Dynamics & Stochastics || Numeration Systems as Dynamical Systems: Introduction

Numeration Systems as Dynamical Systems: IntroductionAuthor(s): Teturo KamaeSource: Lecture Notes-Monograph Series, Vol. 48, Dynamics & Stochastics (2006), pp. 198-211Published by: Institute of Mathematical StatisticsStable URL: http://www.jstor.org/stable/4356373 .

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IMS Lecture Notes?Monograph Series Dynamics & Stochastics Vol. 48 (2006) 198-211 ? Institute of Mathematical Statistics, 2006 DOI: 10.1214/074921706000000220

Numeration systems as dynamical

systems ? introduction

Teturo Kamae1

Matsuyama University

Abstract: A numeration system originally implies a digitization of real numbers, but in this paper it rather implies a compactification of real numbers as a result of the digitization.

By definition, a numeration system with G, where G is a nontrivial closed multiplicative subgroup of R+, is a nontrivial compact metrizable space O admitting a continuous (?a; + ?)-action of (?,?) G G ? ? to ? G O, such that the (? + ?)-action is strictly ergodic with the unique invariant probability measure ?^, which is the unique G-invariant probability measure attaining the topological entropy | logA| of the transformation ? ??? ?a; for any ? f 1.

We construct a class of numeration systems coming from weighted substitutions, which contains those coming from substitutions or ?-expansions with algebraic ?. It also contains those with G = R+.

We obtained an exact formula for the ?-function of the numeration systems coming from weighted substitutions and studied the properties. We found a lot of applications of the numeration systems to the /^-expansions, Fractal geometry or the deterministic self-similar processes which are seen in [10].

This paper is based on [9] changing the way of presentation. The complete version of this paper is in [10].

1. Numeration systems

By a numeration system, we mean a compact metrizable space O with at least 2

elements as follows:

(til) There exists a nontrivial closed multiplicative subgroup G of R+ and a

continuous action ?a; +1 of (?, t) G G x R to u; G O such that ?'(?a; + t) +1' =

?'?a; + X't + f.

(?)2) The (? + ?)-action of t G R to ? G O is strictly ergodic with the unique in-

variant probability measure ??? called the equilibrium measure on O. Consequently, it is invariant under the (?a; + ?)-action of (?, t) G G x R to ? G O as well.

(tt3) For any fixed ?? G G, the transformation ? ???- ??a> on O has the | log ??|-

topological entropy. For any probability measure ? on O other than pq which is

invariant under the Attraction of ? G G to a;, and 1 f ?? G G, it holds that

???) < ???(??) = I log ?01 -

The (? + ?)-action of t G R to ? G O is called the additive action or R-action, while the ?a^^??? of ? G G to a; G O is called the multiplicative action or G-action.

Note that if O is a numeration system, then O is a connected space with the

continuum cardinality. Also, note that the multiplicative group G as above is either

R_i_ or {??; n G ?} for some ? > 1. Moreover, the additive action is faithful, that

is, ? +1 = ? implies t = 0 for any o; G O and t G R.

1Matsuyama University, 790-8578 Japan, e-mail: kamaeGapost.plala.or.jp AMS 2000 subject classifications: primary 37B10. Keywords and phrases: numeration system, weighted substitution, fractal function and set,

self-similar process, ?-function.

198



Numeration systems as dynamical systems 199

This is because if there exist ?? G O and ti f ? such that ???-ti = ??, then

take a sequence ?? in G such that ?? ?? O and ??a>? converges as ? ?? co. Let

a;?? := linin?oo ??a;?. For any t G R, let an be a sequence of integers such that

a????? ?? t as ? ?? co. Then we have

ôo + t ? lim (???? + ??a?*?) ??>?? = lim ?? (?? + Oriti) = lim ??a;? = ??,.

??>?? ??>cx)

Thus, a;^ becomes a fixed point of the (? + t)-action of t G R to o; G O. Since

this action is minimal, we have O = {a;^}, contradicting with that O has at least

2 elements.

