) + a ) = ♯ N ( w ′ ) and i), we get N ( ww ′ ) ≤ N ( w ) + N ( w ′ ) .(2) We need to prove N ( ww ′ ) ≥ N ( w ) + N ( w ′ ) . By ♯ N ( w ′ ) = ♯ ( N ( w ′ ) + a ) , it suffices toprove N ( ww ′ ) ⊃ N ( w ) ∪ ( N ( w ′ ) + a ) . For each k ∈ N ( w ) , obviously k ∈ N ( ww ′ ) . Onthe other hand, if k ∈ ( N ( w ′ ) + a ) , then k − a ∈ N ( w ′ ) , w ′ k − a +1 = 0 and w ′ · · · w ′ k − a ∈ Σ ∗ β .Since w is full, by Proposition 2.7, we get ww ′ · · · w ′ k − a ∈ Σ ∗ β and then k ∈ N ( ww ′ ) .(3) (cid:13) Firstly, we divide ww ′ into three segments. BING LI, YAO-QIANG LI, AND TUOMAS SAHLSTEN i) Let k := τ ′ ( w ) , then ≤ k ≤ a . If k = a , w is full. Then the conclusion followsfrom (2) immediately. Therefore we assumes ≤ k < a in the following proof. Let u (1) := w · · · w k be full and | u (1) | = k . (When k = 0 , we regard u (1) as the emptyword and N ( u (1) ) := 0 .)ii) Consider w k +1 · · · w a w ′ · · · w ′ b ∈ Σ ∗ β (the admissibility follows from ww ′ ∈ Σ ∗ β ).Let k := τ ( w k +1 · · · w a w ′ · · · w ′ b ) ≥ . By the definition of k = τ ′ ( w ) and Propo-sition 2.8, we get k > a − k . In the following, we assume k ≤ a − k + b first. The case k > a − k + b will be considered at the end of the proof. Let u (2) := w k +1 · · · w a w ′ · · · w ′ k + k − a , then | u (2) | = k .iii) Let u (3) := w ′ k + k − a +1 · · · w ′ b . (When k + k − a = b , we regard u (3) as the emptyword and N ( u (3) ) := 0 .)Up to now, we write ww ′ = u (1) u (2) u (3) . w · · · w k | {z } | u (1) | = k w k +1 · · · w a w ′ · · · w ′ k + k − a | {z } | u (2) | = k w ′ k + k − a +1 · · · w ′ b | {z } | u (3) | (cid:13) Estimate N ( ww ′ ) , N ( w ) and N ( w ′ ) .i) N ( ww ′ ) = N ( u (1) u (2) u (3) ) u (1) full ====== by (2) N ( u (1) )+ N ( u (2) u (3) ) u (2) full ====== by (2) N ( u (1) )+ N ( u (2) )+ N ( u (3) ) .ii) N ( w ) u (1) full ====== by (2) N ( u (1) ) + N ( w k +1 · · · w a ) by (1) ≤ N ( u (1) ) + N ( u (2) ) .iii) N ( w ′ ) by (1) ≤ N ( w ′ · · · w ′ k + k − a ) + N ( u (3) ) ≤ M + N ( u (3) ) where the last inequalityfollows from N ( w ′ · · · w ′ k + k − a ) ≤ k + k − a ≤ k = τ ( w k +1 · · · w a w ′ · · · w ′ b ) by Lemma 3.2 ≤ M Combining i), ii) and iii), we get N ( ww ′ ) ≥ N ( w ) + N ( w ′ ) − M .To end the proof, it suffices to consider the case k > a − k + b below. We define u (1) as before and define u (2) := w k +1 · · · w a w ′ · · · w ′ b which is not full. Then | u (2) | = a − k + b .We do not define u (3) . (cid:13) Prove N ( u (2) ) = 0 .By contradiction, we suppose N ( u (2) ) = 0 , then there exists k ∈ N ( u (2) ) , ≤ k < a − k + b such that u (2) k +1 = 0 and u (2)1 · · · u (2) k ∈ Σ ∗ β . By Lemma 2.6, u (2)1 · · · u (2) k +1 is full which contra-dict τ ( u (2) ) = k > a − k + b . (cid:13) Estimate N ( ww ′ ) , N ( w ) and N ( w ′ ) .i) N ( ww ′ ) = N ( u (1) u (2) ) u (1) full ====== by (2) N ( u (1) ) + N ( u (2) ) by (cid:13) ===== N ( u (1) ) .ii) N ( w ) u (1) full ====== by (2) N ( u (1) ) + N ( w k +1 · · · w a ) = N ( u (1) ) where the last equality followsfrom N ( w k +1 · · · w a ) ≤ N ( u (2) ) = 0 .iii) N ( w ′ ) ≤ b ≤ | u (2) | = a − k + b < k = τ ( u (2) ) by Lemma 3.2 ≤ M .Combining i), ii) and iii), we get N ( ww ′ ) ≥ N ( w ) + N ( w ′ ) − M . (cid:3) ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 9 Dynamical properties of the random walk on Σ ∗ β Recall that the random walk ( X n ) in Σ ∗ β defines a probability distribution µ p supportedon the subshift Σ β by setting µ p [ w ] := P ( X | w | = w ) for all w ∈ Σ ∗ and cylinder [ w ] , which then satisfies µ p [0] = p, µ p [1] = 1 − p, and if w / ∈ W , we have µ p [ w
0] = pµ p [ w ] and µ p [ w
1] = (1 − p ) µ p [ w ] . If w ∈ W , we have µ p [ w
0] = µ p [ w ] . Then µ p defines a natural probability measure ν p = π β µ p on [0 , under the natural projec-tion π ( w ) = ∞ X j =1 w j β − j . Remark . (1) By the definition of µ p and ν p , we have ν p ( I ( w )) = µ p [ w ] = p N ( w ) (1 − p ) N ( w ) for any w ∈ Σ ∗ β ; ν p ( I ( w | n )) = µ p [ w | n ] = p N ( w | n ) (1 − p ) N ( w | n ) for any w ∈ Σ β , n ∈ N ; ν p ( I n ( x )) = µ p [ ε ( x, β ) | n ] = p N ( x,n ) (1 − p ) N ( x,n ) for any x ∈ [0 , , n ∈ N . (2) For any w ∈ Σ β , as n → + ∞ , by Lemma 3.3 (2) we get N ( w | n ) → + ∞ and then µ p [ w | n ] → . Proposition 4.2.
The measures µ p , σ kβ µ p , ν p and T kβ ν p have no atoms. That is, µ p ( { w } ) = σ kβ µ p ( { w } ) = ν p ( { x } ) = T kβ ν p ( { x } ) = 0 for any single point w ∈ Σ β , x ∈ [0 , and k ∈ N .Proof. It follows immediately from µ p [ w | n ] → , ♯σ − kβ { w } ≤ k , ♯π − β { x } = 1 and ♯T − kβ { x } ≤ k for any w ∈ Σ β and x ∈ [0 , . (cid:3) Definition 4.3 (Invariance and ergodicity) . Let ( X, F , µ, T ) be a measure-preserving dy-namical system, that is, ( X, F , µ ) is a probability space and µ is T - invariant , i.e., T µ = µ .We say that the probability measure µ is ergodic with respect to T if for every A ∈ F satisfying T − A = A (such a set is called T - invariant ), we have µ ( A ) = 0 or . We also saythat ( X, F , µ, T ) is ergodic.Note that µ p is not σ β -invariant and ν p is not T β -invariant. For example, if β = √ isthe golden ratio, then we have Σ ∗ β = { w ∈ ∞ [ n =1 { , } n : 11 does not appear in w } . Hence µ p [1] = 1 − p, but µ p ( σ − β [1]) = µ p [01] = p (1 − p ) . Correspondingly, ν p [ 1 β ,
1) = 1 − p, but ν p ( T − β [ 1 β , p (1 − p ) . We recall the notion of quasi-invariance.
Definition 4.4 (Quasi-invariance) . Let ( X, F , µ ) be a measure space and T be a measurabletransformation on it. Then(1) µ is quasi-invariant with respect to the transformation T if µ and its image measure T µ are mutually absolutely continuous (i.e. equivalent), that is, µ ≪ T µ ≪ µ ( i.e. T µ ∼ µ ); (2) µ is strongly quasi-invariant with respect to the transformation T if there exists aconstant C > such that C − µ ( A ) ≤ T k µ ( A ) ≤ Cµ ( A ) for any k ∈ N and A ∈ F . We also say µ is C -strongly quasi-invariant if we knowsuch a C . Definition 4.5 (Quasi-Bernoulli) . A measure µ on (Σ β , B (Σ β )) is called quasi-Bernoulli ifthere exists a constant C > such that C − µ [ w ] µ [ w ′ ] ≤ µ [ ww ′ ] ≤ Cµ [ w ] µ [ w ′ ] for every pair w, w ′ ∈ Σ ∗ β satisfying ww ′ ∈ Σ ∗ β . Theorem 4.6.
Let < β ≤ and < p < . Then (1) µ p is quasi-invariant with respect to σ β ; (2) ε (1 , β ) is finite if and only if µ p is quasi-Bernoulli; (3) ε (1 , β ) is finite if and only if µ p is strongly quasi-invariant with respect to σ β . The proof of this is based on the following lemma.
Lemma 4.7.
