Dynamical behavior of alternate base expansions
aa r X i v : . [ m a t h . D S ] F e b DYNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS
ÉMILIE CHARLIER , CÉLIA CISTERNINO , ∗ AND KARMA DAJANI Department of Mathematics, University of Liège, Allée de la Découverte 12, 4000 Liège,Belgium Department of Mathematics, Utrecht University, P.O. Box 80010, 3508TA Utrecht, TheNetherlands
Abstract.
We generalize the greedy and lazy β -transformations for a real base β tothe setting of alternate bases β = ( β , . . . , β p − ) , which were recently introduced by thefirst and second authors as a particular case of Cantor bases. As in the real base case,these new transformations, denoted T β and L β respectively, can be iterated in order togenerate the digits of the greedy and lazy β -expansions of real numbers. The aim of thispaper is to describe the dynamical behaviors of T β and L β . We first prove the existenceof a unique absolutely continuous (with respect to an extended Lebesgue measure, calledthe p -Lebesgue measure) T β -invariant measure. We then show that this unique measureis in fact equivalent to the p -Lebesgue measure and that the corresponding dynamicalsystem is ergodic and has entropy p log( β p − · · · β ) . We then express the density of thismeasure and compute the frequencies of letters in the greedy β -expansions. We obtainthe dynamical properties of L β by showing that the lazy dynamical system is isomorphicto the greedy one. We also provide an isomorphism with a suitable extension of the β -shift. Finally, we show that the β -expansions can be seen as ( β p − · · · β ) -representationsover general digit sets and we compare both frameworks. Mathematics Subject Classification : 11A63, 37E05, 37A45, 28D05
Keywords: Expansions of real numbers, Alternate bases, Greedy algorithm, Lazy algorithm,Measure theory, Ergodic theory, Dynamical systems Introduction
A representation of a nonnegative real number x in a real base β > is an infinitesequence a a a · · · of nonnegative integers such that x = P ∞ i =0 a i β i +1 . Among all β -representations, the greedy and lazy ones play a special role. They can be generatedby iterating the so-called greedy β -transformations T β and lazy β -transformations L β re-spectively. The dynamical properties of T β and L β are now well understood since theseminal works of Rényi [17] and Parry [15]; for example, see [9].In a recent work, the first two authors introduced the notion of expansions of real num-bers in a real Cantor base, that is, an infinite sequence of real bases β = ( β n ) n ≥ satisfying Q ∞ n =0 β n = ∞ [3]. In this initial work, the focus was on the combinatorial properties ofthese expansions. In particular, generalizations of several combinatorial results of real base E-mail address : [email protected], [email protected] and [email protected] . ∗ Corresponding author. expansions were obtained, such as Parry’s criterion for greedy β -expansions or Bertrand-Mathis characterization of sofic β -shifts. The latter result was obtained for the subclass ofperiodic Cantor bases, namely the alternate bases.The aim of this paper is to study the dynamical behaviors of the greedy and lazy expan-sions in an alternate base β = ( β , . . . , β p − , β , . . . , β p − , . . . ) . It is organized as follows.In Section 2, we provide the necessary background on measure theory and on expansions ofreal numbers in a real base. In Section 3, we introduce the greedy and lazy alternate baseexpansions and define the associated transformations T β and L β . Section 4 is concernedwith the dynamical properties of the greedy transformation. We first prove the existenceof a unique absolutely continuous (with respect to an extended Lebesgue measure, calledthe p -Lebesgue measure) T β -invariant measure and then prove that this measure is equiv-alent to the p -Lebesgue measure and that the corresponding dynamical system is ergodic.We then express the density of this measure and compute the frequencies of letters in thegreedy β -expansions. In Section 5 and 6, we prove that the greedy dynamical system isisomorphic to the lazy one, as well as to a suitable extension of the β -shift. In Section 7,we show that the β -expansions can be seen as ( β p − · · · β ) -representations over generaldigit sets and we compare both frameworks.2. Preliminaries
Measure preserving dynamical systems. A probability space is a triplet ( X, F , µ ) where X is a set, F is a σ -algebra over X and µ is a measure on F such that µ ( X ) = 1 .For a measurable transformation T : X → X and a measure µ on F , the measure µ is T -invariant , or equivalently, the transformation T : X → X is measure preserving withrespect to µ , if for all B ∈ F , µ ( T − ( B )) = µ ( B ) . A dynamical system is a quadruple ( X, F , µ, T ) where ( X, F , µ ) is a probability space and T : X → X is a measure preservingtransformation with respect to µ . A dynamical system ( X, F , µ, T ) is ergodic if for all B ∈ F , T − ( B ) = B implies µ ( B ) ∈ { , } , and is exact if T ∞ n =0 { T − n ( B ) : B ∈ F } only contains sets of measure or . Clearly, any exact dynamical system is ergodic.Two dynamical systems ( X, F X , µ X , T X ) and ( Y, F Y , µ Y , T Y ) are (measure preservingly)isomorphic if there exists a µ X -a.e. injective measurable map ψ : X → Y such that µ Y = µ X ◦ ψ − and ψ ◦ T X = T Y ◦ ψ µ X -a.e.For two measures µ and ν on the same σ -algebra F , µ is absolutely continuous withrespect to ν if for all B ∈ F , ν ( B ) = 0 implies µ ( B ) = 0 , and µ and ν are equivalent if theyare absolutely continuous with respect to each other. In what follows, we will be concernedby the Borel σ -algebras B ( A ) , where A ⊂ R . In particular, a measure on B ( A ) is absolutelycontinuous if it is absolutely continuous with respect to the Lebesgue measure λ restrictedto B ( A ) . The Radon-Nikodym theorem states that µ and ν are two probability measuressuch that µ is absolutely continuous with respect to ν , then there exists a ν -integrable map f : X [0 , + ∞ ) such that for all B ∈ F , µ ( B ) = R B f dν . Moreover, the map f is ν -a.e.unique. It is called the density of the measure µ with respect to ν and is usually denoted dµdν .For more details on measure theory and ergodic theory, we refer the reader to [2, 8, 11].2.2. Real base expansions.
Let β be a real number greater than . A β -representation of a nonnegative real number x is an infinite sequence a a a · · · over N such that x = P ∞ i =0 a i β i +1 . For x ∈ [0 , , a particular β -representation of x , called the greedy β -expansionof x , is obtained by using the greedy algorithm . If the first N digits of the β -expansion of x are given by a , . . . , a N − , then the next digit a N is the greatest integer in [[0 , ⌈ β ⌉ − YNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS 3 such that N X n =0 a n β n +1 ≤ x. The greedy β -expansion can also be obtained by iterating the greedy β -transformation T β : [0 , → [0 , , x βx − ⌊ βx ⌋ by setting a n = ⌊ βT nβ ( x ) ⌋ for all n ∈ N . Example 1.
In this example and throughout the paper, ϕ designates the golden ratio,i.e., ϕ = √ . The transformation T ϕ is depicted in Figure 1. ϕ ϕ Figure 1.
The transformation T ϕ .Real base expansions have been studied through various points of view. We refer thereader to [14, Chapter 7] for a survey on their combinatorial properties and [8] for asurvey on their dynamical properties. A fundamental dynamical result is the following.This summarizes results from [15, 17, 18]. Theorem 2.
There exists a unique T β -invariant absolutely continuous probability mea-sure µ β on B ([0 , . Furthermore, the measure µ β is equivalent to the Lebesgue measureon B ([0 , and the dynamical system ([0 , , B ([0 , , µ β , T β ) is ergodic and has entropy log( β ) . Remark 3.
It follows from Theorem 2 that T β is non-singular with respect to the Lebesguemeasure , i.e., for all B ∈ B ([0 , , λ ( B ) = 0 if and only if λ ( T − β ( B )) = 0 .In what follows, we let x β = ⌈ β ⌉ − β − . This value corresponds to the greatest real number that has a β -representation over thealphabet [[0 , ⌈ β ⌉ − . Clearly, we have x β ≥ . The extended greedy β -transformation , stilldenoted T β , is defined in [9] as T β : [0 , x β ) → [0 , x β ) , x ( βx − ⌊ βx ⌋ if x ∈ [0 , βx − ( ⌈ β ⌉ − if x ∈ [1 , x β ) . Note that for all x ∈ (cid:2) ⌈ β ⌉− β , ⌈ β ⌉ β (cid:1) , the two cases of the definition coincide since ⌊ βx ⌋ = ⌈ β ⌉ − . The extended β -transformation restricted to the interval [0 , gives back theclassical greedy β -transformation defined above. Moreover, for all x ∈ [0 , x β ) , there exists N ∈ N such that for all n ≥ N , T nβ ( x ) ∈ [0 , . DYNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS ϕ ϕ ϕ − ϕ − Figure 2.
The extended transformation T ϕ . Example 4.
We continue Example 1. The extended greedy transformation T ϕ is depictedin Figure 2.In the greedy algorithm, each digit is chosen as the largest possible among , , . . . , ⌈ β ⌉− at the considered position. At the other extreme, the lazy algorithm picks the least possibledigit at each step [10]: if the first N digits of the expansion of a real number x ∈ (0 , x β ] are given by a , . . . , a N − , then the next digit a N is the least element in [[0 , ⌈ β ⌉ − suchthat N X n =0 a n β n +1 + ∞ X n = N +1 ⌈ β ⌉ − β n +1 ≥ x, or equivalently, N X n =0 a n β n +1 + x β β N +1 ≥ x. The so-obtained β -representation is called the lazy β -expansion of x . The lazy β -transfor-mation dynamically generating the lazy β -expansion is the transformation L β defined asfollows [8]: L β : (0 , x β ] → (0 , x β ] , x ( βx if x ∈ (0 , x β − βx − ⌈ βx − x β ⌉ if x ∈ ( x β − , x β ] . Observe that for all x ∈ (cid:0) x β − β , x β β (cid:3) , the two cases of the definition coincide since ⌈ βx − x β ⌉ = 0 . Moreover, since L β (cid:0) ( x β − , x β ] (cid:1) = ( x β − , x β ] , the lazy transformation L β canbe restricted to the length-one interval ( x β − , x β ] . Also note that for all x ∈ (0 , x β ] ,there exists N ∈ N such that for all n ≥ N , L nβ ( x ) ∈ ( x β − , x β ] . Furthermore, for all x ∈ ( x β − , x β ] and n ∈ N , we have a n = ⌈ βL nβ ( x ) − x β ⌉ . Example 5.
