1FINITENESS AND PERIODICITY OF BETA EXPANSIONS NUMBER THEORETICAL AND DYNAMICAL OPEN
PROBLEMSSHIGEKI AKIYAMA (NIIGATA UNIVERSITY, JAPAN)

Let y = f (x) be a positive real function. Consider a digital expansion of a real
number x in a form:
x = ε0 + f (ε1 + f (ε2 + f (ε3 + . . . .
given by an algorithm with ε0 = bxc, r0 = x − bxc andεn+1 = bf −1 (rn )c, rn+1 = f −1 (rn ) − bf −1 (rn )c.
This is Rényi’s f -expansion ([24]). It is the usual b-adic expansion when f (x) =x/b and the regular continued fraction when f (x) = 1/x. For f (x) = x/β with a
non-integer β > 1, it is called the β-expansion.

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1

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(a) 3-adic expansion

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1

(b) Continued fraction

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(c) Beta expansion

Figure 1. The graph of f −1 mod 1
Rényi studied ergodic properties of f -expansion for monotone function f withmild growth condition. The images of f −1 consists of ‘full’ copies of intervals [0, 1]
in left two graphs, which implies that the corresponding digits are independent,
while the last case has dependent digits. We start with our fundamental question:
Problem 1. Find a good f -expansion.
Historically, several diﬀerent ‘goodness’ of f -expansion are studied.
• Does it have a nice invariant measure ? We expect a measure equivalent to
the Lebesgue measure with an explicit Radon-Nikodym derivative.
1
Based on the talk on 24 March 2009 at ‘Numeration: Mathematics and Computer Science’(CIRM)

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SHIGEKI AKIYAMA (NIIGATA UNIVERSITY, JAPAN)• Does it have nice ergodic properties (Mixing, unique measure of maximum
entropy) ?
• Does it require little memory to judge admissibility of words (i.e., certain
Markovian property) ?
• Can we characterize periodic orbits and/or ﬁnite orbits ? (The expansion
is ﬁnite, if the digits end up in zeros 0∞ .)• Is there a useful small natural extension ?
• Is there an ‘additive’ dynamical system whose successive induced systems
is described by f -expansion ? (Kamae’s number system [15])• Can we apply this expansion to Diophantine approximation ?
In fact, these questions are intimately related. Hereafter we discuss β-expansion.
This is too simple and we can not expect last two number theoretical ‘goodness’
but others could be achieved for some β’s.
Fix a real number β > 1 and consider a map Tβ (x) = βx − bβxc from [0, 1) toitself. The trajectory of x is written as
a

a

a

a

1
2
3
4
x −→
Tβ (x) −→Tβ2 (x) −→Tβ3 (x) −→
...

with

j
k
ai = βTβi−1 (x) ∈ A := Z ∩ [0, β).

Then x ∈ [0, 1) is uniquely expanded as:
a1
a2
a3
x=
+ 2 + 3 + ...
βββthrough greedy algorithm. We summarize its ergodic properties.
• Rényi (Rényi [24]) showed that it is ergodic with an absolute continuous invariant measure. From this proof, it is weak mixing. It has unique measureof maximum entropy ([28], [12]).
• Its Radon-Nikodym derivative is made explicit (Parry [23]).
• It is ‘exact’ in the sense of Rohlin [25]. This implies mixing of all degree.
• Its natural extension is a Bernoulli shift. (Smorodinski [27] , Fischer [10],Ito-Takahashi [14])• However Tβ itself is not Bernoulli except when β is an integer (KuboMurata-Totoki [17]).
The clue to analyze Markovian property of a piecewise linear transformation, isthe orbit of discontinuity. In the case of beta expansion, the study of the orbits is
reduced to so called ‘expansion of 1’. Beta expansion deﬁnes a map dβ : [0, 1) →AN . This dβ is not surjective when β 6∈ Z. The expansion of 1 is deﬁned bydβ (1−) = limε↓0 dβ (1 − ε). Let c1 = bβc. Then dβ (1−) is the concatenation of c1
and the beta expansion of β − bβc when this expansion is inﬁnite. If it is ﬁnite, i.e.,dβ (β − bβc) = c2 c3 . . . c` , thendβ (1−) = (c1 c2 . . . c`−1 (c` − 1))∞ .
For x = a1 a2 a3 · · · ∈ AN , we havex ∈ dβ ([0, 1)) ⇐⇒ σ n (x) <lex dβ (1−) (n = 0, 1, 2, . . . )where σ(a1 a2 . . . ) = a2 a3 . . . (shift operator) and <lex is the lexicographical order(Parry [23], Ito-Takahashi [14]). Xβ is the closure of dβ ([0, 1)) in AN . This set isclosed and σ(Xβ ) = Xβ . This deﬁnes the beta shift (Xβ , σ). The element in Xβ
is characterized by σ n (x) ≤lex dβ (1−) (∀n).