Let < β ≤ , < p < and w, w ′ ∈ Σ ∗ β with ww ′ ∈ Σ ∗ β . Then (1) µ p [ w ] ≥ µ p [ ww ′ ] ≥ µ p [ w ] µ p [ w ′ ]; (2) when w is full, we have µ p [ ww ′ ] = µ p [ w ] µ p [ w ′ ]; (3) if additionally ε (1 , β ) = ε · · · ε M ∞ with ε M = 0 , then µ p [ ww ′ ] ≤ p − M µ p [ w ] µ p [ w ′ ] . In particular, µ p is quasi-Bernoulli.Proof. It follows from Remark 4.1, Lemma 3.4 and N ( ww ′ ) = N ( w ) + N ( w ′ ) for any ww ′ ∈ Σ ∗ β . (cid:3) Proof of Theorem 4.6. (1) (cid:13) Prove µ p ≪ σ β µ p .Let A ∈ B (Σ β ) with σ β µ p ( A ) = 0 . It suffices to prove µ p ( A ) = 0 . For any ε > , by µ p ( σ − β A ) = inf { X n µ p [ w ( n ) ] : w ( n ) ∈ Σ ∗ β , σ − β A ⊂ [ n [ w ( n ) ] } = 0 , there exists { w ( n ) } ⊂ Σ ∗ β such that σ − β A ⊂ [ n [ w ( n ) ] and X n µ p [ w ( n ) ] < ε. ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 11
Since ε can be small enough such that µ p [0] = p and µ p [1] = 1 − p > ε , we can assume a n := | w ( n ) | ≥ for any n without loss of generality. By the fact that σ β is surjective, weget A = σ β ( σ − β A ) ⊂ σ β ( [ n [ w ( n ) ]) ⊂ [ n σ β [ w ( n ) ] = [ n σ β [ w ( n )1 w ( n )2 · · · w ( n ) a n ] ⊂ [ n [ w ( n )2 · · · w ( n ) a n ] . Therefore µ p ( A ) ≤ X n µ p [ w ( n )2 · · · w ( n ) a n ] ≤ { p, − p } X n µ p [ w ( n )1 ] µ p [ w ( n )2 · · · w ( n ) a n ] ≤ { p, − p } X n µ p [ w ( n ) ] < ε min { p, − p } for any ε > . (cid:13) Prove σ β µ p ≪ µ p .Let B ∈ B (Σ β ) with µ p ( B ) = 0 . It suffices to prove σ β µ p ( B ) = 0 . For any m ∈ N ≥ , wedefine B m := B \ [ ε ∗ · · · ε ∗ m ] .i) Prove that σ β µ p ( B m ) increase to σ β µ p ( B ) . a (cid:13) If ε (1 , β ) is finite, then ε ∗ ε ∗ ε ∗ · · · / ∈ Σ β , [ ε ∗ · · · ε ∗ m ] decrease to ∅ , B m increase to B and σ β µ p ( B m ) increase to σ β µ p ( B ) . b (cid:13) If ε (1 , β ) is infinite, then ε ∗ ε ∗ ε ∗ · · · = ε ε ε · · · = ε ( T β , β ) ∈ Σ β , [ ε ∗ · · · ε ∗ m ] decrease to { ε ∗ ε ∗ ε ∗ · · · } (a single point set), B m increase to ( B \ { ε ∗ ε ∗ ε ∗ · · · } ) and σ β µ p ( B m ) increase to σ β µ p ( B \{ ε ∗ ε ∗ ε ∗ · · · } ) . Since σ β µ p has no atom (by Proposition4.2), we get σ β µ p ( B m ) increase to σ β µ p ( B ) .ii) In order to get σ β µ p ( B ) = 0 , by i) it suffices to prove that for any m ∈ N ≥ , σ β µ p ( B m ) = 0 .Fix m ∈ N ≥ . By µ p ( B m ) ≤ µ p ( B ) = 0 , we get inf n X n µ p [ w ( n ) ] : w ( n ) ∈ Σ ∗ β , B m ⊂ [ n [ w ( n ) ] o = 0 . For any ε > , there exists { w ( n ) } n ∈ N ′ ⊂ Σ ∗ β with B m ⊂ [ n ∈ N ′ [ w ( n ) ] such that X n ∈ N ′ µ p [ w ( n ) ] < ε where N ′ is an index set with cardinality at most countable. Since ε can be smallenough such that δ m := min { µ p [ w ] : w ∈ Σ ∗ β , | w | ≤ m − } > ε, we can assume a n := | w ( n ) | ≥ m for all n ∈ N ′ . Let N := { n ∈ N ′ : w ( n ) | m − = ε ∗ · · · ε ∗ m } ⊂ N ′ . By the fact that for any n ∈ N , [ w ( n ) ] ∩ [ ε ∗ · · · ε ∗ m ] = ∅ and for any n ∈ N ′ \ N , [ w ( n ) ] ⊂ [ ε ∗ · · · ε ∗ m ] , we get B m = B m \ [ ε ∗ · · · ε ∗ m ] ⊂ [ n ∈ N ′ (cid:0) [ w ( n ) ] \ [ ε ∗ · · · ε ∗ m ] (cid:1) = (cid:0) [ n ∈ N (cid:0) [ w ( n ) ] \ [ ε ∗ · · · ε ∗ m ] (cid:1)(cid:1) [ (cid:0) [ n ∈ N ′ \ N (cid:0) [ w ( n ) ] \ [ ε ∗ · · · ε ∗ m ] (cid:1)(cid:1) = [ n ∈ N [ w ( n ) ] and then σ − β B m ⊂ S n ∈ N σ − β [ w ( n ) ] . Let N := { n ∈ N : 1 w ( n ) / ∈ Σ ∗ β } and N := { n ∈ N : 1 w ( n ) ∈ Σ ∗ β } . Then for any n ∈ N , σ − β [ w ( n ) ] = [0 w ( n ) ] and for any n ∈ N , σ − β [ w ( n ) ] = [0 w ( n ) ] ∪ [1 w ( n ) ] . Thus σ − β B m ⊂ (cid:0) [ n ∈ N [0 w ( n ) ] (cid:1) [ (cid:0) [ n ∈ N [1 w ( n ) ] (cid:1) and µ p ( σ − β B m ) ≤ X n ∈ N µ p [0 w ( n ) ] + X n ∈ N µ p [1 w ( n ) ] =: J + J where by Lemma 4.7 (2), J = X n ∈ N pµ p [ w ( n ) ] ≤ p X n ∈ N ′ µ p [ w ( n ) ] < pε. Now we estimate the upper bounded of T .For each n ∈ N ⊂ N , by w ( n )1 · · · w ( n ) m − = ε ∗ ε ∗ · · · ε ∗ m , there exists ≤ k n ≤ m − such that ε ∗ , w ( n )1 = ε ∗ , · · · w ( n ) k n − = ε ∗ k n and w ( n ) k n < ε ∗ k n +1 . Since ε ∗ · · · ε ∗ k n ε ∗ k n +1 ∈ Σ ∗ β , by Lemma 2.6 and Proposition 2.8 (2), we know that both w ( n )1 · · · w ( n ) k n and w ( n )1 · · · w ( n ) k n are full. It follows from Lemma 4.7 (2) that µ p [1 w ( n ) ] = µ p [1 w ( n )1 · · · w ( n ) k n ] µ p [ w ( n ) k n +1 · · · w ( n ) a n ] and µ p [ w ( n ) ] = µ p [ w ( n )1 · · · w ( n ) k n ] µ p [ w ( n ) k n +1 · · · w ( n ) a n ] . Let C m := max n µ p [1 w ] µ p [ w ] : w ∈ Σ ∗ β with w ∈ Σ ∗ β and ≤ | w | ≤ m − o < ∞ . By k n ≤ m − , we get µ p [1 w ( n ) ] ≤ C m µ p [ w ( n ) ] for any n ∈ N . This implies J = X n ∈ N µ p [1 w ( n ) ] ≤ C m X n ∈ N µ p [ w ( n ) ] ≤ C m X n ∈ N ′ µ p [ w ( n ) ] < C m ε. Therefore µ p ( σ − β B m ) < ( p + C m ) ε for any < ε < δ m . We conclude that σ β µ p ( B m ) =0 .(2) ⇒ follows from Lemma 4.7. ⇐ (By contradiction) Assume that ε (1 , β ) = ε ε ε · · · is infinite. By ε ε · · · = ε ( T β , β ) ∈ Σ β and Lemma 3.3 (2), we get N ( ε ε · · · ) = + ∞ . Then for any N ∈ N , there exists n ∈ N such that N ( ε ε · · · ε n ) ≥ N . Let w := ε = 1 and w ′ := ε ε · · · ε n . Then ww ′ = ε · · · ε n and obviously N ( ww ′ ) = 0 = 0 + N − N ≤ N ( w ) + N ( w ′ ) − N. ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 13
By Remark 4.1 (1) and N ( ww ′ ) = N ( w ) + N ( w ′ ) , we get µ p [ ww ′ ] ≥ p − N µ p [ w ] µ p [ w ′ ] . Since for any N ∈ N , there exists w, w ′ which satisfy the above inequality and p − N can bearbitrary large, we know that µ p is not quasi-Bernoulli.(3) ⇐ (By contradiction) Assume that ε (1 , β ) = ε ε ε · · · is infinite. By ε ε · · · = ε ( T β , β ) ∈ Σ β and Lemma 3.3 (2), we get N ( ε ε · · · ) = + ∞ . Then for any N ∈ N , thereexists n ∈ N such that N ( ε ε · · · ε n ) ≥ N . Let w := ε · · · ε n . Then σ β µ p [ w ] = µ p [0 w ] + µ p [1 w ] ≥ µ p [ ε ε · · · ε n ] = p N ( ε ··· ε n ) (1 − p ) N ( ε ··· ε n ) = (1 − p ) N ( ε ··· ε n ) and µ p [ w ] = p N ( w ) (1 − p ) N ( w ) ≤ p N (1 − p ) N ( ε ··· ε n ) . Thus σ β µ p [ w ] ≥ (1 − p ) p − N µ p [ w ] . Since for any N ∈ N , there exists w which satisfy the above inequality and (1 − p ) p − N canbe arbitrary large, we know that µ p is not strongly quasi-invariant. ⇒ Let ε (1 , β ) = ε · · · ε M ∞ with ε M = 0 and c = p − M > . (cid:13) Prove c − µ p [ w ] ≤ σ kβ µ p [ w ] ≤ cµ p [ w ] for all k ∈ N and w ∈ Σ ∗ β .Notice that σ − kβ [ w ] = [ u ··· u k w ∈ Σ ∗ β [ u · · · u k w ] is a disjoint union.i) Estimate the upper bound of σ kβ µ p [ w ] : µ p σ − kβ [ w ] = X u ··· u k w ∈ Σ ∗ β µ p [ u · · · u k w ] a (cid:13)≤ X u ··· u k w ∈ Σ ∗ β p − M µ p [ u · · · u k ] µ p [ w ] ≤ p − M X u ··· u k ∈ Σ ∗ β µ p [ u · · · u k ] µ p [ w ]= p − M µ p [ w ] . where a (cid:13) follows from Lemma 4.7.ii) Estimate the lower bound of σ kβ µ p [ w ] : µ p σ − kβ [ w ] = X u ··· u k w Σ ∗ β µ p [ u · · · u k w ] ≥ X u ··· u k − M M Σ ∗ β µ p [ u · · · k − M M w ] . (Without loss of generality, we assume k ≥ M . Otherwise, we consider k w instead of u · · · u k − M M w ). By Proposition 2.10, u · · · u k − m M is full for any u · · · u k − m ∈ Σ ∗ β .Then by Proposition 2.7 (4), we get u · · · u k − M M w ∈ Σ ∗ β ⇐⇒ u · · · u k − M ∈ Σ ∗ β . Therefore µ p σ − kβ [ w ] ≥ X u ··· u k − M ∈ Σ ∗ β µ p [ u · · · u k − M M w ] b (cid:13) = X u ··· u k − M ∈ Σ ∗ β µ p [ u · · · u k − M M ] µ p [ w ] c (cid:13)≥ X u ··· u k − M ∈ Σ ∗ β µ p [ u · · · u k − M ] p M µ p [ w ]= p M µ p [ w ] where b (cid:13) and c (cid:13) follow from Lemma 4.7 (2) and (1) respectively. (cid:13) Prove c − µ p ( B ) ≤ σ kβ µ p ( B ) ≤ cµ p ( B ) for all k ∈ N and B ∈ B (Σ β ) .Let C := { [ w ] : w ∈ Σ ∗ β } ∪ { ∅ } , C Σ f := { S ni =1 C i : C , · · · , C n ∈ C are disjoint, n ∈ N } and G := { B ∈ B (Σ β ) : c − µ p ( B ) ≤ σ kβ µ p ( B ) ≤ cµ p ( B ) for all k ∈ N } . Then C is a semi-algebra, C Σ f is the algebra generated by C (by Theorem 2.17 (1)) and G isa monotone class. Since in (cid:13) we have already C ⊂ G , it is obvious that C Σ f ⊂ G ⊂ B (Σ β ) .By Monotone Class Theorem (Theorem 2.15), we get G = B (Σ β ) . (cid:3) By Theorem 4.6, we get the following.