The lazy transformation L ϕ is depicted in Figure 3.It is proven in [9] that there is an isomorphism between the greedy and the lazy β -transformations. As a direct consequence of this property, an analogue of Theorem 2 isobtained for the lazy transformation restricted to the interval ( x β − , x β ] . YNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS 5 ϕ − − ϕ − − ϕ − ϕ − ϕ − − ϕ ϕ − − ϕ Figure 3.
The transformation L ϕ .3. Alternate base expansions
Let p be a positive integer and β = ( β , . . . , β p − ) be a p -tuple of real numbers greaterthan . Such a p -tuple β is called an alternate base and p is called its length . A β -representation of a nonnegative real number x is an infinite sequence a a a · · · over N such that x = a β + a β β + · · · + a p − β p − · · · β (1) + a p β ( β p − · · · β ) + a p +1 β β ( β p − · · · β ) + · · · + a p − ( β p − · · · β ) + · · · We use the convention that for all n ∈ Z , β n = β n mod p and β ( n ) = ( β n , . . . , β n + p − ) .Therefore, the equality (1) can be rewritten x = + ∞ X n =0 a n Q nk =0 β k . The alternate bases are particular cases of Cantor real bases, which were introduced andstudied in [3].In this paper, our aim is to study the dynamics behind some distinguished represen-tation in alternate bases, namely the greedy and lazy β -expansions. First, we recall thenotion of greedy β -expansions defined in [3] and we introduce the greedy β -transformationdynamically generating the digits of the greedy β -expansions. Second, we introduce thenotion of lazy β -expansions and the corresponding lazy β -transformation.3.1. The greedy β -expansion. For x ∈ [0 , , a distinguished β -representation, calledthe greedy β -expansion of x , is obtained from the greedy algorithm . If the first N digitsof the greedy β -expansion of x are given by a , . . . , a N − , then the next digit a N is thegreatest integer in [[0 , ⌈ β N ⌉ − such that N X n =0 a n Q nk =0 β k ≤ x. The greedy β -expansion can also be obtained by alternating the β i -transformations: forall x ∈ [0 , and n ∈ N , a n = ⌊ β n (cid:0) T β n − ◦ · · · ◦ T β ( x ) (cid:1) ⌋ . The greedy β -expansion of x is DYNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS denoted d β ( x ) . In particular, if p = 1 then it corresponds to the usual greedy β -expansionas defined in Section 2.2. Example 6.
Consider the alternate base β = ( √ , √ ) already studied in [3]. Thegreedy β -expansions are obtained by alternating the transformations T √ and T √ ,which are both depicted in Figure 4. Moreover, in Figure 5 we see the computation of thefirst five digits of the greedy β -expansion of √ . β β β Figure 4.
The transformations T √ (blue) and T √ (green). Figure 5.
The first five digits of the greedy β -expansion of √ are for β = ( √ , √ ) .We now define the greedy β -transformation by(2) T β : [[0 , p − × [0 , → [[0 , p − × [0 , , ( i, x ) (cid:0) ( i + 1) mod p, T β i ( x ) (cid:1) . The greedy β -transformation generates the digits of the greedy β -expansions as follows.For all x ∈ [0 , and n ∈ N , the digit a n of d β ( x ) is equal to ⌊ β n π (cid:0) T n β (0 , x ) (cid:1) ⌋ where π : N × R → R , ( n, x ) x. As in Section 2.2, the greedy β -transformation can be extended to an interval of realnumbers bigger than [0 , . To do so, we define(3) x β = ∞ X n =0 ⌈ β n ⌉ − Q nk =0 β k . It can be easily seen that ≤ x β < ∞ . This value corresponds to the greatest real numberthat has a β -representation a a a · · · such that each letter a n belongs to the alphabet YNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS 7 [[0 , ⌈ β n ⌉ − . Moreover, for all n ∈ Z ,(4) x β ( n ) = x β ( n +1) + ⌈ β n ⌉ − β n . We define the extended greedy β -transformation , still denoted T β , by T β : p − [ i =0 (cid:0) { i } × [0 , x β ( i ) ) (cid:1) → p − [ i =0 (cid:0) { i } × [0 , x β ( i ) ) (cid:1) , ( i, x ) ((cid:0) ( i + 1) mod p, β i x − ⌊ β i x ⌋ (cid:1) if x ∈ [0 , (cid:0) ( i + 1) mod p, β i x − ( ⌈ β i ⌉ − (cid:1) if x ∈ [1 , x β ( i ) ) . The greedy β -expansion of x ∈ [0 , x β ) is obtained by alternating the p maps π ◦ T β ◦ δ i (cid:12)(cid:12) [0 ,x β ( i ) ) : [0 , x β ( i ) ) → [0 , x β ( i +1) ) for i ∈ [[0 , p − , where δ i : R → { i } × R , x ( i, x ) . Proposition 7.
For all x ∈ [0 , x β ) and n ∈ N , we have π ◦ T n β ◦ δ ( x ) = β n − · · · β x − n − X k =0 β n − · · · β k +1 c k where ( c , . . . , c n − ) is the lexicographically greatest n -tuple in Q n − k =0 [[0 , ⌈ β k ⌉ − such that P n − k =0 β n − ··· β k +1 c k β n − ··· β ≤ x .Proof. We proceed by induction on n . The base case n = 0 is immediate. Now, supposethat the result is satisfied for some n ∈ N . Let x ∈ [0 , x β ) . Let ( c , . . . , c n − ) is thelexicographically greatest n -tuple in Q n − k =0 [[0 , ⌈ β k ⌉ − such that P n − k =0 β n − ··· β k +1 c k β n − ··· β ≤ x .Then it is easily seen that for all m < n , ( c , . . . , c m ) is the lexicographically greatest ( m + 1) -tuple in Q mk =0 [[0 , ⌈ β k ⌉ − such that P mk =0 β m ··· β k +1 c k β m ··· β ≤ x . Now, set y = π ◦ T n β ◦ δ ( x ) . Then y ∈ [0 , x β ( n ) ) and by induction hypothesis, we obtain that y = β n − · · · β x − P n − k =0 β n − · · · β k +1 c k . Then, by setting c n = ( ⌊ β n y ⌋ if y ∈ [0 , ⌈ β n ⌉ − if y ∈ [1 , x β ( n ) ) we obtain that π ◦ T n +1 β ◦ δ ( x ) = β n · · · β x − P nk =0 β n · · · β k +1 c k . In order to conclude,we have to show thata) P nk =0 β n ··· β k +1 c k β n ··· β ≤ x b) ( c , . . . , c n ) is the lexicographically greatest ( n + 1) -tuple in Q nk =0 [[0 , ⌈ β k ⌉ − suchthat a) holds.By definition of c n , we have c n ≤ β n y . Therefore, n X k =0 β n · · · β k +1 c k = β n n − X k =0 β n − · · · β k +1 c k + c n = β n ( β n − · · · β x − y ) + c n ≤ β n · · · β x. This shows that a) holds.Let us show b) by contradiction. Suppose that there exists ( c ′ , . . . , c ′ n ) ∈ Q nk =0 [[0 , ⌈ β k ⌉− such that ( c ′ , . . . , c ′ n ) > lex ( c , . . . , c n ) and P nk =0 β n ··· β k +1 c ′ k β n ··· β ≤ x . Then there exists m ≤ n such that c ′ = c , . . . , c ′ m − = c m − and c ′ m ≥ c m + 1 . We again consider two cases. First, DYNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS suppose that m < n . Since ( c ′ , . . . , c ′ m ) > lex ( c , . . . , c m ) , we get P mk =0 β m ··· β k +1 c ′ k β m ··· β > x . Butthen n X k =0 β n · · · β k +1 c ′ k ≥ β n · · · β m +1 m X k =0 β m · · · β k +1 c ′ k > β n · · · β x, a contradiction. Second, suppose that m = n . Then β n · · · β x ≥ n X k =0 β n · · · β k +1 c ′ k ≥ n − X k =0 β n · · · β k +1 c k + c n + 1 , hence β n y ≥ c n + 1 . If y ∈ [0 , then c n + 1 = ⌊ β n y ⌋ + 1 > β n y , a contradiction.Otherwise, y ∈ [1 , x β ( n ) ) and c n + 1 = ⌈ β n ⌉ . But then c ′ n ≥ ⌈ β n ⌉ , which is impossible since c ′ n ∈ [[0 , ⌈ β n ⌉ − . This shows b) and ends the proof. (cid:3) The restriction of the extended greedy β -transformation to the domain [[0 , p − × [0 , gives back the greedy β -transformation initially defined in 2. Moreover, for all ( i, x ) ∈ S p − i =0 (cid:0) { i } × [0 , x β ( i ) ) (cid:1) , there exists N ∈ N such that for all n ≥ N , T n β ( i, x ) ∈ [[0 , p − × [0 , . Example 8.
Let β = ( √ , √ ) be the alternate base of Example 6. The maps π ◦ T β ◦ δ (cid:12)(cid:12) [0 ,x β ) : [0 , x β ) → [0 , x β (1) ) and π ◦ T β ◦ δ (cid:12)(cid:12) [0 ,x β (1) ) : [0 , x β (1) ) → [0 , x β ) are de-picted in Figure 6. x β (1) x β (1) x β x β β β β Figure 6.