FINITENESS AND PERIODICITY OF BETA EXPANSIONS - NUMBER THEORETICAL AND DYNAMICAL OPEN PROBLEM√
Example 1. dβ (1−) = (10)∞ holds for β = (1 + 5)/2. dβ ([0, 1)) is the set of
inﬁnite words on {0, 1} that 11 and the tail (10)∞ are forbidden. Xβ is exactly theset of inﬁnite words on {0, 1} without 11.
Example 2. If β > 1 is the root of x3 − x2 − x − 1, then dβ (1−) = (110)∞holds. dβ ([0, 1)) is the set of inﬁnite words on {0, 1} that 111 and the tail (110)∞
are forbidden. Xβ is exactly the set of inﬁnite words on {0, 1} without 111.
From this dβ (1−) we can introduce symbolic dynamical classiﬁcation of β.
• β is a simple Parry number if dβ (1−) is purely periodic. It is equivalent
to the fact that Xβ is a subshift of ﬁnite type (SFT).
• β is a Parry number if dβ (1−) is eventually periodic. This is exactly thecase when Xβ is soﬁc, i.e., it is a factor of SFT.
• β is a Delone number if dβ (1−) has bounded run of 0’s. Xβ has ‘speciﬁed’
property.
Many questions on this classiﬁcation remain open (Blanchard [7]). We pick up
a few of them:
Problem 2. (Salem Periodicity Problem 1) Is there a non-Parry Salem number? How about x6 − 3x5 − x4 − 7x3 − x2 − 3x + 1 (Boyd [8]) ?
Problem 3. √Is there an algebraic Delone number except Parry numbers ? Whatabout 3/2 or 2 ?
If β is a Pisot number, then dβ (x) is eventually periodic for x ∈ Q(β) ∩ [0, 1).
Conversely if dβ (x) is eventually periodic for x ∈ Q ∩ [0, 1) then β must be a Pisot
or Salem number (Schmidt [26]). Therefore ‘Pisot’ implies ‘Parry’.
Problem 4. (Salem Periodicity Problem 2) For Salem β, is dβ (x) eventuallyperiodic for all x ∈ Q(β) ∩ [0, 1) ?
The answer is expected negative and there are possible counter-examples (Boyd[8, 9]) in degree ≥ 6.
Problem 5. For Salem β, is there a way to prove that dβ (x) is non-periodic for aﬁxed x ?
For Salem number of degree 4, the above periodicity conjecture seems valid bothby numerical experiments and by heuristic consideration.
We say that the expansion of x is ﬁnite, if dβ (x) ends up in 0∞ . Clearly in thiscase x ∈ Z[1/β]. Frougny-Solomyak [11] studied the property(F) dβ (x) is ﬁnite for all x ∈ Z[1/β] ∩ [0, 1).
This implies that β is a Pisot number. Converse is not true. For e.g, if β has
a positive conjugate other than itself, then it can not satisfy (F). The problem tocharacterize β’s with (F) is open for degree ≥ 3, and it is transformed into thefollowing shift radix system problem. The vector (r0 , r1 , . . . , rd−1 ) ∈ Rd gives ashift radix system, SRS, if the integer sequence deﬁned by
0 ≤ r0 an + r1 an+1 + · · · + rd−1 an+d−1 + an+d < 1always falls into a trivial cycle 0d → 0d .
Example 3. (1/2, 1) gives a shift radix system. For e.g.,(−5, 3) → (3, 0) → (0, −1) → (−1, 1) → (1, 0) → (0, 0) → (0, 0)