Corollary 4.8.
Let < β ≤ and < p < . Then(1) ν p is quasi-invariant with respect to T β ;(2) ε (1 , β ) is finite if and only if ν p is strongly quasi-invariant with respect to T β . Theorem 4.9.
Let < β ≤ and < p < . If ε (1 , β ) is finite, then there exists a unique T β -ergodic probability measure m p on ([0 , , B [0 , equivalent to ν p , where m p is definedby m p ( B ) := lim n →∞ n n − X k =0 T kβ ν p ( B ) for B ∈ B [0 , . The proof of this is based on the following lemmas.
Lemma 4.10 ([5]) . Let ( X, B , µ ) be a probability space and T be a measurable transformationon X satisfying µ ( T − E ) = 0 whenever E ∈ B with µ ( E ) = 0 . If there exists a constant M such that for any E ∈ B and any n ≥ , n n − X k =0 µ ( T − k E ) ≤ M µ ( E ) , then for any real integrable function f on X , the limit lim n →∞ n n − X k =0 f ( T k x ) exists for µ -almost every x ∈ X . Lemma 4.11.
Let < β ≤ and < p < .(1) If B ∈ B (Σ β ) with σ − β B = B , then µ p ( B ) = 0 or .(2) If B ∈ B [0 , with T − β B = B , then ν p ( B ) = 0 or . ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 15
Proof. (1) Let F := { w ∈ Σ ∗ β : w is full } . (cid:13) Let w ∈ F with | w | = n . We prove µ p ([ w ] ∩ σ − nβ A ) = µ p [ w ] µ p ( A ) for any A ∈ B (Σ β ) as below.Since w is full and [ ww ′ ] = [ w ] ∩ σ − nβ [ w ′ ] for any w ′ ∈ Σ ∗ β , we get µ p ([ w ] ∩ σ − nβ [ w ′ ]) = µ p [ ww ′ ] by Lemma 4.7 (2) ============ µ p [ w ] µ p [ w ′ ] . Let C := { [ w ′ ] : w ′ ∈ Σ ∗ β } ∪ { ∅ } and G := { A ∈ B (Σ β ) : µ p ([ w ] ∩ σ − nβ A ) = µ p [ w ] µ p ( A ) } . Then we have already got C ⊂ G ⊂ B (Σ β ) . Since C is a π -system, G is a λ -system and C generates B (Σ β ) , by Dynkin’s π - λ Theorem 2.16, we get G = B (Σ β ) . (cid:13) We use B c to denote the complement of B in Σ β . For any δ > , by Lemma 2.17 andLemma 2.14, there exists a countable disjoint union of full cylinders E δ = S i [ w ( i ) ] with { w ( i ) } ⊂ F such that µ p ( B c △ E δ ) < δ . (cid:13) Let B ∈ B (Σ β ) with σ − β B = B . Then B = σ − nβ B and by (cid:13) we get µ p ( B ∩ [ w ]) = µ p ( σ − nβ B ∩ [ w ]) = µ p ( B ) µ p [ w ] for any w ∈ F . Thus µ p ( B ∩ E δ ) = µ p ( B ∩ [ i [ w ( i ) ]) = X i µ p ( B ∩ [ w ( i ) ]) = X i µ p ( B ) µ p [ w ( i ) ] = µ p ( B ) µ p ( E δ ) . Let a = µ p (( B ∪ E δ ) c ) , b = µ p ( B ∩ E δ ) , c = µ p ( B \ E δ ) and d = µ p ( E δ \ B ) . Then b = ( b + c )( b + d ) , a + b < δ (by (cid:13) ) and a + b + c + d = 1 . By ( b + c )( a + d − δ ) ≤ ( b + c )( b + d ) = b < δ, we get ( b + c )( a + d ) < (1 + b + c ) δ ≤ δ which implies µ p ( B ) µ p ( B c ) ≤ δ for any δ > . Therefore µ p ( B ) = 0 or µ p ( B c ) = 0 .(2) follows from (1). In fact, let B ∈ B [0 , with T − β B = B . By σ − β π − β B = π − β T − β B = π − β B ∈ B (Σ β ) and (1), we get µ p ( π − β B ) = 0 or , i.e., ν p ( B ) = 0 or . (cid:3) Proof of Theorem 4.9. (1) For any n ∈ N and B ∈ B [0 , , define m np ( B ) := 1 n n − X k =0 ν p ( T − kβ B ) . Then m np is a probability measure on ([0 , , B [0 , . By Corollary 4.8, there exists c > such that c − ν p ( B ) ≤ m np ( B ) ≤ cν p ( B ) for any B ∈ B [0 , and n ∈ N . (4.1)(2) For any B ∈ B [0 , , prove that lim n →∞ m np ( B ) exists. In fact, lim n →∞ m np ( B ) = lim n →∞ n n − X k =0 ˆ T − kβ B dν p = lim n →∞ ˆ n n − X k =0 B ( T kβ x ) dν p ( x )= ˆ lim n →∞ n n − X k =0 B ( T kβ x ) dν p ( x ) . The last equality follows from Dominate Convergence Theorem where the ν p -a.e. existenceof lim n →∞ n n − P k =0 B ( T kβ x ) follows from Lemma 4.10, the strongly quasi-invariance of ν p and (1).(3) For any B ∈ B [0 , , define m p ( B ) := lim n →∞ m np ( B ) . Then m p is a probability measureon ([0 , , B [0 , .(4) m p ∼ ν p on B [0 , follows from (4.1) and the definition of m p .(5) Prove that m p is T β -invariant.For any B ∈ B [0 , and n ∈ N , we have m np ( T − β B ) = 1 n n X k =1 ν p ( T − kβ B ) = n + 1 n m n +1 p ( B ) − ν p ( B ) n . As n → ∞ , we get m p ( T − β B ) = m p ( B ) .(6) Prove that ([0 , , B [0 , , m p , T β ) is ergodic.Let B ∈ B [0 , such that T − β B = B . Then by Lemma 4.11 (2), we get ν p ( B ) = 0 or ν p ( B c ) = 0 which implies m p ( B ) = 0 or m p ( B c ) = 0 since m p ∼ ν p . Noting that m p is T β -invariant, we know that m p is ergodic with respect to T β .(7) Prove that such m p is unique on B [0 , .Let m ′ p be a T β -ergodic probability measure on ([0 , , B [0 , equivalent to ν p . Then forany B ∈ B [0 , , by the Birkhoff Ergodic Theorem, we get m p ( B ) = ˆ B dm p = lim n →∞ n n − X k =0 B ( T kβ x ) for m p -a.e. x ∈ [0 , and m ′ p ( B ) = ˆ B dm ′ p = lim n →∞ n n − X k =0 B ( T kβ x ) for m ′ p -a.e. x ∈ [0 , . Since m p ∼ ν p ∼ m ′ p , there exists x ∈ [0 , such that m p ( B ) = lim n →∞ n P n − k =0 B ( T kβ x ) = m ′ p ( B ) . (cid:3) Modified lower local dimension related to β -expansions Let ν be a finite measure on R n . The lower local dimension of ν at x ∈ R n is defined by dim loc ν ( x ) := lim r → log ν ( B ( x, r ))log r , where B ( x, r ) is the closed ball centered on x with radius r . Theoretically, we can use thelower local dimension to estimate the upper and lower bounds of the Hausdorff dimension(see [6] for definition) by the following proposition. Proposition 5.1 ([7] Proposition 2.3) . Let s ≥ , E ⊂ R n be a Borel set and ν be a finiteBorel measure on R n . (1) If dim loc ν ( x ) ≤ s for all x ∈ E then dim H E ≤ s . (2) If dim loc ν ( x ) ≥ s for all x ∈ E and ν ( E ) > then dim H E ≥ s . But in the definition of the lower local dimension, the Bernoulli-type measure of a ball ν p ( B ( x, r )) is difficult to estimate. Therefore, we use the measure of a cylinder ν ( I n ( x )) instead of ν p ( B ( x, r )) to define the modified lower local dimension related to β -expansions of a measure at a point. ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 17
Definition 5.2.