The maps π ◦ T β ◦ δ (cid:12)(cid:12) [0 ,x β ) (blue) and π ◦ T β ◦ δ (cid:12)(cid:12) [0 ,x β (1) ) (green) with β = ( √ , √ ) .3.2. The lazy β -expansion. As in the real base case, in the greedy β -expansion, eachdigit is chosen as the largest possible at the considered position. Here, we define and studythe other extreme β -representation, called the lazy β -expansion , taking the least possibledigit at each step. For x ∈ [0 , x β ) , if the first N digits of the lazy β -expansion of x aregiven by a , . . . , a N − , then the next digit a N is the least element in [[0 , ⌈ β N ⌉ − suchthat N X n =0 a n Q nk =0 β k + ∞ X n = N +1 ⌈ β n ⌉ − Q nk =0 β k ≥ x, YNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS 9 or equivalently, N X n =0 a n Q nk =0 β k + x β ( N ) Q Nk =0 β k ≥ x. This algorithm is called the lazy algorithm . For all N ∈ N , we have N X n =0 a n Q nk =0 β k ≤ x, which implies that the lazy algorithm converges, that is, x = ∞ X n =0 a n Q nk =0 β k . We now define the lazy β -transformation by L β : p − [ i =0 (cid:0) { i } × (0 , x β ( i ) ] (cid:1) → p − [ i =0 (cid:0) { i } × (0 , x β ( i ) ] (cid:1) , ( i, x ) ((cid:0) ( i + 1) mod p, β i x (cid:1) if x ∈ (0 , x β ( i ) − (cid:0) ( i + 1) mod p, β i x − ⌈ β i x − x β ( i +1) ⌉ (cid:1) if x ∈ ( x β ( i ) − , x β ( i ) ] . The lazy β -expansion of x ∈ (0 , x β ] is obtained by alternating the p maps π ◦ L β ◦ δ i (cid:12)(cid:12) (0 ,x β ( i ) ] : (0 , x β ( i ) ] → (0 , x β ( i +1) ] for i ∈ [[0 , p − . The following proposition is the analogue of Proposition 7 for the lazy β -transformation, which can be proved in a similar fashion. Proposition 9.
For all x ∈ (0 , x β ] and n ∈ N , we have π ◦ L n β ◦ δ ( x ) = β n − · · · β x − n − X i =0 β n − · · · β i +1 c i where ( c , . . . , c n − ) is the lexicographically least n -tuple in Q n − k =0 [[0 , ⌈ β k ⌉ − such that P n − i =0 β n − ··· β i +1 c i β n − ··· β + P ∞ m = n ⌈ β m ⌉− Q mk =0 β k ≥ x . Note that for each i ∈ [[0 , p − , L β (cid:0) { i } × ( x β ( i ) − , x β ( i ) ] (cid:1) ⊆ { ( i + 1) mod p } × ( x β ( i +1) − , x β ( i +1) ] . Therefore, the lazy β -transformation can be restricted to the domain S p − i =0 (cid:0) { i } × ( x β ( i ) − , x β ( i ) ] . The (restricted) lazy β -transformation generates the digits of the lazy β -expan-sions of real numbers in the interval ( x β − , x β ] as follows. For all x ∈ ( x β − , x β ] and n ∈ N , the digit a n in the lazy β -expansion of x is equal to ⌈ β n π (cid:0) L n β (0 , x ) (cid:1) − x β ( n +1) ⌉ .Finally, observe that for all ( i, x ) ∈ S p − i =0 ( { i } × (0 , x β ( i ) ]) , there exists N ∈ N such that forall n ≥ N , L n β ( i, x ) ∈ S p − i =0 (cid:0) { i } × ( x β ( i ) − , x β ( i ) ] . Example 10.
Consider again the length- alternate base β = ( √ , √ ) from Ex-amples 6 and 8. We have x β = √ ≃ . and x β (1) = √ ≃ . . The maps π ◦ L β ◦ δ (cid:12)(cid:12) (0 ,x β ] : (0 , x β ] → (0 , x β (1) ] and π ◦ L β ◦ δ (cid:12)(cid:12) (0 ,x β (1) ] : (0 , x β (1) ] → (0 , x β ] are de-picted in Figure 7. In Figure 8 we see the computation of the first five digits of the lazy β -expansion of √ . x β (1) x β x β (1) − β x β − β x β − β x β (1) x β x β (1) − x β − Figure 7.
The maps π ◦ L β ◦ δ (cid:12)(cid:12) (0 ,x β ] (blue) and π ◦ L β ◦ δ (cid:12)(cid:12) (0 ,x β (1) ] (green) with β = ( √ , √ ) . Figure 8.
The first five digits of the lazy β -expansion of √ are for β = ( √ , √ ) .3.3. A note on Cantor bases.
The greedy algorithm described in Sections 3.1 and 3.2is well defined in the extended context of Cantor bases, i.e., sequences of real numbers β = ( β n ) n ∈ N greater than such that the product Q ∞ n =0 β n is infinite [3]. In this case, thegreedy algorithm converge on [0 , : for all x ∈ [0 , , the computed digits a n are such that P ∞ n =0 a n Q nk =0 β k = x . Therefore, the value x β defined as in (3) is greater than or equal to .However, it might be that x β = ∞ . For example, it is the case for the Cantor base givenby β n = 1 + n +1 for all n ∈ N .Note that the restriction of the transformation π ◦ T n β ◦ δ to the unit interval [0 , coincide with the composition T β n − ◦· · ·◦ T β . Thus, when restricted to [0 , , Proposition 7can be reformulated as follows. Proposition 11.
For all x ∈ [0 , and n ∈ N , we have T β n − ◦ · · · ◦ T β ( x ) = β n − · · · β x − n − X k =0 β n − · · · β k +1 c k YNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS 11 where ( c , . . . , c n − ) is the lexicographically greatest n -tuple in Q n − k =0 [[0 , ⌈ β k ⌉ − such that P n − k =0 β n − ··· β k +1 c k β n − ··· β ≤ x . For all k ∈ [[0 , n − , the transformation L β k is defined on (0 , x β k ] and can be restrictedto ( x β k − , x β k ] . So, the restricted transformations L β , . . . , L β n − cannot be composedto one another in general. Therefore, even if the lazy algorithm can be defined for Cantorbases, provided that x β < ∞ , we cannot state an analogue of Proposition 11 in terms ofthe lazy transformations for Cantor bases.Even though this paper is mostly concerned with alternate bases, let us emphasize thatsome results are indeed valid for any sequence ( β n ) n ∈ N ∈ ( R > ) N , and hence for any Cantorbase. This is the case of Proposition 11, Theorem 14, Corollary 15 and Proposition 25.4. Dynamical properties of T β In this section, we study the dynamics of the greedy β -transformation. First, we gen-eralize Theorem 2 to the transformation T β on [[0 , p − × [0 , . Second, we extend theobtained result to the extended transformation T β . Third, we provide a formula for thedensities of the measures found in the first two parts. Finally, we compute the frequenciesof the digits in the greedy β -expansions.4.1. Unique absolutely continuous T β -invariant measure. In order to generalizeTheorem 2 to alternate bases, we start by recalling a result of Lasota and Yorke.
Theorem 12. [13, Theorem 4]
Let T : [0 , → [0 , be a transformation for which thereexists a partition [ a , a ) , . . . , [ a K − , a K ) of the interval [0 , with a < · · · < a K such thatfor each k ∈ [[0 , K − , T (cid:12)(cid:12) [ a k ,a k +1 ) is convex, T ( a k ) = 0 , T ′ ( a k ) > and T ′ (0) > . Thenthere exists a unique T -invariant absolutely continuous probability measure. Furthermore,its density is bounded and decreasing, and the corresponding dynamical system is exact. We then prove a stability lemma.
Lemma 13.
Let I be the family of transformations T : [0 , → [0 , for which there exist apartition [ a , a ) , . . . , [ a K − , a K ) of the interval [0 , with a < · · · < a K and a slope s > such that for all k ∈ [[0 , K − , a k +1 − a k ≤ s and for all x ∈ [ a k , a k +1 ) , T ( x ) = s ( x − a k ) .Then I is closed under composition.Proof. Let
S, T ∈ I . Let [ a , a ) , . . . , [ a K − , a K ) and [ b , b ) , . . . , [ b L − , b L ) be partitionsof the interval [0 , with a < · · · < a K , b < · · · < b L , and let s, t > such thatfor all k ∈ [[0 , K − , a k +1 − a k ≤ s , for all ℓ ∈ [[0 , L − , b ℓ +1 − b ℓ ≤ t and for all x ∈ [0 , , S ( x ) = s ( x − a k ) if x ∈ [ a k , a k +1 ) and T ( x ) = t ( x − b ℓ ) if x ∈ [ b ℓ , b ℓ +1 ) . Foreach k ∈ [[0 , K − , define L k to be the greatest ℓ ∈ [[0 , L − such that a k + b ℓ s < a k +1 .Consider the partition h a + b s , a + b s (cid:17) , . . . , h a + b L − s , a + b L s (cid:17) , h a + b L s , a (cid:17) ... h a K − + b s , a K − + b s (cid:17) , . . . , h a K − + b L K − − s , a K − + b L K − s (cid:17) , h a K − + b L K − s , a K (cid:17) of the interval [0 , . For each k ∈ [[0 , K − and ℓ ∈ [[0 , L k − , a k + b ℓ +1 s − a k − b ℓ s ≤ ts and a k +1 − a k − b Lk s = ( a k +1 − a k − b Lk +1 s )+ b Lk +1 − b Lk s ≤ ts . Now, let x ∈ [0 , and k ∈ [[0 , K − be such that x ∈ [ a k , a k +1 ) . Then S ( x ) = s ( x − a k ) ∈ [0 , . We distinguish two cases: either there exists ℓ ∈ [[0 , L k − such that x ∈ [ a k + b ℓ s , a k + b ℓ +1 s ) , or x ∈ [ a k + b Lk s , a k +1 ) .In the former case, S ( x ) ∈ [ b ℓ , b ℓ +1 ) and T ◦ S ( x ) = t ( S ( x ) − b ℓ ) = ts ( x − ( a k + b ℓ s )) . Inthe latter case, since a k +1 − a k ≤ b Lk +1 s , we get that S ( x ) ∈ [ b L k , b L k +1 ) and hence that T ◦ S ( x ) = t ( S ( x ) − b L k ) = ts ( x − ( a k + b Lk s )) . This shows that the composition T ◦ S belongs to I . (cid:3) The following theorem provides us with the main tool for the construction of a T β -invariant measure. Theorem 14.