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SHIGEKI AKIYAMA (NIIGATA UNIVERSITY, JAPAN)

by the recurrence:
0 ≤ an /2 + an+1 + an+2 < 1.
Indeed any orbits ends up in the trivial cycle (0, 0) → (0, 0).
A necessary condition for SRS (r0 , r1 , . . . , rd−1 ) is that xd + rd−1 xd−1 + · · · + r0
is semi-contractive, i.e., all roots of the polynomial has modulus not greater thanone. If it is contractive, i.e., all roots have modulus less than one, then the orbits
must be eventually periodic. This is a simple fact on a contractive transformation
acting on the lattice.
Take a Pisot number β and its minimal polynomial p(x) which is factorized intop(x) = (x − β)(xd−1 + rd−1 xd−1 + · · · + r1 x + r0 )in C. Then one can prove that β has property (F) if and only if (r0 , r1 , . . . , rd−1 )gives a SRS. Semi-contractive cases correspond to the above-stated Salem periodicity problems.

(1,2)

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5.10
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5.1 5.11

0.5(−1,0)
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T.4.27-1(1,−2)(a) 2d SRS Approximation(b) Known regions

In Figure (a), the isosceles triangle surrounded by broken edges is the semicontractive region while the shaded part gives an approximation of SRS parameters.
White (mostly polygonal) regions in Figure (b) are shown (in pretty diﬀerent ways!)
to be in SRS and dark grey parts are not in SRS. Light grey parts are not settled.
Problem 6. Must SRS polynomials be contractive ?
This is true for d ≤ 2. We know that r0 6= 1 for SRS in general.

FINITENESS AND PERIODICITY OF BETA EXPANSIONS - NUMBER THEORETICAL AND DYNAMICAL OPEN PROBLEM

Problem 7. Is SRS region connected ?
This seems likely by observing the approximations. However a little skeptic
40 30
evidence exists as well. P.Surer found a cut point ( 41
, 41 ) of SRS region in d = 2 !
Problem 8. Prove (or disprove) that {(x, y) | 0 < 2x < y < x + 1} is a SRS region.
This seems rather tough and a brand new idea is expected for this possible SRS
region in d = 2. Figure (b) reads one can substitute 0 by 0.8. The essential diﬃcultyof this problem exists around (1, 2).
Problem 9. Let |γ| < 2. Prove that each integer sequence {an }n satisfying 0 ≤an + γan+1 + an+2 < 1 is eventually periodic.
This easy-looking problem seems quite hard except for trivial cases γ = 0, ±1.
This ‘Salem type periodicity’ problem is related to the orbit of piecewise isometry
and aﬃrmative answers are known for certain quadratic γ’s (Kouptsov-LowensteinVivaldi [16], Akiyama-Brunotte-Pethő-Steiner [4]).
Let us switch our topic and brieﬂy discuss dual tilings due to Thurston. It is
intimately related to the explicit construction of an algebraic natural extension of βexpansion (see [3] for detail.) To this matter, it is important to extend β-expansion
to the other direction. We introduce a set of β-integers:
¯
( n
)
¯
X
i ¯
ai β ¯ n ∈ N, an an−1 . . . a0 ∈ dβ ([0, 1))
Zβ =
¯
i=0