Let β > and ν be a finite measure on [0 , . The modified lower localdimension of ν at x ∈ [0 , is defined by dim βloc ν ( x ) := lim n →∞ log ν ( I n ( x ))log | I n ( x ) | where I n ( x ) is the cylinder of order n containing x .Combining Proposition 5.1 (1) and the following proposition, we can estimate the upperbound of the Hausdorff dimension by the modified lower local dimension. Proposition 5.3.
Let β > and ν be a finite measure on [0 , . Then for any x ∈ [0 , , dim βloc ( ν, x ) ≥ dim loc ( ν, x ) . Proof.
For any x ∈ [0 , and n ∈ N . Let r n := | I n ( x ) | , then I n ( x ) ⊂ B ( x, r n ) , ν ( I n ( x )) ≤ ν ( B ( x, r n )) and − log ν ( I n ( x )) ≥ − log ν ( B ( x, r n )) . We get − log ν ( I n ( x )) − log | I n ( x ) | ≥ − log ν ( B ( x, r n )) − log r n . Therefore lim n →∞ log ν ( I n ( x ))log | I n ( x ) | ≥ lim n →∞ log ν ( B ( x, r n ))log r n ≥ dim loc ν ( x ) . (cid:3) Remark . The reverse inequality in Proposition 5.3, i.e., dim βloc ( ν, x ) ≤ dim loc ( ν, x ) is notalways true. For example, let β be the golden ratio ( √ / , x = β − and ν = ν p bethe ( p, − p ) Bernoulli-type measure with < p < / . For any n ∈ N , let r n = | I n ( x ) | and J n be the left consecutive cylinder of I n ( x ) with the same order n . When n ≥ , wehave r n = β − n ≥ | J n | and B ( x, r n ) ⊃ J n . Then ν p ( B ( x, r n )) ≥ ν p ( J n ) ≥ p (1 − p ) n − and ν p ( I n ( x )) = (1 − p ) p n − which implies dim βloc ν p ( x ) = lim n →∞ log(1 − p ) p n − log β − n = − log p log β and dim loc ν p ( x ) ≤ lim n →∞ log ν p ( B ( x, r n ))log r n ≤ lim n →∞ log p (1 − p ) n − log β − n = − log(1 − p )log β . When < p < / , we have dim βloc ( ν p , x ) > dim loc ( ν p , x ) .Though the reverse inequality in Proposition 5.3 is not always true, we are going toestablish the following theorem for estimating both of the upper and lower bounds of theHausdorff dimension by the modified lower local dimension of a finite measure. Theorem 5.5.
Let β > , s ≥ , E ⊂ [0 , be a Borel set and ν be a finite Borel measureon [0 , . (1) If dim βloc ν ( x ) ≤ s for all x ∈ E , then dim H E ≤ s . (2) If dim βloc ν ( x ) ≥ s for all x ∈ E and ν ( E ) > , then dim H E ≥ s .Proof. (1) follows from Proposition 5.1 (1) and Proposition 5.3.(2) follows from the following Lemma 5.7. In fact, if s = 0 , dim H E ≥ s is obvious. If s > , let < t < s . For any x ∈ E , by lim n →∞ log ν ( I n ( x ))log | I n ( x ) | ≥ s > t , there exists N ∈ N suchthat any n > N implies log ν ( I n ( x ))log | I n ( x ) | > t and ν ( I n ( x )) < | I n ( x ) | t . So lim n →∞ ν ( I n ( x )) | I n ( x ) | t ≤ < . For any < ε < t , by Lemma 5.7, we get H t − ε ( E ) > (where H s ( E ) denotes the classical s -dimension Hausdorff measure of a set E .) and then dim H E ≥ t − ε . So dim H E ≥ t forany t < s . Therefore dim H E ≥ s . (cid:3) Remark . The statement (2) in Theorem 5.5 obviously implies the Proposition 1.3 in [1]which is called the modified mass distribution principle.
Lemma 5.7.
Let β > , s > , c > , E ⊂ [0 , be a Borel set and ν be a finite Borelmeasure on [0 , . If lim n →∞ ν ( I n ( x )) | I n ( x ) | s < c for all x ∈ E , then for any < ε < s , H s − ε ( E ) ≥ c − ν ( E ) .Proof. It follows immediately from Lemma 5.9 and Lemma 5.8. (cid:3)
For establishing this lemma, we need the followings.Let β > , s ≥ and E ⊂ [0 , . For any δ > , we define H s,βδ ( E ) := inf n X k | J k | s : | J k | ≤ δ, E ⊂ [ k J k , { J k } are countable cylinders o . It is increasing as δ ց . We call H s,β ( E ) := lim δ → H s,βδ ( E ) the s -dimension Hausdorffmeasure of E related to the cylinder net of β . Lemma 5.8.
Let β > , s > and E ⊂ [0 , . Then for any < ε < s , H s,β ( E ) ≤ H s − ε ( E ) .Proof. Fix < ε < s .(1) Choose δ > as below.Since β ( n +1) ε → ∞ much faster than β s n → ∞ as n → ∞ , there exists n ∈ N such thatfor any n > n , β s n ≤ β ( n +1) ε . By − log δ log β − → ∞ as δ → + , there exists δ > smallenough such that − log δ log β − > n . Then for any n > − log δ log β − , we will have β s n ≤ β ( n +1) ε .(2) In order to arrive at the conclusion, it suffices to prove for any < δ < δ , H s,ββδ ( E ) ≤H s − εδ ( E ) .Fix < δ < δ . Let { U i } be a δ -cover of E , i.e., < | U i | ≤ δ and E ⊂ ∪ i U i . Then foreach U i , there exists n i ∈ N such that β − n i − < | U i | ≤ β − n i . By Proposition 2.12, U i can becovered by at most n i cylinders I i, , I i, , · · · , I i, n i of order n i . Noting that | I i,j | ≤ β − n i < β | U i | ≤ βδ and E ⊂ [ i n i [ j =1 I i,j , we get H s,ββδ ( E ) ≤ X i n i X j =1 | I i,j | s ≤ X i n i β n i s ( ⋆ ) ≤ X i β ( n i +1)( s − ε ) < X i | U i | s − ε . Taking inf on the right, we conclude that H s,ββδ ( E ) ≤ H s − εδ ( E ) .( ( ⋆ ) is because β ni +1 < | U i | < δ implies n i > − log δ log β − and then n i β s ≤ β ( n i +1) ε .) (cid:3) Lemma 5.9.
Let β > , s ≥ , c > , E ⊂ [0 , be a Borel set and ν be a finite Borelmeasure on [0 , . If lim n →∞ ν ( I n ( x )) | I n ( x ) | s < c for all x ∈ E , then H s,β ( E ) ≥ c − ν ( E ) .Proof. For any δ > , let E δ := { x ∈ E : | I n ( x ) | < δ implies ν ( I n ( x )) < c | I n ( x ) | s } .(1) Prove that when δ ց , E δ ր E as below. (cid:13) If < δ < δ , then obviously E δ ⊃ E δ . (cid:13) It suffices to prove E = S δ> E δ . ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 19 ⊃ follows from E ⊃ E δ , ∀ δ > . ⊂ Let x ∈ E . By lim n →∞ ν ( I n ( x )) | I n ( x ) | s < c , there exists N x ∈ N such that any n > N x willhave ν ( I n ( x )) < c | I n ( x ) | s . Let δ x = | I N x ( x ) | , then | I n ( x ) | < δ x will imply n > N x and ν ( I n ( x )) < c | I n ( x ) | s . Therefore x ∈ E δ x ⊂ S δ> E δ .(2) Fix δ > . Let { J k } k ∈ K be countable cylinders such that | J k | < δ and S k ∈ K J k ⊃ E ⊃ E δ .Let K ′ = { k ∈ K : J k ∩ E δ = ∅ } . For any k ∈ K ′ , there exists x k ∈ J k ∪ E δ . By thedefinition of E δ , we get ν ( J k ) < c | J k | s . So ν ( E δ ) ≤ ν ( [ k ∈ K ′ J k ) ≤ X k ∈ K ′ ν ( J k ) < X k ∈ K ′ c | J k | s ≤ c X k ∈ K | J k | s . Taking inf on the right, we get ν ( E δ ) ≤ c H s,βδ ( E ) ≤ c H s,β ( E ) . Let δ → on the left, by E δ ր E , we conclude that ν ( E ) ≤ c H s,β ( E ) . (cid:3) Hausdorff dimension of some level sets
We apply the Bernoulli-type measures and the modified lower local dimension relatedto β -expansions to give some new results on the Hausdorff dimension of level sets in thissection.For < β ≤ and ≤ p ≤ , consider the following level sets F p := n x ∈ [0 ,
1) : lim n →∞ ♯ { ≤ k ≤ n : ε k ( x, β ) = 0 } n = p o ,F p := n x ∈ [0 ,
1) : lim n →∞ ♯ { ≤ k ≤ n : ε k ( x, β ) = 0 } n = p o ,F p := n x ∈ [0 ,
1) : lim n →∞ ♯ { ≤ k ≤ n : ε k ( x, β ) = 0 } n = p o . Obviously, F p = F p ∩ F p . Theorem 6.1 (Upper bound of the Hausdorff dimension of level sets) . Let < β ≤ and ≤ p ≤ . Then dim H F p ≤ min { dim H F p , dim H F p } ≤ max { dim H F p , dim H F p } ≤ − p log p − (1 − p ) log(1 − p )log β . In particular, dim H F = dim H F = dim H F = dim H F = dim H F = dim H F = 0 .Proof. First, we consider < p < .For any x ∈ [0 , and n ∈ N , it follows from ν p ( I n ( x )) = p N ( x,n ) (1 − p ) N ( x,n ) that − log ν p ( I n ( x )) = N ( x, n )( − log p ) + N ( x, n )( − log(1 − p )) ≤ ( n − N ( x, n ))( − log p ) + N ( x, n )( − log(1 − p )) . By | I n ( x ) | ≤ β − n , we get − log ν p ( I n ( x )) − log | I n ( x ) | ≤ (1 − N ( x,n ) n )( − log p ) + N ( x,n ) n ( − log(1 − p ))log β . (6.1) (1) For any x ∈ F p , it follows from lim n →∞ (1 − N ( x,n ) n ) = p and lim n →∞ N ( x,n ) n = 1 − p that lim n →∞ log ν p ( I n ( x ))log | I n ( x ) | ≤ lim n →∞ (1 − N ( x,n ) n )( − log p ) + lim n →∞ N ( x,n ) n ( − log(1 − p ))log β = − p log p − (1 − p ) log(1 − p )log β . By Theorem 5.5 (1), we get dim H F p ≤ − p log p − (1 − p ) log(1 − p )log β . (2) For any x ∈ F p , it follows from lim n →∞ (1 − N ( x,n ) n ) = p and lim n →∞ N ( x,n ) n = 1 − p that lim n →∞ log ν p ( I n ( x ))log | I n ( x ) | ≤ lim n →∞ (1 − N ( x,n ) n )( − log p ) + lim n →∞ N ( x,n ) n ( − log(1 − p ))log β = − p log p − (1 − p ) log(1 − p )log β . By Theorem 5.5 (1), we get dim H F p ≤ − p log p − (1 − p ) log(1 − p )log β . Therefore, by F p = F p ∩ F p , we get dim H F p ≤ min { dim H F p , dim H F p } ≤ max { dim H F p , dim H F p } ≤ − p log p − (1 − p ) log(1 − p )log β . Before proving dim H F = dim H F = dim H F = dim H F = dim H F = dim H F = 0 ,we establish the following. Lemma 6.2.