For all n ∈ N ≥ and all β , . . . , β n − > , there exists a unique ( T β n − ◦· · · ◦ T β ) -invariant absolutely continuous probability measure µ on B ([0 , . Furthermore,the measure µ is equivalent to the Lebesgue measure on B ([0 , , its density is bounded anddecreasing, and the dynamical system ([0 , , B ([0 , , µ, T β n − ◦ · · · ◦ T β ) is exact and hasentropy log( β n − · · · β ) .Proof. The existence of a unique ( T β n − ◦ · · · ◦ T β ) -invariant absolutely continuous proba-bility measure µ on B ([0 , , the fact that its density is bounded and decreasing, and theexactness of the corresponding dynamical system follow from Theorem 12 and Lemma 13.With a similar argument as in [6], we can conclude that dµdλ > λ -a.e. on [0 , . It followsthat µ is equivalent to the Lebesgue measure on B ([0 , . Moreover, the entropy equals log( β n − · · · β ) since T β n − ◦ · · · ◦ T β is a piecewise linear transformation of constant slope β n − · · · β [7, 18]. (cid:3) The following consequence of Theorem 14 will be useful for proving our generalizationof Theorem 2.
Corollary 15.
Let n ∈ N ≥ and β , . . . , β n − > . Then for all B ∈ B ([0 , such that ( T β n − ◦ · · · ◦ T β ) − ( B ) = B , we have λ ( B ) ∈ { , } . For each i ∈ [[0 , p − , we let µ β ,i denote the unique ( T β i − ◦ · · · ◦ T β i − p ) -invariantabsolutely continuous probability measure given by Theorem 14. We use the conventionthat for all n ∈ Z , µ β ,n = µ β ,n mod p . Let us define a probability measure µ β on the σ -algebra T p = ( p − [ i =0 ( { i } × B i ) : ∀ i ∈ [[0 , p − , B i ∈ B ([0 , ) over [[0 , p − × [0 , as follows. For all B , . . . , B p − ∈ B ([0 , , we set µ β p − [ i =0 ( { i } × B i ) ! = 1 p p − X i =0 µ β ,i ( B i ) . We now study the properties of the probability measure µ β . Lemma 16.
For i ∈ [[0 , p − , we have µ β ,i = µ β ,i − ◦ T − β i − .Proof. Let i ∈ [[0 , p − . By definition of µ β ,i , it suffices to show that µ β ,i − ◦ T − β i − is a ( T β i − ◦ · · · ◦ T β i − p ) -invariant absolutely continuous probability measure on B ([0 , . First,we have µ β ,i − (cid:0) T − β i − ([0 , (cid:1) = µ β ,i − ([0 , . Second, for all B ∈ B ([0 , , we have µ β ,i − ◦ T − β i − (cid:0) ( T β i − ◦ · · · ◦ T β i − p ) − ( B ) (cid:1) = µ β ,i − (cid:0) ( T β i − ◦ · · · ◦ T β i − p ◦ T β i − p − ) − ( B ) (cid:1) = µ β ,i − (cid:0) ( T β i − ◦ · · · ◦ T β i − p − ) − ( T − β i − ( B )) (cid:1) = µ β ,i − (cid:0) T − β i − ( B ) (cid:1) . YNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS 13
Third, for all B ∈ B ([0 , such that λ ( B ) = 0 , we get that λ ( T − β i − ( B )) = 0 by Remark 3,and hence that µ β ,i − ( T − β i − ( B )) = 0 since µ β ,i − is absolutely continuous. (cid:3) Proposition 17.
The measure µ β is T β -invariant.Proof. For all B , . . . , B p − ∈ B ([0 , , µ β T − β p − [ i =0 ( { i } × B i ) !! = µ β p − [ i =0 T − β ( { i } × B i ) ! = µ β p − [ i =0 (cid:0) { ( i −
1) mod p } × T − β i − ( B i ) (cid:1)! = 1 p p − X i =0 µ β ,i − ( T − β i − ( B i ))= 1 p p − X i =0 µ β ,i ( B i )= µ β p − [ i =0 ( { i } × B i ) ! where we applied Lemma 16 for the fourth equality. (cid:3) Corollary 18.
The quadruple (cid:0) [[0 , p − × [0 , , T p , µ β , T β (cid:1) is a dynamical system. Let us define a new measure λ p over the σ -algebra T p . For all B , . . . , B p − ∈ B ([0 , ,we set λ p p − [ i =0 ( { i } × B i ) ! = 1 p p − X i =0 λ ( B i ) . We call this measure the p -Lebesgue measure on T p . Proposition 19.
The measure µ β is equivalent to the p -Lebesgue measure on T p .Proof. This follows from the fact that the p measures µ β , , . . . , µ β ,p − are equivalent to theLebesgue measure λ on B ([0 , . (cid:3) Next, we compute the entropy of the dynamical system (cid:0) [[0 , p − × [0 , , T p , µ β , T β (cid:1) .To do so, we consider the p induced transformations T β ,i : { i } × [0 , → { i } × [0 , , ( i, x ) T p β ( i, x ) for i ∈ [[0 , p − . Note that indeed, for all ( i, x ) ∈ [[0 , p − × [0 , , the first returnof ( i, x ) to { i } × [0 , is equal to p . Thus T β ,i = T p β (cid:12)(cid:12) { i }× [0 , . As is well know [7], foreach i ∈ [[0 , p − , the induced transformation T β ,i is measure preserving with respectto the measure ν β ,i on the σ -algebra {{ i } × B : B ∈ B ([0 , } defined as follows: for all B ∈ B ([0 , , ν β ,i ( { i } × B ) = pµ β ( { i } × B ) . Lemma 20.
For every i ∈ [[0 , p − , the map δ i (cid:12)(cid:12) [0 , : [0 , → { i } × [0 , , x ( i, x ) defines an isomorphism between the dynamical systems (cid:0) [0 , , B ([0 , , µ β ,i , T β i − ◦ · · · ◦ T β i − p (cid:1) and (cid:0) { i } × [0 , , {{ i } × B : B ∈ B ([0 , } , ν β ,i , T β ,i (cid:1) Proof.
This is a straightforward verification. (cid:3)
Proposition 21.
The entropy h β of the dynamical system (cid:0) [[0 , p − × [0 , , T p , µ β , T β (cid:1) is p log( β p − · · · β ) .Proof. Let i ∈ [[0 , p − . By Abramov’s formula [1], we have h β = µ β ( { i } × [0 , h β ,i = 1 p h β ,i . where h β ,i denotes the entropy of the induced dynamical system (cid:0) { i }× [0 , , {{ i }× B : B ∈B ([0 , } , ν β ,i , T β ,i ) . Since the entropy is an isomorphism invariant, it follows from Theo-rem 14 and Lemma 20 that h β ,i = log( β p − · · · β ) . Hence the conclusion. (cid:3) Finally, we prove that any T β -invariant set has p -Lebesgue measure or . Proposition 22.
For all A ∈ T p such that T − β ( A ) = A , we have λ p ( A ) ∈ { , } .Proof. Let B , . . . , B p − be sets in B ([0 , such that T − β p − [ i =0 ( { i } × B i ) ! = p − [ i =0 ( { i } × B i ) . This implies that(5) T − β i − ( B i ) = B ( i −
1) mod p for all i ∈ [[0 , p − . We use the convention that B n = B n mod p for all n ∈ Z . An easy induction yields that forall i ∈ [[0 , p − and n ∈ N , ( T β i − ◦ · · · ◦ T β i − n ) − ( B i ) = B i − n . In particular, for n = p , weget that for each i ∈ [[0 , p − , ( T β i − ◦ · · · ◦ T β i − p ) − ( B i ) = B i . By Corollary 15, for each i ∈ [[0 , p − , λ ( B i ) ∈ { , } . By definition of λ p , in order to conclude, it suffices to showthat either λ ( B i ) = 0 for all i ∈ [[0 , p − , or λ ( B i ) = 1 for all i ∈ [[0 , p − . From (5) andRemark 3, we get that for each i ∈ [[0 , p − , λ ( B i ) = 0 if and only if λ ( B i +1 ) = 0 . Theconclusion follows. (cid:3) We are now able to state the announced generalization of Theorem 2 to alternate bases.
Theorem 23.
The measure µ β is the unique T β -invariant probability measure on T p that isabsolutely continuous with respect to λ p . Furthermore, µ β is equivalent to λ p on T p and thedynamical system ([[0 , p − × [0 , , T p , µ β , T β ) is ergodic and has entropy p log( β p − · · · β ) .Proof. By Propositions 17 and 19, µ β is a T β -invariant probability measure that is abso-lutely continuous with respect to λ p on B ([0 , . Then we get from Proposition 22 thatfor all A ∈ T p such that T − β ( A ) = A , we have µ β ( A ) ∈ { , } . Therefore, the dynamicalsystem ([[0 , p − × [0 , , T p , µ β , T β ) is ergodic. Now, we obtain that the measure µ β isunique as a well-known consequence of the Ergodic Theorem, see [7, Theorem 3.1.2]. Theequivalence between µ β and λ p and the entropy of the system were already obtained inPropositions 19 and 21. (cid:3) For p greater than , the dynamical system ([[0 , p − × [0 , , T p , µ β , T β ) is not exact eventhough for all i ∈ [[0 , p − , the dynamical systems ([0 , , B ([0 , , µ β ,i , T β i − ◦ · · · ◦ T β i − p ) are exact. It suffices to note that the dynamical system ([[0 , p − × [0 , , T p , µ β , T p β ) is notergodic for p > . Indeed, T − p β ( { } × [0 , { } × [0 , whereas µ β ( { } × [0 , p . YNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS 15
Extended measure.