This set is a Delone set (relatively dense and uniformly discrete) in R ∩ [0, ∞) if
and only if β is a Delone number. Let β be a Pisot number of degree d with r1 real
conjugates and 2r2 complex conjugates. We assume that β is a unit. Consider anembeddingΦ : Q(β) → Rr1 −1 × Cr2 ' Rd−1
deﬁned by x 7→ (x(2) , . . . , x(r1 ) , x(r1 +1) , . . . , x(r1 +r2 ) ) where x(i) are the non trivial
Galois conjugates of x. Since β is Pisot, the set Φ(Zβ ) becomes compact, which iscalled the central tile. This central tile has a natural self-similar structure arosefrom multiplication/division by β. In fact, we can construct a covering of Rd−1 asin Figure 2 but we are not sure that it always gives a tiling, a covering of degree 1.
Since β is Pisot, dβ (1−) is eventually periodic and consequently the corresponding symbolic dynamics is soﬁc. Under the weak ﬁnite condition:(W) For any z ∈ Z[β] ∩ [0, 1), there are x, y with ﬁnite expansion such thatz = x − y,this construction surely gives a tiling of Rd−1 by ﬁnitely many tiles up to translation,
which is a geometric realization of a soﬁc shift Xβ ([2]). The condition (F) is clearlystronger than (W). Under (F), the origin is an inner point of the central tile. Thiscondition (W) seems to hold for all Pisot numbers.
This (W) is understood as a special form of Pisot conjecture in substitutive
dynamical system. An arithmetic natural extension of ([0, 1), Tβ ) in Rd isconstructed by dual tiles as in Figure 3.

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SHIGEKI AKIYAMA (NIIGATA UNIVERSITY, JAPAN)

Figure 2. [110]∞ : Tribonacci CaseAs this Φ is basically composed by Galois conjugates, number theoretical information is preserved. Thanks to this advantage, one can characterize the set of periodic
expansions as Q(β)-points in the fractal shape (Ito-Rao [13]). The same idea works
for non-unit β’s by using p-adic embedding ( see Berthé-Siegel [6]).
Let τ be an irreducible Pisot substitution on {0, 1, . . . , d−1}. Pisot conjecturestates that the dynamical system generated by shift closure of a ﬁx point of τ has
pure discrete spectrum. Many equivalent coincidence conditions are known.
Here is one of them (a joint work with J.Y. Lee):
Problem 10. In the ﬁxed point of τ , is there a non empty word w and a relatively
dense occurrences of patterns (of inﬁnite size) of the form:
wx1 wx2 wx3 . . .
where xi are arbitrary words satisfying |wx1 wx2 . . . wxm | = |τ m−1 (wy)| for someﬁxed y ?
One can show this criterion from (W) for β-substitution.
To attack (W) or coincidence, it is worthy to consider a little weaker problem ofalgebraic nature, which is called ‘Height reducing problem’.

FINITENESS AND PERIODICITY OF BETA EXPANSIONS - NUMBER THEORETICAL AND DYNAMICAL OPEN PROBLEM

Figure 3. Natural extension
Problem 11. Let β be a Pisot number and put B = {0, ±1, ±2, . . . , ±bβc}. Prove
Z[1/β] ∩ [0, 1) = B[1/β] ∩ [0, 1).
Since we do not care that coeﬃcients in the right side can be represented asa diﬀerence of admissible words, this is a weaker statement than (W). One cancompare it with the same type of question in expanding case:
Problem 12. Let α be an algebraic integer whose all conjugates are greaterthan one in modulus and N (α) be the absolute norm of α over Q. Set B ={0, ±1, ±2, . . . , ±(N (α) − 1)}. Is there an easy proof of Z[α] = B[α] ?
To prove tiling property in a diﬀerent setting, Lagarias-Wang [21, 20, 22] gave
an indirect proof of this fact using Wavelet analysis. Class number problems are
related ([18, 19]).
To ﬁnish this note, I wish to mention my favorite conjecture posed in [1] inJapanese. Let β be a Parry number, i.e.,dβ (1−) = a1 a2 . . . am [am+1 am+2 . . . am+` ]∞with am 6= am+` .