Let < β ≤ and < p < .(1) Let F ≤ p := n x ∈ [0 ,
1) : lim n →∞ ♯ { ≤ k ≤ n : ε k ( x, β ) = 0 } n ≤ p o . Then dim H F ≤ p ≤ − p log p − log(1 − p )log β . (2) Let F ≥ p := n x ∈ [0 ,
1) : lim n →∞ ♯ { ≤ k ≤ n : ε k ( x, β ) = 0 } n ≥ p o . Then dim H F ≥ p ≤ − log p − (1 − p ) log(1 − p )log β . Proof. (1) For any x ∈ F ≤ p , it follows from (6.1), lim n →∞ (1 − N ( x,n ) n ) ≤ p and N ( x,n ) n ≤ ∀ n ∈ N ) that lim n →∞ log ν p ( I n ( x ))log | I n ( x ) | ≤ − p log p − log(1 − p )log β . ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 21
By Theorem 5.5 (1), we get dim H F ≤ p ≤ − p log p − log(1 − p )log β . (2) For any x ∈ F ≥ p , it follows from (6.1), lim n →∞ N ( x,n ) n ≤ − p and − N ( x,n ) n ≤ ∀ n ∈ N ) that lim n →∞ log ν p ( I n ( x ))log | I n ( x ) | ≤ − log p − (1 − p ) log(1 − p )log β . By Theorem 5.5 (1), we get dim H F ≥ p ≤ − log p − (1 − p ) log(1 − p )log β . (cid:3) Now we prove dim H F = dim H F = dim H F = dim H F = dim H F = dim H F = 0 .(1) For any < p < , F = F ⊂ F ⊂ F ≤ p implies dim H F = dim H F ≤ dim H F ≤ dim H F ≤ p . Let p → , by Lemma 6.2 (1), we get dim H F = dim H F = dim H F = 0 .(2) For any < p < , F = F ⊂ F ⊂ F ≥ p implies dim H F = dim H F ≤ dim H F ≤ dim H F ≥ p . Let p → , by Lemma 6.2 (2), we get dim H F = dim H F = dim H F = 0 . (cid:3) We give the Hausdorff dimensions of these three kinds of level sets for a class of β . Theorem 6.3.
Let < β < , m ∈ N ≥ such that ε (1 , β ) = 10 m ∞ .(1) If ≤ p < m +1 m +2 , then F p = F p = F p = ∅ and dim H F p = dim H F p = dim H F p = 0 .(2) If m +1 m +2 ≤ p ≤ , then dim H F p = dim H F p = dim H F p = ( mp − m + p ) log( mp − m + p ) − ( mp − m + 2 p −
1) log( mp − m + 2 p − − (1 − p ) log(1 − p )log β . In particular, dim H F m +1 m +2 = dim H F m +1 m +2 = dim H F m +1 m +2 = dim H F = dim H F = dim H F =0 .Remark . Take m = 0 in Theorem 6.3. We get the well-known result (see for example[9]) dim H F p = p log p − (2 p −
1) log(2 p − − (1 − p ) log(1 − p )log β where β = √ is the golden ratio and ≤ p ≤ . Proof of Theorem 6.3. (1) For any x ∈ [0 , , by Lemma 2.3, each digit in ε ( x, β ) must be followed by at least ( m + 1) consecutive s. Thus lim n →∞ N ( x, n ) n ≤ m + 2 and then lim n →∞ ♯ { ≤ k ≤ n : ε k ( x, β ) = 0 } n ≥ m + 1 m + 2 for any x ∈ [0 , . If ≤ p < m +1 m +2 , we get F p = F p = F p = ∅ .(2) (cid:13) First, we consider m +1 m +2 < p < .For any x ∈ [1 , and n ∈ N , by Proposition 2.11, we get n log β − log c ≤ − log | I n ( x ) | ≤ n log β . Let q := mp − m +2 p − mp − m + p . Then < q < since m +1 m +2 < p < . Let ν q be the ( q, − q ) Bernoullimeasure on [0 , . It follows from − log ν q ( I n ( x )) = N ( x, n )( − log q ) + N ( x, n )( − log(1 − q )) that N ( x,n ) n ( − log q ) + N ( x,n ) n ( − log(1 − q ))log β − log cn ≤ log ν q ( I n ( x ))log | I n ( x ) | ≤ N ( x,n ) n ( − log q ) + N ( x,n ) n ( − log(1 − q ))log β . (6.2)Taking lim n →∞ , we get dim βloc ν q ( x ) = lim n →∞ N ( x,n ) n ( − log q ) + N ( x,n ) n ( − log(1 − q ))log β . i) Prove dim H F p ≤ (1 − ( m +2)(1 − p ))( − log q )+(1 − p )( − log(1 − q ))log β .For any x ∈ F p , we have lim n →∞ N ( x,n ) n = 1 − p and then by Lemma 6.5, lim n →∞ N ( x,n ) n =1 − ( m + 2)(1 − p ) . Thus dim βloc ν q ( x ) ≤ lim n →∞ N ( x,n ) n ( − log q ) + lim n →∞ N ( x,n ) n ( − log(1 − q ))log β = (1 − ( m + 2)(1 − p ))( − log q ) + (1 − p )( − log(1 − q ))log β . Then we apply Theorem 5.5 (1).ii) Prove dim H F p ≤ (1 − ( m +2)(1 − p ))( − log q )+(1 − p )( − log(1 − q ))log β .For any x ∈ F p , we have lim n →∞ N ( x,n ) n = 1 − p and then by Lemma 6.5, lim n →∞ N ( x,n ) n =1 − ( m + 2)(1 − p ) . Thus dim βloc ν q ( x ) ≤ lim n →∞ N ( x,n ) n ( − log q ) + lim n →∞ N ( x,n ) n ( − log(1 − q ))log β = (1 − ( m + 2)(1 − p ))( − log q ) + (1 − p )( − log(1 − q ))log β . Then we apply Theorem 5.5 (1).iii) Prove dim H F p ≥ (1 − ( m +2)(1 − p ))( − log q )+(1 − p )( − log(1 − q ))log β .For any x ∈ F p , we have lim n →∞ N ( x,n ) n = 1 − p and then by Lemma 6.5, lim n →∞ N ( x,n ) n =1 − ( m + 2)(1 − p ) . Thus dim βloc ν q ( x ) = (1 − ( m + 2)(1 − p ))( − log q ) + (1 − p )( − log(1 − q ))log β . By Theorem 5.5 (2), it suffices to prove ν q ( F p ) = 1 > .By ε k ( x, β ) = 0 ⇔ ⌊ βT k − β x ⌋ = 0 ⇔ ≤ T k − β x ≤ β ⇔ [0 , β ) ( T k − β x ) = 1 , we get n ♯ { ≤ k ≤ n : ε k ( x, β ) = 0 } = 1 n n X k =1 [0 , β ) ( T k − β x ) . ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 23
Since ([0 , , B [0 , , m q , T β ) is ergodic and the indicator function [0 , β ) is m q -integrable,it follows from the Birkhoff Ergodic Theorem that lim n →∞ n n X k =1 [0 , β ) ( T k − β x ) = ˆ [0 , β ) dm q = m q [0 , β ) by ======== Lemma 6.6 m (1 − q ) + 1( m + 1)(1 − q ) + 1 by the ====== def. of q p for m q - a.e. x ∈ [0 , . Therefore m q ( F p ) = 1 . By m q ∼ ν q , we get ν q ( F p ) = 1 > .Combining i), ii) iii) and F p = F p ∩ F p , we get dim H F p = dim H F p = dim H F p = (1 − ( m + 2)(1 − p ))( − log q ) + (1 − p )( − log(1 − q ))log β . We draw the conclusion by q = mp − m +2 p − mp − m + p . (cid:13) For p = 1 , it follows from Theorem 6.1 that dim H F = dim H F = dim H F = 0 . (cid:13) Prove dim H F m +1 m +2 = dim H F m +1 m +2 = dim H F m +1 m +2 = 0 .By lim n →∞ ♯ { ≤ k ≤ n : ε k ( x,β )=0 } n ≥ m +1 m +2 for any x ∈ [0 , in (1), we get F m +1 m +2 = F m +1 m +2 . Since F m +1 m +2 ⊂ F m +1 m +2 , it suffices to prove dim F m +1 m +2 = 0 .For m +1 m +2 < p < , let q := mp − m +2 p − mp − m + p . Then < q < . For any x ∈ F ≤ p (see Lemma 6.2(1) for definition), we have lim n →∞ N ( x,n ) n ≥ − p and then by Lemma 6.5, lim n →∞ N ( x,n ) n ≤ − ( m + 2)(1 − p ) . It follows from N ( x,n ) n ≤ ∀ n ∈ N ) and (6.2) that lim n →∞ log ν q ( I n ( x ))log | I n ( x ) | ≤ − (1 − ( m + 2)(1 − p )) log q + log(1 − q )log β for any x ∈ F ≤ p . By Theorem 5.5 (1) and the definition of q , we get dim H F ≤ p ≤ − ( mp − m + 2 p −
1) log( mp − m + 2 p − − ( mp − m + 2 p −
1) log( mp − m + p ) + log(1 − q )log β . For any m +1 m +2 < p < , F m +1 m +2 ⊂ F ≤ p implies dim H F m +1 m +2 ≤ dim H F ≤ p . Let p → m +1 m +2 , then q → and we get dim H F m +1 m +2 = 0 . (cid:3) Lemma 6.5.