In order to study the dynamics of the extended greedy β -transformation, we extend the definitions of the measures µ β and λ p . First, we define anew σ -algebra T β on S p − i =0 (cid:0) { i } × [0 , x β ( i ) ) (cid:1) as follows: T β = ( p − [ i =0 ( { i } × B i ) : ∀ i ∈ [[0 , p − , B i ∈ B ([0 , x β ( i ) )) ) . Second, we extend the domain of the measures µ β and λ p to T β (while keeping the samenotation) as follows. For A ∈ T β , we set µ β ( A ) = µ β (cid:0) A ∩ (cid:0) [[0 , p − × [0 , (cid:1)(cid:1) and λ p ( A ) = λ p (cid:0) A ∩ (cid:0) [[0 , p − × [0 , (cid:1)(cid:1) . Theorem 24.
The measure µ β is the unique T β -invariant probability measure on T β thatis absolutely continuous with respect to λ p . Furthermore, µ β is equivalent to λ p on T β and the dynamical system ( S p − i =0 (cid:0) { i } × [0 , x β ( i ) ) (cid:1) , T β , µ β , T β ) is ergodic and has entropy p log( β p − · · · β ) .Proof. Clearly, µ β is a probability measure on T β . For all A ∈ T β , we have µ β ( T − β ( A )) = µ β (cid:0) T − β ( A ) ∩ ([[0 , p − × [0 , (cid:1) = µ β (cid:0) T − β (cid:0) A ∩ ([[0 , p − × [0 , (cid:1) ∩ ([[0 , p − × [0 , (cid:1) = µ β (cid:0) T − β (cid:0) A ∩ ([[0 , p − × [0 , (cid:1)(cid:1) = µ β (cid:0) A ∩ ([[0 , p − × [0 , (cid:1) = µ β ( A ) where we used Proposition 17 for the fourth equality. This shows that µ β is T β -invariant on T β . The conclusion then follows from the fact that the identity map from [[0 , p − × [0 , to S p − i =0 (cid:0) { i } × [0 , x β ( i ) ) (cid:1) defines an isomorphism between the dynamical systems ([[0 , p − × [0 , , F p , µ β , T β ) and ( S p − i =0 (cid:0) { i } × [0 , x β ( i ) ) (cid:1) , T β , µ β , T β ) . (cid:3) Densities.
In the next proposition, we express the density of the unique measuregiven in Theorem 14.
Proposition 25.
Consider n ∈ N ≥ and β , . . . , β n − > . Suppose that • K is the number of not onto branches of T β n − ◦ · · · ◦ T β • for j ∈ [[1 , K ]] , c j is the right-hand side endpoint of the domain of the j -th not ontobranche of T β n − ◦ · · · ◦ T β • T : [0 , → [0 , is the transformation defined by T ( x ) = T β n − ◦ · · · ◦ T β ( x ) for x / ∈ { c , . . . , c K } and T ( c j ) = lim x → c − j T β n − ◦ · · · ◦ T β ( x ) for j ∈ [[1 , K ]] • S is the matrix defined by S = ( S i,j ) ≤ i,j, ≤ K where S i,j = ∞ X m =1 δ ( T m ( c i ) > c j )( β n − · · · β ) m , where δ ( P ) equals when the property P is satisfied and otherwise • is not an eigenvalue of S • d = 1 and (cid:0) d · · · d K (cid:1) = (cid:0) · · · (cid:1) ( − S + Id K ) − • C = R (cid:16) d + P Kj =1 d j P ∞ m =1 χ [0 ,Tm ( cj )] ( β n − ··· β ) m (cid:17) dλ is the normalization constant. Then the density of the ( T β n − ◦ · · · ◦ T β ) -invariant measure given by Theorem 14 withrespect to the Lebesgue measure is (6) C d + K X j =1 d j ∞ X m =1 χ [0 ,T m ( c j )] ( β n − · · · β ) m ! . Proof.
This is an application of the formula given in [12]. (cid:3)
Note that the only hypothesis in the statement of Proposition 25 is that is not aneigenvalue of the matrix S . In [12] Gora conjectured that this condition is equivalent tothe exactness of the dynamical system, which is a property we know to be satisfied byTheorem 14. Example 26.
Consider once again the alternate base β = ( √ , √ ) . The composi-tion T β ◦ T β is depicted in Figure 9. Since β = β − , keeping the notation of Proposi- β β β β β +1 β β β Figure 9.
The composition T β ◦ T β with β = ( √ , √ ) .tion 25, we have K = 3 , c = β , c = β and c = 1 . We have T ( c ) = T ( c ) = T ( c ) = c .Therefore, all elements in S equal , d = d = d = d = 1 and C = 1+ β ( β β − = 1+ β .The density of the unique absolutely continuous ( T β ◦ T β ) -invariant probability measureis C (cid:16) β χ [0 , β ] (cid:17) . For example, µ (cid:0) [0 , β ) (cid:1) = √ . Moreover, it can be checked that µ (cid:0) ( T β ◦ T β ) − [0 , β ) (cid:1) = µ (cid:0) [0 , β ) (cid:1) .We obtain a formula for the density dµ β dλ p by using the densities dµ β ,i dλ for i ∈ [[0 , p − given in Proposition 25. We first need a lemma. Lemma 27.
For all i ∈ [[0 , p − , all sets B ∈ B ([0 , and all B ([0 , -measurablefunctions f : [0 , → [0 , ∞ ) , the map f ◦ π : [[0 , p − × [0 , → [0 , ∞ ) is T p -measurableand Z { i }× B f ◦ π dλ p = 1 p Z B f dλ. Proof.
This follows from the definition of the Lebesgue integral via simple functions. (cid:3)
YNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS 17
Proposition 28.
The density dµ β dλ p of µ β with respect to the p -Lebesgue measure on T p is (7) p − X i =0 dµ β ,i dλ ◦ π ! · χ { i }× [0 , . Proof.
Let A ∈ T p and let B , . . . , B p − ∈ B ([0 , such that A = S p − i =0 ( { i } × B i ) . Itfollows from Lemma 27 that Z A p − X i =0 dµ β ,i dλ ◦ π ! · χ { i }× [0 , dλ p = p − X i =0 Z { i }× B i dµ β ,i dλ ◦ π dλ p = 1 p p − X i =0 Z B i dµ β ,i dλ dλ = 1 p p − X i =0 µ β ,i ( B i )= µ β ( A ) . (cid:3) Note that the formula (7) also holds for the extended measures µ β and λ p on T β .4.4. Frequencies.
We now turn to the frequencies of the digits in the greedy β -expansionsof real numbers in the interval [0 , . Recall that the frequency of a digit d occurring inthe greedy β -expansion a a a · · · of a real number x in [0 , is equal to lim n →∞ n { ≤ k < n : a k = d } , provided that this limit converges. Proposition 29.
For λ -almost all x ∈ [0 , , the frequency of any digit d occurring in thegreedy β -expansion of x exists and is equal to p p − X i =0 µ β ,i (cid:16)(cid:2) dβ i , d +1 β i (cid:1) ∩ [0 , (cid:17) . Proof.
Let x ∈ [0 , and let d be a digit occurring in d β ( x ) = a a a · · · . Then for all k ∈ N , a k = d if and only if π ( T k β (0 , x )) ∈ [ dβ k , d +1 β k ) ∩ [0 , . Moreover, since for all k ∈ N , T k β (0 , x ) ∈ { k mod p } × [0 , , we have χ [ dβk , d +1 βk ) ∩ [0 , (cid:0) π (cid:0) T k β (0 , x ) (cid:1)(cid:1) = χ { k mod p }× (cid:0) [ dβk , d +1 βk ) ∩ [0 , (cid:1)(cid:0) T k β (0 , x ) (cid:1) = p − X i =0 χ { i }× (cid:0) [ dβi , d +1 βi ) ∩ [0 , (cid:1)(cid:0) T k β (0 , x ) (cid:1) . Therefore, if it exists, the frequency of d in d β ( x ) is equal to lim n →∞ n n − X k =0 p − X i =0 χ { i }× (cid:0) [ dβi , d +1 βi ) ∩ [0 , (cid:1)(cid:0) T k β (0 , x ) (cid:1) . Yet, for each i ∈ [[0 , p − and for µ β -almost all y ∈ [[0 , p − × [0 , , we have lim n →∞ n n − X k =0 χ { i }× (cid:0) [ dβi , d +1 βi ) ∩ [0 , (cid:1)(cid:0) T k β ( y ) (cid:1) = Z [[0 ,p − × [0 , χ { i }× (cid:0) [ dβi , d +1 βi ) ∩ [0 , (cid:1) dµ β = µ β (cid:16) { i } × (cid:0)(cid:2) dβ i , d +1 β i (cid:1) ∩ [0 , (cid:1)(cid:17) = 1 p µ β ,i (cid:16)(cid:2) dβ i , d +1 β i (cid:1) ∩ [0 , (cid:17) where we used Theorem 23 and the Ergodic Theorem for the first equality. The conclusionnow follows from Proposition 19. (cid:3) Isomorphism between greedy and lazy β -transformations In this section, we show that(8) φ β : p − [ i =0 (cid:0) { i } × [0 , x β ( i ) ) (cid:1) → p − [ i =0 (cid:0) { i } × (0 , x β ( i ) ] (cid:1) , ( i, x ) (cid:0) i, x β ( i ) − x (cid:1) defines an isomorphism between the greedy β -transformation and the lazy β -transfor-mation.We consider the σ -algebra L β = ( p − [ i =0 ( { i } × B i ) : ∀ i ∈ [[0 , p − , B i ∈ B (cid:0) (0 , x β ( i ) ] (cid:1)) on S p − i =0 (cid:0) { i } × (cid:0) , x β ( i ) (cid:3)(cid:1) . Theorem 30.