Problem 13. (Dynamical Norm Conjecture) Prove that N (β) = |am − am+` |.
This is supported by extensive numerical computation. When β is a simple-Parry
Pisot unit, this conjecture implies that the associated tiles are connected. Here thesimpleness is necessary since tiles could be disconnected in degree 4 (Akiyama-Gjini[5]) for non-simple Parry cases.
References
1. S. Akiyama, The dynamical norm conjecture and Pisot tilings, SūrikaisekikenkyūshoKōkyūroku (1999), no. 1091, 241–250.
2. S. Akiyama, On the boundary of self aﬃne tilings generated by Pisot numbers, J. Math. Soc.
Japan 54 (2002), no. 2, 283–308.
3.
, Pisot number system and its dual tiling, Physics and Theoretical Computer Science,
IOS Press, 2007, pp. 133–154.
4. S. Akiyama, H. Brunotte, A. Pethő, and W. Steiner, Periodicity of certain piecewise aﬃneplanar maps, Tsukuba J. Math. 32 (2008), no. 1, 1–55.

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5. S. Akiyama and N. Gjini, Connectedness of number theoretic tilings, Discrete Math. Theor.
Comput. Sci. 7 (2005), no. 1, 269–312.
6. V. Berthé and A. Siegel, Purely periodic β-expansions in the Pisot non-unit case, J. Number
Theory 127 (2007), no. 2, 153–172.
7. F. Blanchard, β-expansions and symbolic dynamics, Theoret. Comput. Sci. 65 (1989), no. 2,
131–141.
8. D. W. Boyd, Salem numbers of degree four have periodic expansions, Number theory, Walterde Gruyter, 1989, pp. 57–64.
9.
, On the beta expansion for Salem numbers of degree 6, Math. Comp. 65 (1996),
861–875.
10. R. Fischer, Ergodische Theorie von Ziﬀernentwicklungen in Wahrscheinlichkeitsräumen,
Math. Z. 128 (1972), 217–230.
11. Ch. Frougny and B. Solomyak, Finite beta-expansions, Ergodic Theory Dynam. Systems 12(1992), no. 4, 713–723.
12. F. Hofbauer, β-shifts have unique maximal measure, Monatsh. Math. 85 (1978), no. 3, 189–
198.
13. Sh. Ito and H. Rao, Purely periodic β-expansions with Pisot unit base, Proc. Amer. Math.
Soc. 133 (2005), no. 4, 953–964.
14. Sh. Ito and Y. Takahashi, Markov subshifts and realization of β-expansions, J. Math. Soc.
Japan 26 (1974), no. 1, 33–55.
15. T. Kamae, Numeration systems, fractals and stochastic processes, Israel J. Math. 149 (2005),
87–135, Probability in mathematics.
16. K. Koupstov, J. H. Lowenstein, and F. Vivaldi, Quadratic rational rotations of the torus anddual lattice maps, Nonlinearity 15 (2002), 1795–1842.
17. I. Kubo, H. Murata, and H. Totoki, On the isomorphism problem for endomorphisms ofLebesgue spaces. II, Publ. Res. Inst. Math. Sci. 9 (1973/74), 297–303.
18. J. C. Lagarias and Y. Wang, Haar bases for L2 (Rn ) and algebraic number theory, J. Number
Theory 57 (1996), no. 1, 181–197.
19.
, Corrigendum/addendum: “Haar bases for L2 (Rn ) and algebraic number theory” [J.
Number Theory 57 (1996), no. 1, 181–197, J. Number Theory 76 (1999), no. 2, 330–336.
20. J.C. Lagarias and Y. Wang, Integral self-aﬃne tiles in Rn I. Standard and nonstandard digit
sets, J. London Math. Soc. 54 (1996), no. 2, 161–179.
21.
, Self-aﬃne tiles in Rn , Adv. Math. 121 (1996), 21–49.
22.
, Integral self-aﬃne tiles in Rn II. Lattice tilings, J. Fourier Anal. Appl. 3 (1997),
no. 1, 83–102.
23. W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960),
401–416.
24. A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad.
Sci. Hungar. 8 (1957), 477–493.
25. V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Math.
25 (1961), 499–530.
26. K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London
Math. Soc. 12 (1980), 269–278.
27. M. Smorodinsky, β-automorphisms are Bernoulli shifts, Acta Math. Acad. Sci. Hungar. 24(1973), 273–278.
28. Y. Takahashi, Isomorphisms of β-automorphisms to Markov automorphisms, Osaka J. Math.
10 (1973), 175–184.