Let < β < and m ∈ N ≥ such that ε (1 , β ) = 10 m ∞ . Then for any x ∈ [0 , and n ≥ m + 2 , we have n ≤ N ( x, n ) + ( m + 2) N ( x, n ) ≤ n + m + 1 .Proof. Let w ∈ Σ nβ . It suffices to prove n (1) ≤ N ( w ) + ( m + 2) N ( w ) (2) ≤ n + m + 1 .(1) Write N ( w ) := { ≤ k ≤ n : w k − w k = 10 } , N ( w ) := { ≤ k ≤ n : w k − w k − w k = 100 } , · · · , N m +1 ( w ) := { m + 2 ≤ k ≤ n : w k − m − · · · w k = 10 m +1 } and let N ( w ) := ♯ N ( w ) , N ( w ) := ♯ N ( w ) , · · · , N m +1 ( w ) := ♯ N m +1 ( w ) . Noting that by Proposition 2.10, u m +1 is full for any u ∈ Σ ∗ β and then u m +1 is admissible,we get { ≤ k ≤ n : w k = 0 } = ( N ( w ) + 1) ∪ N ( w ) ∪ N ( w ) ∪ · · · ∪ N m +1 which is a disjoint union. Thus ♯ { ≤ k ≤ n : w k = 0 } = N ( w ) + N ( w ) + N ( w ) + · · · + N m +1 ( w ) and then n = N ( w ) + N ( w ) + N ( w ) + · · · + N m +1 ( w ) + N ( w ) . By N ( w ) , N ( w ) , · · · , N m +1 ( w ) ≤ N ( w ) , we get n ≤ N ( w ) + ( m + 2) N ( w ) .(2) If N ( w ) = 0 , the conclusion is obvious. If N ( w ) ≥ , except for the last digit in w ,by Lemma 2.3, the other s must be followed by at least (m+1) consecutive s, and non ofthese s can be replaced by to get an admissible word. Therefore N ( w ) + ( m + 1)( N ( w ) −
1) + N ( w ) ≤ n, i.e., N ( w ) + ( m + 2) N ( w ) ≤ n + m + 1 . (cid:3) Lemma 6.6.
Let < β < , m ∈ N ≥ such that ε (1 , β ) = 10 m ∞ and < p < . Then m p [0 , β ) = m (1 − p ) + 1( m + 1)(1 − p ) + 1 where m p is given by Theorem 4.9.Proof. Notice that m p [0 , β ) = 1 − m p [ β , where m p [ 1 β ,
1) = lim n →∞ n n − X k =0 ν p T − kβ [ 1 β ,
1) = lim n →∞ n n − X k =0 µ p σ − kβ [1] by Theorem 4.9. For any k ∈ N ≥ , let a k := µ p σ − kβ [1] = X u ··· u k ∈ Σ ∗ β µ p [ u · · · u k and b k := µ p σ − kβ [0 m +1 ] = X u ··· u k m +1 ∈ Σ ∗ β µ p [ u · · · u k m +1 ] . By Theorem 4.9, the limits a := lim n →∞ n n − X k =0 a k and b := lim n →∞ n n − X k =0 b k exist.(1) Prove a = (1 − p ) b . Write b k +1 = X u ··· u k m +1 ∈ Σ ∗ β µ p [ u · · · u k m +1 ] + X u ··· u k m +1 ∈ Σ ∗ β µ p [ u · · · u k m +1 ]= X u ··· u k m +1 ∈ Σ ∗ β µ p [ u · · · u k m +1
0] + X u ··· u k ∈ Σ ∗ β µ p [ u · · · u k m +1 ] . On the one hand, by Proposition 2.10, u · · · u k m +1 is full and then u · · · u k m +1 ∈ Σ ∗ β .On the other hand, by Lemma 2.3, for any ≤ s ≤ m , u · · · u k s m − s / ∈ Σ ∗ β and then [ u · · · u k m +1 ] = [ u · · · u k . Thus, it follows from the definition of µ p that b k +1 = p X u ··· u k m +1 ∈ Σ ∗ β µ p [ u · · · u k m +1 ] + X u ··· u k ∈ Σ ∗ β µ p [ u · · · u k
1] = pb k + a k . Let n → ∞ in n n − X k =0 b k +1 = p · n n − X k =0 b k + 1 n n − X k =0 a k . ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 25
We get b = pb + a .(2) Prove b + ( m + 1) a = 1 . It follows from (cid:16) [ u ··· u k m +1 ∈ Σ ∗ β [ u · · · u k m +1 ] (cid:17) ∪ (cid:16) [ u ··· u k ∈ Σ ∗ β [ u · · · u k (cid:17) ∪ (cid:16) [ u ··· u k +1 ∈ Σ ∗ β [ u · · · u k +1 (cid:17) ∪ · · · ∪ (cid:16) [ u ··· u k + m ∈ Σ ∗ β [ u · · · u k + m (cid:17) = (cid:16) [ u ··· u k m +1 ∈ Σ ∗ β [ u · · · u k m +1 ] (cid:17) ∪ (cid:16) [ u ··· u k m ∈ Σ ∗ β [ u · · · u k m ] (cid:17) ∪ (cid:16) [ u ··· u k +1 m − ∈ Σ ∗ β [ u · · · u k +1 m − ] (cid:17) ∪ · · · ∪ (cid:16) [ u ··· u k + m ∈ Σ ∗ β [ u · · · u k + m (cid:17) = Σ β that b k + a k + a k +1 + · · · + a k + m = 1 . Let n → ∞ in n n − X k =0 b k + 1 n n − X k =0 a k + 1 n n − X k =0 a k +1 + · · · + 1 n n − X k =0 a k + m = 1 . We get b + a + a + · · · + a = 1 .(3) It follows from (1) and (2) that a = − p ( m +1)(1 − p )+1 . Therefore m p [0 , β ) = 1 − a = m (1 − p ) + 1( m + 1)(1 − p ) + 1 . (cid:3) Proofs of the examples
Let M σ (Σ β ) be the set of σ -invariant probability Borel measure on (Σ β , B (Σ β )) and M T β ([0 , be the set of T β -invariant probability Borel measure on ([0 , , B [0 , . Weneed the following. Definition 7.1 ( k -step Markov measure) . Let k ∈ N and µ ∈ M σ (Σ β ) . We call µ a k -step Markov measure if there exists an × k probability vector p = ( p ( i ··· i k ) ) i , ··· ,i k =0 , (i.e., P i , ··· ,i k =0 , p ( i ··· i k ) = 1 and p ( i ··· i k ) ≥ for all i , · · · , i k ∈ { , } ) and a k × k stochastic matrix P = ( P ( i ··· i k )( j ··· j k ) ) i , ··· ,i k ,j , ··· ,j k =0 , (i.e., P j , ··· ,j k =0 , P ( i ··· i k )( j ··· j k ) = 1 forall i , · · · , i k ∈ { , } and P ( i ··· i k )( j ··· j k ) ≥ for all i , · · · , i k , j , · · · , j k ∈ { , } ) with pP = p such that µ [ i · · · i k ] = p ( i ··· i k ) for all i , · · · , i k ∈ { , } and µ [ i · · · i n ] = p ( i ··· i k ) P ( i ··· i k )( i ··· i k +1 ) P ( i ··· i k +1 )( i ··· i k +2 ) · · · P ( i n − k ··· i n − )( i n − k +1 ··· i n ) for all i , · · · , i n ∈ { , } and n > k .We prove the following useful lemma for self-contained (see also [10, Observation 6.2.7]). Lemma 7.2.
Let k ≥ and µ ∈ M σ (Σ β ) . If µ [ w · · · w n + k +1 ] µ [ w · · · w n + k ] = µ [ w n +1 · · · w n + k +1 ] µ [ w n +1 · · · w n + k ] (7.1) for all w · · · w n + k +1 ∈ Σ n + k +1 β and n ≥ , then µ is a k -step Markov measure. Proof.