The map φ β is an isomorphism between the dynamical systems (cid:0) S p − i =0 (cid:0) { i }× [0 , x β ( i ) ) (cid:1) , T β , µ β , T β (cid:1) and (cid:0) S p − i =0 (cid:0) { i } × (0 , x β ( i ) ] (cid:1) , L β , µ β ◦ φ − β , L β (cid:1) .Proof. Clearly, φ β is a bijective map. Hence, we only have to show that φ β ◦ T β = L β ◦ φ β .Let ( i, x ) ∈ S p − i =0 (cid:0) { i } × [0 , x β ( i ) ) (cid:1) . First, suppose that x ∈ [0 , . Then φ β ◦ T β ( i, x ) = (cid:0) ( i + 1) mod p, x β ( i +1) − β i x + ⌊ β i x ⌋ (cid:1) and L β ◦ φ β ( i, x ) = (cid:0) ( i + 1) mod p, β i ( x β ( i ) − x ) − ⌈ β i ( x β ( i ) − x ) − x β ( i +1) ⌉ (cid:1) . Second, suppose that x ∈ [1 , x β ( i ) ) . Then φ β ◦ T β ( i, x ) = (cid:0) ( i + 1) mod p, x β ( i +1) − β i x + ⌊ β i ⌋ − (cid:1) and L β ◦ φ β ( i, x ) = (cid:0) ( i + 1) mod p, β i ( x β ( i ) − x ) (cid:1) . In both cases, we easily get that φ β ◦ T β ( i, x ) = L β ◦ φ β ( i, x ) by using (4). (cid:3) Thanks to Theorem 30, we obtain an analogue of Theorem 24 for the lazy β -transfor-mation. Theorem 31.
The measure µ β ◦ φ − β is the unique L β -invariant probability measure on L β that is absolutely continuous with respect to λ p ◦ φ − β . Furthermore, µ β ◦ φ − β is equivalentto λ p ◦ φ − β on L β and the dynamical system (cid:0) S p − i =0 (cid:0) { i } × (0 , x β ( i ) ] (cid:1) , L β , µ β ◦ φ − β , L β (cid:1) isergodic and has entropy p log( β p − · · · β ) . YNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS 19
Similarly, we have an analogue of Theorem 23 for the lazy β -transformation, by consid-ering the σ -algebra L ′ β = ( p − [ i =0 ( { i } × B i ) : ∀ i ∈ [[0 , p − , B i ∈ B (cid:0) ( x β ( i ) − , x β ( i ) ] (cid:1)) . Theorem 32.
The measure µ β ◦ φ − β is the unique L β -invariant probability measure on L ′ β that is absolutely continuous with respect to λ p ◦ φ − β . Furthermore, µ β ◦ φ − β is equivalent to λ p ◦ φ − β on L ′ β and the dynamical system (cid:0) S p − i =0 (cid:0) { i } × ( x β ( i ) − , x β ( i ) ] (cid:1) , L ′ β , µ β ◦ φ − β , L β (cid:1) is ergodic and has entropy p log( β p − · · · β ) . Remark 33.
We deduce from Theorem 30 that if the greedy β -expansion of a real number x ∈ [0 , x β ) is a a a · · · , then the lazy β -expansion of x β − x is ( ⌈ β ⌉ − − a )( ⌈ β ⌉ − − a )( ⌈ β ⌉ − − a ) · · · . 6. Isomorphism with the β -shift The aim of this section is to generalize the isomorphism between the greedy β -transfor-mation and the β -shift to the framework of alternate bases. We start by providing somebackground of the real base case.Let D β denote the set of all greedy β -expansions of real numbers in the interval [0 , .The β -shift is the set S β defined as the topological closure of D β with respect to the prefixdistance of infinite words. For an alphabet A , we let C A denote the σ -algebra generatedby the cylinders C A ( a , . . . , a ℓ − ) = { w ∈ A N : w [0] = a , . . . , w [ ℓ −
1] = a ℓ − } for all ℓ ∈ N and a , . . . , a ℓ − ∈ A , where the notation w [ k ] designates the letter at position k in the infinite word w , and we call σ A : A N → A N , a a a · · · 7→ a a a · · · the shift operator over A . If no confusion is possible, we simply write σ instead of σ A .Then the map ψ β : [0 , → S β , x d β ( x ) defines an isomorphism between the dynamicalsystems ([0 , , B ([0 , , µ β , T β ) and ( S β , { C ∩ S β : C ∈ C A β } , µ β ◦ ψ − β , σ | S β ) where A β denote the alphabet of digits [[0 , ⌈ β ⌉ − .Now, let us extend the previous notation to the framework of alternate bases. Let A β denote the alphabet [[0 , max i ∈ [[0 ,p − ⌈ β i ⌉ − , let D β denote the subset of A N β made of all greedy β -expansions of real numbers in [0 , and let S β denote the topological closure of D β withrespect to the prefix distance of infinite words: D β = { d β ( x ) : x ∈ [0 , } and S β = D β . The following lemma was proved in [3].
Lemma 34.
For all n ∈ N , if w ∈ S β ( n ) then σ ( w ) ∈ S β ( n +1) . Consider the σ -algebra G β = ( p − [ i =0 (cid:0) { i } × ( C i ∩ S β ( i ) ) (cid:1) : C i ∈ C A β ) on S p − i =0 ( { i } × S β ( i ) ) . We define σ p : p − [ i =0 ( { i } × S β ( i ) ) → p − [ i =0 ( { i } × S β ( i ) ) , ( i, w ) (( i + 1) mod p, σ ( w )) ψ β : [[0 , p − × [0 , → p − [ i =0 ( { i } × S β ( i ) ) , ( i, x ) ( i, d β ( i ) ( x )) . Note that the transformation σ p is well defined by Lemma 34. Theorem 35.
The map ψ β defines an isomorphism between the dynamical systems (cid:0) [[0 , p − × [0 , , T p , µ β , T β (cid:1) and p − [ i =0 ( { i } × S β ( i ) ) , G β , µ β ◦ ψ − β , σ p ! . Proof.
It is easily seen that ψ β ◦ T β = σ p ◦ ψ β and that ψ β is injective. (cid:3) However, since ψ β is not surjective, it does not define a topological isomorphism. Remark 36.
In view of Theorem 35, the set S p − i =0 ( { i } × S β ( i ) ) can be seen as the β -shift , that is, the generalization of the β -shift to alternate bases. However, in the previouswork [3], what we called the β -shift is the union S p − i =0 S β ( i ) . This definition was motivatedby the following combinatorial result : the set S p − i =0 S β ( i ) is sofic if and only if for every i ∈ [[0 , p − , the quasi-greedy β ( i ) -representation of is ultimately periodic. In summary,we can say that there are two ways to extend the notion of β -shift to alternate bases β ,depending on the way we look at it: either as a dynamical object or as a combinatorialobject.Thanks to Theorem 35, we obtain an analogue of Theorem 23 for the transformation σ p . Theorem 37.
The measure ρ β is the unique σ p -invariant probability measure on G β thatis absolutely continuous with respect to λ p ◦ ψ − β . Furthermore, ρ β is equivalent to λ p ◦ ψ − β on G β and the dynamical system (cid:0) S p − i =0 ( { i } × S β ( i ) ) , G β , ρ β , σ p (cid:1) is ergodic and has entropy p log( β p − · · · β ) . Remark 38.
Let D ′ β denote the subset of A N β made of all lazy β -expansions of real numbersin ( x β − , x β ] and let S ′ β denote the topological closure of D ′ β with respect to the prefixdistance of infinite words. From Remark 33, it is easily seen that θ β : p − [ i =0 ( { i }× S β ( i ) ) → p − [ i =0 ( { i }× S ′ β ( i ) ) , ( i, a a · · · ) ( i, ( ⌈ β i ⌉− − a )( ⌈ β i +1 ⌉− − a ) · · · ) defines a isomorphism from (cid:0) S p − i =0 ( { i } × S β ( i ) ) , G β , ρ β , σ p (cid:1) to (cid:0) S p − i =0 ( { i } × S ′ β ( i ) ) , G ′ β , ρ β ◦ θ − β , σ ′ p (cid:1) where G ′ β = ( p − [ i =0 (cid:0) { i } × ( C i ∩ S ′ β ( i ) ) (cid:1) : C i ∈ C A β ) σ ′ p : p − [ i =0 ( { i } × S ′ β ( i ) ) → p − [ i =0 ( { i } × S ′ β ( i ) ) , ( i, w ) (( i + 1) mod p, σ ( w )) . YNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS 21
We then deduce from Theorem 30 and 35 that θ β ◦ ψ β ◦ φ − β is an isomorphism from (cid:0) S p − i =0 (cid:0) { i } × ( x β ( i ) − , x β ( i ) ] (cid:1) , L β , µ β ◦ φ − β , L β (cid:1) to (cid:0) S p − i =0 ( { i } × S ′ β ( i ) ) , G ′ β , ρ β ◦ θ − β , σ ′ p (cid:1) where here φ β denoted the restricted map φ β : p − [ i =0 (cid:0) { i } × [0 , (cid:1) → p − [ i =0 (cid:0) { i } × ( x β ( i ) − , x β ( i ) ] (cid:1) , ( i, x ) (cid:0) i, x β ( i ) − x (cid:1) . It is easy to check that, as expected, that for all ( i, x ) ∈ S p − i =0 (cid:0) { i } × ( x β ( i ) − , x β ( i ) ] , wehave θ β ◦ ψ β ◦ φ − β ( i, x ) = ( i, ℓ β ( i ) ( x )) where ℓ β ( x ) denoted the lazy β -expansion of x .7. β -expansions and ( β p − · · · β , ∆ β ) -expansions By rewriting Equality (1) from Section 3 as x = β p − · · · β a + β p − · · · β a + · · · + a p − β p − · · · β (9) + β p − · · · β a p + β p − · · · β a p +1 + · · · + a p − ( β p − · · · β ) + · · · we can see the greedy and lazy β -expansions of real numbers as ( β p − · · · β ) -representationsover the digit set ∆ β = ( p − X i =0 β p − · · · β i +1 c i : ∀ i ∈ [[0 , p − , c i ∈ [[0 , ⌈ β i ⌉ − ) . In this section, we examine some cases where by considering the greedy (resp. lazy) β -expansion and rewriting it as (9), the obtained representation is the greedy (resp. lazy) ( β p − · · · β , ∆ β ) -expansion. We first recall the formalism of β -expansions of real numbersover a general digit set [16].7.1. Real base expansions over general digit sets.