For any i , · · · , i k ∈ { , } , let p ( i ··· i k ) := µ [ i · · · i k ] . Then p = ( p ( i ··· i k ) ) i , ··· ,i k =0 , is a × k probability vector. We define a k × k stochastic matrix P = ( P ( i ··· i k )( j ··· j k +1 ) ) i , ··· ,i k ,j , ··· ,j k +1 =0 , as follows.i) If there exists integer t with ≤ t ≤ k such that i t = j t , let P ( i i ··· i k )( j ··· j k j k +1 ) := 0; ii) If µ [ i · · · i k ] = 0 , let P ( i ··· i k )( i ··· i k +1 ) := µ [ i · · · i k +1 ] µ [ i · · · i k ] for i k +1 = 0 , iii) If µ [ i · · · i k ] = 0 , let P ( i ··· i k )( i ··· i k := 1 and P ( i ··· i k )( i ··· i k := 0 . Then P j , ··· ,j k +1 =0 , P ( i ··· i k )( j ··· j k +1 ) = 1 for all i , · · · , i k ∈ { , } and pP = p . Since for all s ≥ and i , · · · , i s + k ∈ { , } we have µ [ i · · · i k + s ] = µ [ i · · · i k ] µ [ i · · · i k +1 ] µ [ i · · · i k ] µ [ i i · · · i k +2 ] µ [ i i · · · i k +1 ] · · · µ [ i · · · i s · · · i s + k ] µ [ i · · · i s · · · i s + k − ] by ===== ( . ) µ [ i · · · i k ] µ [ i · · · i k +1 ] µ [ i · · · i k ] µ [ i · · · i k +2 ] µ [ i · · · i k +1 ] · · · µ [ i s · · · i s + k ] µ [ i s · · · i s + k − ]= p ( i ··· i k ) P ( i ··· i k )( i ··· i k +1 ) P ( i ··· i k +1 )( i ··· i k +2 ) · · · P ( i s ··· i s + k − )( i s +1 ··· i s + k ) , by definition we know that µ is a k -step Markov measure. (cid:3) Proof of Example 1.3.
Let p ∈ (0 , and λ := lim n →∞ n P n − k =0 µ p ◦ σ − k . Then λ is σ -invariantand m p = λ ◦ π − β . It suffices to prove h λ ( σ ) = sup n h µ ( σ ) : µ ∈ M σ (Σ β ) and µ [0] = λ [0] o . Let a := λ [0] . By [11, Theorem 1.2 and Proposition 4.2], it suffices to prove that λ is aunique ( m + 1) -step Markov measure (see [10, 11] for definition) in M σ (Σ β ) taking value a on [0] .(1) Prove the uniqueness. Noting that k / ∈ Σ ∗ β for all ≤ k ≤ m, (7.2)we get σ − i [1] = [0 i for all ≤ i ≤ m + 1 . Let µ ∈ M σ (Σ β ) with µ [0] = a . Then we have µ [0 i
1] = µ [1] = 1 − a for all ≤ i ≤ m + 1 . For i, j ∈ { , , · · · , m + 1 } , by (7.2) we get [0 i j ] = [0 i . Thus µ [0 i j ] = µ [0 i
1] = 1 − a for all ≤ i, j ≤ m + 1 . For k ∈ { , · · · , m + 2 } , also by (7.2) we get Σ β = [0 k ] ∪ S k − i =0 [0 i k − i − ] , which implies µ [0 k ] = 1 − k − X i =0 µ [0 i k − i − ] = 1 − k (1 − a ) = ka − k + 1 . The above calculation means that all the measures in M σ (Σ β ) taking value a on [0] arethe same on all the cylinders with order no larger than m + 2 . Since ( m + 1) -step Markov ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 27 measures only depend on their values on the cylinders with order no larger than m + 2 , theuniqueness of λ follows.(2) Prove that λ is an ( m + 1) -step Markov measure. Let k := m + 1 . By Lemma 7.2, itsuffices to check (7.1). (cid:13) For any n ≥ and w · · · w n + k +1 ∈ Σ n + k +1 β , prove µ p [ w · · · w n + k +1 ] µ p [ w · · · w n + k ] = µ p [ w n +1 · · · w n + k +1 ] µ p [ w n +1 · · · w n + k ] . In fact, this follows from p N ( w ··· w n + k +1 ) · (1 − p ) N ( w ··· w n + k +1 ) p N ( w ··· w n + k ) · (1 − p ) N ( w ··· w n + k ) = p N ( w ··· w n + k +1 ) − N ( w ··· w n + k ) · (1 − p ) N ( w n + k +1 )( ⋆ ) = p N ( w n +1 ··· w n + k +1 ) − N ( w n +1 ··· w n + k ) · (1 − p ) N ( w n + k +1 ) = p N ( w n +1 ··· w n + k +1 ) · (1 − p ) N ( w n +1 ··· w n + k +1 ) p N ( w n +1 ··· w n + k ) · (1 − p ) N ( w n +1 ··· w n + k ) , where ( ⋆ ) can be proved as follows. If w n + k +1 = 1 , then ( ⋆ ) is obviously true. If w n + k +1 = 0 ,then N ( w · · · w n + k +1 ) − N ( w · · · w n + k ) = (cid:26) if w · · · w n + k ∈ Σ ∗ β if w · · · w n + k / ∈ Σ ∗ β and N ( w n +1 · · · w n + k +1 ) − N ( w n +1 · · · w n + k ) = (cid:26) if w n +1 · · · w n + k ∈ Σ ∗ β if w n +1 · · · w n + k / ∈ Σ ∗ β . By w · · · w n + k ∈ Σ ∗ β and ε (1 , β ) = 10 k − ∞ , we know w · · · w n + k ∈ Σ ∗ β ⇔ w n +1 · · · w n + k = 0 k ⇔ w n +1 · · · w n + k ∈ Σ ∗ β . Thus N ( w · · · w n + k +1 ) − N ( w · · · w n + k ) = N ( w n +1 · · · w n + k +1 ) − N ( w n +1 · · · w n + k ) . (cid:13) For any n ≥ and w · · · w n + k +1 ∈ Σ n + k +1 β , prove µ p ◦ σ − [ w · · · w n + k +1 ] µ p ◦ σ − [ w · · · w n + k ] = µ p ◦ σ − [ w n +1 · · · w n + k +1 ] µ p ◦ σ − [ w n +1 · · · w n + k ] = µ p [ w n +1 · · · w n + k +1 ] µ p [ w n +1 · · · w n + k ] . By w · · · w n + k +1 ∈ Σ ∗ β and ε (1 , β ) = 10 k − ∞ , we get w · · · w n + k +1 ∈ Σ ∗ β ⇔ w · · · w k = 0 k ⇔ w · · · w n + k ∈ Σ ∗ β , which implies µ p ◦ σ − [ w · · · w n + k +1 ] µ p ◦ σ − [ w · · · w n + k ] = ( µ p [0 w ··· w n + k +1 ]+ µ p [1 w ··· w n + k +1 ] µ p [0 w ··· w n + k ]+ µ p [1 w ··· w n + k ] if w · · · w n + k +1 ∈ Σ ∗ βµ p [0 w ··· w n + k +1 ] µ p [0 w ··· w n + k ] if w · · · w n + k +1 / ∈ Σ ∗ β by (cid:13) ===== µ p [ w n +1 · · · w n + k +1 ] µ p [ w n +1 · · · w n + k ] . By w · · · w n + k +1 ∈ Σ ∗ β and ε (1 , β ) = 10 k − ∞ , we get w n +1 · · · w n + k +1 ∈ Σ ∗ β ⇔ w n +1 · · · w n + k = 0 k ⇔ w n +1 · · · w n + k ∈ Σ ∗ β , which implies µ p ◦ σ − [ w n +1 · · · w n + k +1 ] µ p ◦ σ − [ w n +1 · · · w n + k ] = ( µ p [0 w n +1 ··· w n + k +1 ]+ µ p [1 w n +1 ··· w n + k +1 ] µ p [0 w n +1 ··· w n + k ]+ µ p [1 w n +1 ··· w n + k ] if w n +1 · · · w n + k +1 ∈ Σ ∗ βµ p [0 w n +1 ··· w n + k +1 ] µ p [0 w n +1 ··· w n + k ] if w n +1 · · · w n + k +1 / ∈ Σ ∗ β by (cid:13) ===== µ p [ w n +1 · · · w n + k +1 ] µ p [ w n +1 · · · w n + k ] . (cid:13) Repeat the above process. By induction, we can get that for any j ≥ , n ≥ and w · · · w n + k +1 ∈ Σ n + k +1 β , we have µ p ◦ σ − j [ w · · · w n + k +1 ] µ p ◦ σ − j [ w · · · w n + k ] = µ p ◦ σ − j [ w n +1 · · · w n + k +1 ] µ p ◦ σ − j [ w n +1 · · · w n + k ] = µ p [ w n +1 · · · w n + k +1 ] µ p [ w n +1 · · · w n + k ] , and then λ [ w · · · w n + k +1 ] λ [ w · · · w n + k ] = lim s →∞ s P s − j =0 µ p ◦ σ − j [ w · · · w n + k +1 ]lim s →∞ s P s − j =0 µ p ◦ σ − j [ w · · · w n + k ]= lim s →∞ P s − j =0 µ p ◦ σ − j [ w · · · w n + k +1 ] P s − j =0 µ p ◦ σ − j [ w · · · w n + k ]= µ p [ w n +1 · · · w n + k +1 ] µ p [ w n +1 · · · w n + k ]= lim s →∞ P s − j =0 µ p ◦ σ − j [ w n +1 · · · w n + k +1 ] P s − j =0 µ p ◦ σ − j [ w n +1 · · · w n + k ]= lim s →∞ s P s − j =0 µ p ◦ σ − j [ w n +1 · · · w n + k +1 ]lim s →∞ s P s − j =0 µ p ◦ σ − j [ w n +1 · · · w n + k ] = λ [ w n +1 · · · w n + k +1 ] λ [ w n +1 · · · w n + k ] . Therefore λ satisfies (7.1). (cid:3) Proof of Example 1.4.
Let p ∈ (0 , and λ := lim n →∞ n P n − k =0 µ p ◦ σ − k . Then λ is σ -invariantand m p = λ ◦ π − β . It suffices to prove h λ ( σ ) < sup n h µ ( σ ) : µ ∈ M σ (Σ β ) and µ [0] = λ [0] o . By the fact that P := { [0] , [1] } is a partition generator of B (Σ β ) , we know h λ ( σ ) = h λ ( σ, P ) .Since H λ ( P (cid:12)(cid:12) W nk =1 σ − k P ) decreases as n increases, by [17, Theorem 4.14] we get h λ ( σ ) ≤ H λ (cid:16) P (cid:12)(cid:12)(cid:12) _ k =1 σ − k P (cid:17) ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 29 where H λ (cid:16) P (cid:12)(cid:12)(cid:12) σ − P _ σ − P (cid:17) = H λ (cid:16) P (cid:12)(cid:12)(cid:12) σ − (cid:0) P _ σ − P (cid:1)(cid:17) = − X P ∈P ,Q ∈P W σ − P λ ( P ∩ σ − Q ) log λ ( P ∩ σ − Q ) λ ( σ − Q )= − X i ,i ,i ∈{ , } λ [ i i i ] log λ [ i i i ] λ ( σ − [ i i ])= X i ,i ,i ∈{ , } λ [ i i i ] log λ [ i i ] − X i ,i ,i ∈{ , } λ [ i i i ] log λ [ i i i ]= X i ,i ∈{ , } λ [ i i ] log λ [ i i ] − X i ,i ,i ∈{ , } λ [ i i i ] log λ [ i i i ]= X i ,i ∈{ , } λ [ i i ] log λ [ i i ] − X i ,i ,i ∈{ , } λ [ i i i ] log λ [ i i i ]= X i ,i ,i ∈{ , } λ [ i i i ] log λ [ i i ] − X i ,i ,i ∈{ , } λ [ i i i ] log λ [ i i i ]= − X i ,i ,i ∈{ , } λ [ i i i ] log λ [ i i i ] λ [ i i ] , where is regarded as . It follows from [111] = ∅ that [110] = [11] and h λ ( σ ) ≤ − X i ,i ,i ∈{ , } i i =11 λ [ i i i ] log λ [ i i i ] λ [ i i ]= − X i ,i ∈{ , } λ [ i i ] log λ [ i i ] λ [ i − λ [010] log λ [010] λ [01] − λ [011] log λ [011] λ [01] . For i ∈ { , } , we have − λ [00 i ] log λ [00 i ] λ [00] − λ [10 i ] log λ [10 i ] λ [10]= λ [0] (cid:16) λ [00] λ [0] (cid:0) − λ [00 i ] λ [00] log λ [00 i ] λ [00] (cid:1) + λ [10] λ [0] (cid:0) − λ [10 i ] λ [10] log λ [10 i ] λ [10] (cid:1)(cid:17) ≤ − λ [0 i ] log λ [0 i ] λ [0] where the last inequality follows from Lemma 7.3. Thus h λ ( σ ) ≤ − λ [00] log λ [00] λ [0] − λ [01] log λ [01] λ [0] − λ [010] log λ [010] λ [01] − λ [011] log λ [011] λ [01] . (7.3)Since λ is a σ -invariant probability measure, we have λ [0] + λ [1] = 1 , λ [00] + λ [01] = λ [0] , λ [01] + λ [11] = λ [1] , λ [010] + λ [011] = λ [01] and λ [011] + λ [111] = λ [11] where λ [111] = 0 .Let a := λ [0] and b := λ [01] . Then by a simple calculation we get λ [00] = a − b, λ [010] = 2 b + a − and λ [011] = 1 − a − b. It follows from (7.3) that h λ ( σ ) ≤ a log a − ( a − b ) log( a − b ) − (1 − a − b ) log(1 − a − b ) − (2 b + a −
1) log(2 b + a − . By Lemma 7.4, we know a = p − (1 − p ) ≥ . For x ∈ [ − a , min { a, − a } ] , we define thefunction f a ( x ) := a log a − ( a − x ) log( a − x ) − (1 − a − x ) log(1 − a − x ) − (2 x + a −
1) log(2 x + a − . Then h λ ( σ ) ≤ f a ( b ) . By calculating the derivative, it is straightforward to see that f a isstrictly increasing on h − a , − a + √− a + 12 a − i and strictly decreasing on h − a + √− a + 12 a − , min { a, − a } i . By Lemma 7.4, it is not difficult to check b = − a + √− a +12 a − . Thus h λ ( σ ) < max f a ( x ) .By [11, Proposition 4.2 and Remark 1.4], we have max f a ( x ) = sup n h µ ( σ ) : µ ∈ M σ (Σ β ) , µ [0] = a o . Therefore h λ ( σ ) < sup n h µ ( σ ) : µ ∈ M σ (Σ β ) and µ [0] = a o . (cid:3) The following lemma follows immediately from the convexity of the function x log x . Lemma 7.3.
Let ϕ : [0 , ∞ ) → R be defined by ϕ ( x ) = (cid:26) if x = 0; − x log x if x > . Then for all x, y ∈ [0 , ∞ ) and a, b ≥ with a + b = 1 , aϕ ( x ) + bϕ ( y ) ≤ ϕ ( ax + by ) . The equality holds if and only if x = y , a = 0 or b = 0 . Lemma 7.4.
Let β ∈ (1 , be a pseudo-golden ratio, i.e., ε (1 , β ) = 1 m ∞ for some m ∈ N ≥ and < p < . Then m p [0 , β ) = p − (1 − p ) m and m p [ 1 β + · · · + 1 β m − ,
1) = p (1 − p ) m − − (1 − p ) m . Proof. (1) By Theorem 4.9, we get m p [0 , β ) = lim n →∞ n n − X k =0 ν p T − kβ [0 , β ) = lim n →∞ n n − X k =0 µ p σ − kβ [0] . For any k ≥ , it follows from Σ ∗ β = { w ∈ ∞ [ n =1 { , } n : 1 m does not appear in w } ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 31 that Σ β = [ u ··· u k + m ∈ Σ ∗ β [ u · · · u k + m ]= (cid:16) [ u ··· u k + m − ∈ Σ ∗ β [ u · · · u k + m − (cid:17) ∪ (cid:16) [ u ··· u k + m − ∈ Σ ∗ β [ u · · · u k + m − (cid:17) ∪ (cid:16) [ u ··· u k + m − ∈ Σ ∗ β [ u · · · u k + m − ] (cid:17) ∪ · · · ∪ (cid:16) [ u ··· u k ∈ Σ ∗ β [ u · · · u k m − ] (cid:17) and then µ p σ − ( k + m − β [0] + X u ··· u k + m − ∈ Σ ∗ β µ p [ u · · · u k + m − X u ··· u k + m − ∈ Σ ∗ β µ p [ u · · · u k + m − ] + · · · + X u ··· u k ∈ Σ ∗ β µ p [ u · · · u k m − ]= µ p σ − ( k + m − β [0] + (1 − p ) X u ··· u k + m − ∈ Σ ∗ β µ p [ u · · · u k + m − − p ) X u ··· u k + m − ∈ Σ ∗ β µ p [ u · · · u k + m −
0] + · · · + (1 − p ) m − X u ··· u k ∈ Σ ∗ β µ p [ u · · · u k ]= µ p σ − ( k + m − β [0] + (1 − p ) µ p σ − ( k + m − β [0] + · · · + (1 − p ) m − µ p σ − kβ [0] Thus n n − X k =0 µ p σ − ( k + m − β [0] + (1 − p ) 1 n n − X k =0 µ p σ − ( k + m − β [0] + · · · + (1 − p ) m − n n − X k =0 µ p σ − kβ [0] = 1 . Taking n → ∞ , we get m p [0 , β ) + (1 − p ) m p [0 , β ) + · · · + (1 − p ) m − m p [0 , β ) = 1 . Therefore m p [0 , β ) = p − (1 − p ) m .(2) By Theorem 4.9, we get m p [ 1 β + · · · + 1 β m − ,
1) = lim n →∞ n n − X k =0 ν p T − kβ [ 1 β + · · · + 1 β m − ,
1) = lim n →∞ n n − X k =0 µ p σ − kβ [1 m − ] . For any k ≥ , it follows from σ − ( k +1) β [1 m − ] = [ u ··· u k u k +1 m − ∈ Σ ∗ β [ u · · · u k u k +1 m − ] = [ u ··· u k ∈ Σ ∗ β [ u · · · u k m − ] that µ p σ − ( k +1) β [1 m − ] = X u ··· u k ∈ Σ ∗ β µ p [ u · · · u k m − ]= (1 − p ) m − X u ··· u k ∈ Σ ∗ β µ p [ u · · · u k − p ) m − µ p σ − kβ [0] . Thus n n − X k =0 µ p σ − ( k +1) β [1 m − ] = 1 n n − X k =0 (1 − p ) m − µ p σ − kβ [0] . Taking n → ∞ , we get m p [ 1 β + · · · + 1 β m − ,
1) = (1 − p ) m − m p [0 , β ) = p (1 − p ) m − − (1 − p ) m . (cid:3) Acknowledgement.
The first and the third author thank Tony Samuel for useful discussions.The first author was supported by NSFC 11671151 and Guangdong Natural Science Foun-dation 2018B0303110005. The second author is grateful to the Oversea Study Program ofGuangzhou Elite Project.
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School of Mathematics, South China University of Technology, Guangzhou, 510641, P.R.China
E-mail address : [email protected] School of Mathematics, South China University of Technology, Guangzhou, 510641, P.R.China
E-mail address : [email protected] ANDOM WALKS ASSOCIATED TO BETA-SHIFTS 33
Institut de Mathématiques de Jussieu - Paris Rive Gauche, Sorbonne Université - CampusPierre et Marie Curie, Paris, 75005, France
E-mail address : [email protected] School of Mathematics, University of Manchester, Manchester, M13 9PL, UK
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