Consider an arbitrary finite set ∆ = { d , d , . . . , d m } ⊂ R where d < d < · · · < d m . Then a ( β, ∆) -representation ofa real number x in the interval [0 , d m β − ) is an infinite sequence a a a · · · over ∆ such that x = P ∞ n =0 a n β n +1 . Such a set ∆ is called an allowable digit set for β if(10) max k ∈ [[0 ,m − ( d k +1 − d k ) ≤ d m β − . In this case, the greedy ( β, ∆) -expansion of a real number x ∈ [0 , d m β − ) is defined recursivelyas follows: if the first N digits of the greedy ( β, ∆) -expansion of x are given by a , . . . , a N − ,then the next digit a N is the greatest element in ∆ such that N X n =0 a n β n +1 ≤ x. The greedy ( β, ∆) -expansion can also be obtained by iterating the greedy ( β, ∆) -transfor-mation T β, ∆ : [0 , d m β − ) → [0 , d m β − ) , x ( βx − d k if x ∈ [ d k β , d k +1 β ) , k ∈ [[0 , m − βx − d m if x ∈ [ d m β , d m β − ) as follows: for all n ∈ N , a n is the greatest digit d in ∆ such that dβ ≤ T nβ, ∆ ( x ) [5]. Example 39.
Consider the digit set ∆ = { , , ϕ + ϕ , ϕ } . It is easily checked that ∆ isan allowable digit set for ϕ . The greedy ( ϕ, ∆) -transformation T ϕ, ∆ : [0 , ϕ ϕ − ) → [0 , ϕ ϕ − ) , x ϕx if x ∈ [0 , ϕ ) ϕx − if x ∈ [ ϕ , ϕ ) ϕx − ( ϕ + ϕ ) if x ∈ [1 + ϕ , ϕ ) ϕx − ϕ if x ∈ [ ϕ, ϕ ϕ − ) is depicted in Figure 10. ϕ ϕ ϕ ϕ ϕ − ϕ ϕ ϕ ϕ − Figure 10.
The transformation T ϕ, ∆ for ∆ = { , , ϕ +1 ϕ , ϕ } .Similarly, if ∆ is an allowable digit set for β , then the lazy ( β, ∆) -expansion of a realnumber x ∈ (0 , d m β − ] is defined recursively as follows: if the first N digits of the lazy ( β, ∆) -expansion of x are given by a , . . . , a N − , then the next digit a N is the least element in ∆ such that N X n =0 a n β n +1 + ∞ X n = N +1 d m β n +1 ≥ x. The lazy ( β, ∆) -transformation L β, ∆ : (0 , d m β − ] → (0 , d m β − ] , x ( βx if x ∈ (0 , d m β − − d m β ] βx − d k if x ∈ ( d m β − − d m − d k − β , d m β − − d m − d k β ] , k ∈ [[1 , m ]] can be used to obtain the digits of the lazy ( β, ∆) -expansions: for all n ∈ N , a n is the leastdigit d in ∆ such that dβ + P ∞ k =1 d m β k +1 ≥ L nβ, ∆ ( x ) [5].In [5], it is shown that if ∆ is an allowable digit set for β then so is the set e ∆ := { , d m − d m − , . . . , d m − d , d m } and φ β, ∆ : [0 , d m β − ) → (0 , d m β − ] , x d m β − − x is a bicontinuous bijection satisfying L β, e ∆ ◦ φ β, ∆ = φ β, ∆ ◦ T β, ∆ . YNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS 23
Example 40.
Consider the digit set e ∆ where ∆ is the digit set from Example 39. We get e ∆ = { , − ϕ , ϕ, ϕ } . The lazy ( ϕ, e ∆) -transformation L ϕ, e ∆ : (0 , ϕ ϕ − ] → (0 , ϕ ϕ − ] , x ϕx if x ∈ (0 , ϕϕ − ] ϕx − (1 − ϕ ) if x ∈ ( ϕϕ − , ϕ +3 ϕ ] ϕx − ϕ if x ∈ ( ϕ +3 ϕ , ϕ − ϕ − ] ϕx − ϕ if x ∈ ( ϕ − ϕ − , ϕ ϕ − ] is depicted in Figure 11. It is conjugate to the greedy ( ϕ, ∆) -transformation T ϕ, ∆ by φ ϕ, ∆ : [0 , ϕ ϕ − ) → (0 , ϕ ϕ − ] , x ϕ ϕ − − x . ϕϕ − ϕ +3 ϕ ϕ − ϕ − ϕ ϕ − ϕ − ϕ − ϕ ϕ − Figure 11.
The transformation L ϕ, e ∆ for ∆ = { , , ϕ + ϕ , ϕ } .7.2. Comparison between β -expansions and ( β p − · · · β , ∆ β ) -expansions. The digitset ∆ β has cardinality at most Q p − i =0 ⌈ β i ⌉ and can be rewritten ∆ β = im( f β ) where f β : p − Y i =0 [[0 , ⌈ β i ⌉ − → R , ( c , . . . , c p − ) p − X i =0 β p − · · · β i +1 c i . Note that f β is not injective in general. Let us write ∆ β = { d , d . . . , d m } with d 0) = 0 , d = f β (0 , . . . , , 1) = 1 and d m = f β ( ⌈ β ⌉− , . . . , ⌈ β p − ⌉− . In what follows, we suppose that Q p − i =0 [[0 , ⌈ β i ⌉− is equippedwith the lexicographic order: ( c , . . . , c p − ) < lex ( c ′ , . . . , c ′ p − ) if there exists i ∈ [[0 , p − such that c = c ′ , . . . , c i − = c ′ i − and c i < c ′ i . Lemma 41. The set ∆ β is an allowable digit set for β p − · · · β .Proof. We need to check Condition (10). We have d = 0 and d m = f β ( ⌈ β ⌉ − , . . . , ⌈ β p − ⌉ − ≥ p − X i =0 β p − · · · β i +1 ( β i − 1) = β p − · · · β − , Therefore, it suffices to show that for all k ∈ [[0 , m − , d k +1 − d k ≤ . Thus, we only haveto show that f ( c ′ , . . . , c ′ p − ) − f ( c , . . . , c p − ) ≤ where ( c , . . . , c p − ) and ( c ′ , . . . , c ′ p − ) arelexicographically consecutive elements of Q p − i =0 [[0 , ⌈ β i ⌉ − . For such p -tuples, there exists j ∈ [[0 , p − such that c = c ′ , . . . , c j − = c ′ j − , c j = c ′ j − , c j +1 = ⌈ β j +1 ⌉ − , . . . , c p − = ⌈ β p − ⌉ − and c ′ j +1 = · · · = c ′ p − = 0 . Then f ( c ′ , . . . , c ′ p − ) − f ( c , . . . , c p − ) = β p − · · · β j +1 − p − X i = j +1 β p − · · · β i +1 ( ⌈ β i ⌉ − ≤ β p − · · · β j +2 − p − X i = j +2 β p − · · · β i +1 ( ⌈ β i ⌉ − ... ≤ β p − − ( ⌈ β p − ⌉ − ≤ . (cid:3) Since x β = d m β p − ··· β − , it follows from Lemma 41 that every point in [0 , x β ) admits agreedy ( β p − · · · β , ∆ β ) -expansion.Let us restate Proposition 7 when n equals p in terms of the map f β . Lemma 42. For all x ∈ [0 , x β ) , we have π ◦ T p β ◦ δ ( x ) = β p − · · · β x − f β ( c ) where c is the lexicographically greatest p -tuple in Q p − i =0 [[0 , ⌈ β i ⌉ − such that f β ( c ) β p − ··· β ≤ x . Proposition 43. For all x ∈ [0 , x β ) , we have T β p − ··· β , ∆ β ( x ) ≤ π ◦ T p β ◦ δ ( x ) and L β p − ··· β , ∆ β ( x ) ≥ π ◦ L p β ◦ δ ( x ) .Proof. Let x ∈ [0 , x β ) . On the one hand, T β p − ··· β , ∆ β ( x ) = β p − · · · β x − d where d is the greatest digit in ∆ β such that dβ p − ··· β ≤ x . On the other hand, by Lemma 42, π ◦ T p β ◦ δ ( x ) = β p − · · · β x − f β ( c ) where c is the greatest p -tuple in Q p − i =0 [[0 , ⌈ β i ⌉ − such that f β ( c ) β p − ··· β ≤ x . By definition of d , we get d ≥ f β ( c ) . Therefore, we obtain that T β p − ··· β , ∆ β ( x ) ≤ π ◦ T p β ◦ δ ( x ) . The inequality L β p − ··· β , ∆ β ( x ) ≥ π ◦ L p β ◦ δ ( x ) thenfollows from Theorem 30. (cid:3) In what follows, we provide some conditions under which the inequalities of Proposi-tion 43 happen to be equalities. Proposition 44. The transformations T β p − ··· β , ∆ β and π ◦ T p β ◦ δ (cid:12)(cid:12) [0 ,x β ) coincide if andonly if the transformations L β p − ··· β , ∆ β and π ◦ L p β ◦ δ (cid:12)(cid:12) (0 ,x β ] do.Proof. We only show the forward direction, the backward direction being similar. Supposethat T β p − ··· β , ∆ β = π ◦ T p β ◦ δ (cid:12)(cid:12) [0 ,x β ) and let x ∈ (0 , x β ] . Since x β = d m β p − ··· β − and ∆ β = f ∆ β , we successively obtain that L β p − ··· β , ∆ β ( x ) = L β p − ··· β , ∆ β ◦ φ β p − ··· β , ∆ β ( x β − x )= φ β p − ··· β , ∆ β ◦ T β p − ··· β , ∆ β ( x β − x )= φ β p − ··· β , ∆ β ◦ π ◦ T p β ◦ δ ( x β − x )= π ◦ φ β ◦ T p β ◦ δ ( x β − x )= π ◦ L p β ◦ φ β ◦ δ ( x β − x ) YNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS 25 = π ◦ L p β ◦ δ ( x ) . (cid:3) The next result provides us with a sufficient condition under which the transformations T β p − ··· β , ∆ β and π ◦ T p β ◦ δ (cid:12)(cid:12) [0 ,x β ) coincide. Here, the non-decreasingness of the map f β refers to the lexicographic order: for all c, c ′ ∈ Q p − i =0 [[0 , ⌈ β i ⌉ − , c < lex c ′ = ⇒ f β ( c ) ≤ f β ( c ′ ) . Theorem 45. If the map f β is non-decreasing then T β p − ··· β , ∆ β = π ◦ T p β ◦ δ (cid:12)(cid:12) [0 ,x β ) .Proof. We keep the same notation as in the proof of Proposition 43. Let c ′ ∈ Q p − i =0 [[0 , ⌈ β i ⌉− such that d = f β ( c ′ ) . By definition of c , we get c ≥ lex c ′ . Now, if f β is non-decreasingthen f β ( c ) ≥ f β ( c ′ ) = d . Hence the conclusion. (cid:3) The following example shows that considering the length- p alternate base β = ( β, . . . , β ) with p ∈ N ≥ , it may happen that T β p , ∆ β differs from π ◦ T p β ◦ δ (cid:12)(cid:12) [0 ,x β ) . This result wasalready proved in [4]. Example 46. Consider the alternate base β = ( ϕ , ϕ , ϕ ) . Then ∆ β = { ϕ c + ϕ c + c : c , c , c ∈ { , , }} . In [4, Proposition 2.1], it is proved that T β n , ∆ β = T nβ for all n ∈ N if and only if f β is non-decreasing. Since f β (0 , , 2) = 2 ϕ + 2 > ϕ = f β (1 , , ,the tranformations T ϕ , ∆ β and π ◦ T β ◦ δ (cid:12)(cid:12) [0 ,x β ) differ by [4, Proposition 2.1].Whenever f β is not non-decreasing, the transformations T β p − ··· β , ∆ β and π ◦ T p β ◦ δ (cid:12)(cid:12) [0 ,x β ) can either coincide or not. The following two examples illustrate both cases. In particular,Example 48 shows that the sufficient condition given in Theorem 45 is not necessary. Example 47. Consider the alternate base β = ( ϕ, ϕ, √ . Then ∆ β = {√ ϕc + √ c + c : c , c ∈ { , } , c ∈ { , , }} . However, f β (0 , , 2) = √ ≃ . and f β (1 , , 0) = √ ϕ ≃ . . It can be easily check that there exists x ∈ [0 , x β ) such that T √ ϕ , ∆ β ( x ) = π ◦ T β ◦ δ ( x ) . For example, we can compute T √ ϕ , ∆ β (0 . ≃ . and π ◦ T β ◦ δ (0 . ≃ . . The transformations T √ ϕ , ∆ β and π ◦ T β ◦ δ (cid:12)(cid:12) [0 ,x β ) are depicted in Figure 12, wherethe red lines show the images of the interval (cid:2) √ √ ϕ , √ ϕ +1 √ ϕ (cid:1) ≃ [0 . , . , that is where thetwo transformations differ. Similarly, the transformations L √ ϕ , ∆ β and π ◦ L β ◦ δ (cid:12)(cid:12) (0 ,x β ] are depicted in Figure 13. As illustrated in red, the two transformations differ on theinterval φ √ ϕ , ∆ β (cid:16)(cid:2) √ √ ϕ , √ ϕ +1 √ ϕ (cid:1)(cid:17) ≃ (0 . , . . Example 48. Consider the alternate base β = ( , , . We have ∆ β = [[0 , . Themap f β is not non-decreasing since we have f β (0 , , 3) = 7 and f β (1 , , 0) = 6 . However, T , ∆ β = π ◦ T β ◦ δ (cid:12)(cid:12) [0 ,x β ) and L , ∆ β = π ◦ L β ◦ δ (cid:12)(cid:12) [0 ,x β ) . The transformation T , ∆ β isdepicted in Figure 14.The next example illustrates that it may happen that the transformations T β p − ··· β , ∆ β and π ◦ T p β ◦ δ (cid:12)(cid:12) [0 ,x β ) indeed coincide on [0 , but not on [0 , x β ) . Example 49. Consider the alternate base β = ( √ , √ , √ ) . Then f β (0 , , > f β (1 , , and it can be checked that the maps T √ , ∆ β and π ◦ T β ◦ δ (cid:12)(cid:12) [0 ,x β ) differ on the interval (cid:2) f β (0 , , β β β , f β (1 , , β β β (cid:1) ≃ [1 . , . . However, the two maps coincide on [0 , . x β x β 11 0 x β x β Figure 12. The transformations T √ ϕ , ∆ β (left) and π ◦ T β ◦ δ (cid:12)(cid:12) [0 ,x β ) (right) with β = ( ϕ, ϕ, √ . x β x β x β − x β − x β x β x β − x β − Figure 13. The transformations L √ ϕ , ∆ β (left) and π ◦ L β ◦ δ (cid:12)(cid:12) [0 ,x β ) (right) with β = ( ϕ, ϕ, √ .Finally, we provide a necessary and sufficient condition for the map f β to be non-decreasing. Proposition 50. The map f β is non-decreasing if and only if for all j ∈ [[1 , p − , (11) p − X i = j β p − · · · β i +1 ( ⌈ β i ⌉ − ≤ β p − · · · β j . Proof. If the map f β is non-decreasing then for all j ∈ [[1 , p − , p − X i = j β p − · · · β i +1 ( ⌈ β i ⌉ − 1) = f β (0 , . . . , , , ⌈ β j ⌉ − , . . . , ⌈ β p − ⌉ − YNAMICAL BEHAVIOR OF ALTERNATE BASE EXPANSIONS 27 x β x β Figure 14. The transformations T , ∆ β where β = ( , , . ≤ f β (0 , . . . , , , , . . . , β p − · · · β j . Conversely, suppose that (11) holds for all j ∈ [[1 , p − and that ( c , . . . , c p − ) and ( c ′ , . . . , c ′ p − ) are p -tuples in Q p − i =0 [[0 , ⌈ β i ⌉ − such that ( c , . . . , c p − ) < lex ( c ′ , . . . , c ′ p − ) .Then there exists j ∈ [[0 , p − such that c = c ′ , . . . , c j − = c ′ j − and c j ≤ c ′ j − . We get f β ( c , . . . , c p − ) ≤ j X i =0 β p − · · · β i +1 c ′ i − β p − · · · β j +1 + p − X i = j +1 β p − · · · β i +1 ( ⌈ β i ⌉ − ≤ j X i =0 β p − · · · β i +1 c ′ i ≤ f β ( c ′ , . . . , c ′ p − ) . (cid:3) Corollary 51. If p = 2 then T β β , ∆ β = π ◦ T β ◦ δ (cid:12)(cid:12) [0 ,x β ) . In particular, T β β , ∆ β (cid:12)(cid:12) [0 , = T β ◦ T β .Proof. This follows from Theorem 45 and Proposition 50. (cid:3) Example 52. Consider once more the alternate base β = ( √ , √ ) from Example 6.Then ∆ β = { , , β , β + 1 , β , β + 1 } and x β = β +1 β β − = √ . The transformations π ◦ T β ◦ δ (cid:12)(cid:12) [0 ,x β ) and π ◦ L β ◦ δ (cid:12)(cid:12) (0 ,x β ] are depicted in Figure 15. By Corollary 51, theycoincides with T β β , ∆ β and L β β , ∆ β respectively.8. Acknowledgment We thank Julien Leroy for suggesting Lemma 16, which allowed us to simplify severalproofs. Célia Cisternino is supported by the FNRS Research Fellow grant 1.A.564.19F. x β x β x β − x β − x β x β Figure 15. The transformations π ◦ T β ◦ δ (cid:12)(cid:12) [0 ,x β ) (left) and π ◦ L β ◦ δ (cid:12)(cid:12) (0 ,x β ] (right) for β = ( √ , √ ) . References [1] L. M. Abramov. The entropy of a derived automorphism. Dokl. Akad. Nauk SSSR , 128:647–650, 1959.[2] A. Boyarsky and P. Góra. Laws of chaos . Probability and its Applications. Birkhäuser Boston, Inc.,Boston, MA, 1997. Invariant measures and dynamical systems in one dimension.[3] É. Charlier and C. Cisternino. Expansions in Cantor real bases. Submitted in 2020, arXiv: 2102.07722.[4] K. Dajani, M. de Vries, V. 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