An example of a numeration system is the set {0,1}Z with the product topology divided by the closed equivalence relation ~ such that

(...,a_2,a_?;a?,a?,a2,...) ~ (... ,/3-2,0-?? A)?AiA2, ? ? ?)

if and only if there exists N G Z U {?00} satisfying that an = ?n (Vn > ?), aN = ?? + 1 and an = 0, ?? = 1 (Vn < N) or the same statement with a

and ? exchanged. Let O(2) := {0,1}Z/ ~ and the equivalence class containing

(...,a_2,a_i;ao,ai,a2,-..) ? {0,1}Z is denoted by S?=-??a?2? G O(2). Then,

O(2) is an additive topological group with the addition as follows:

00 00 00

n= ? oo n=?00 n= ? oo

if and only if there exists (..., n_2,n_i; 770, r/i,7/2,...) G {0,1}Z satisfying that

2r/n+1 -f 7n = an ?+? /3n -f r/n (Vn G ?).

This is isomorphic to the 2-adic solenoidal group which is by definition the pro-

jective limit of the projective system ? : R/Z -+ R/Z with ?(a) = 2a (a G R/Z). Moreover, R is imbedded in O(2) continuously as a dense additive subgroup in

the way that a nonnegative real number a is identified with ?^??=_00 cxn2n such

that a = X)n=_00 an2n and an = 0 (Vn > N) for some N G ?, while a negative real number ?a with a as above is identified with X^^L.^l

? an)2n. Then, R

acts additively to O(2) by this addition. Furthermore, G := {2fc; k G ?} acts

multiplicatively to O(2) by

CX) 00

2fc ? a?2"= ? an_fc2". n= ? cx) n= ? oo

Thus, we have a group of actions on O(2) satisfying ((tl), (tf2)and (fl3) with G :=

{2k; fc G ?} and the equilibrium measure (1/2, l/2)z.

Theorem 1.1. O(2) is a numeration system with G = {2n; n G ?}.

We can express O(2) in the following different way. By a partition of the upper half plane EI := {z = ? + iy\ y > 0}, we mean a disjoint family of open sets

such that the union of their closures coincides with H. Let us consider the space

O(2)' of partitions a; of EI by open squares of the form (xi,X2) x (^1,2/2) with

X2 - x\ = ?/2 - y\ = y\ and yi G G such that (xx,x2) ? (2/1,2/2) ? ? implies

(xi,(xi +X2)/2) x (2/1/2,7/1) G a; (type 0) and

(0*1 +*2)/2,x2) x (yi/2,2/i) Go; (type 1).



200 T. Kamae

y

I I I I 1 I 1 I I I I I M I 1 ? ? ? ? I I I I I I I I I I I I I I I I I I I 13

Fig 1. The tiling corresponding to ? ? 01.101 ?

An example of o; G O(2)' is shown in Figure 1. For ? G O(2)', let (a?,a?,...) be

the sequence of the types defined in (1) of the squares in ? intersecting with the

half vertical line from +0 + i to +0 + ?co and let (a_?, ct-2,...) be the sequence of the types of the squares in ? intersecting with the line segment from +0 + i to

+0. Then, ? is identified with S??:=_00 an2n. Note that replacing +0 by -0, we get

E!?L-oo?*2n suchthat (... ,a_2,a_?;a0,a?,...) ~ (... ,?-2,?-?\?^,??,...). The topology on O(2)' is defined so that ?? G O(2)' converges to a; G O(2)' as

? ?? co if for every R G ?, there exist Rn G ?? such that limn_>oo ?(?, Rn) = 0, where ? is the Hausdorff metric between sets R, R' C EI

p(R, R') := max{sup inf \z-z'\, sup inf \z - z'\).

z?RZfeR' z'^R'Z^R (2)

For ? G O(2)', t G R and ? G {2n; n G R}, ? + ? G O(2); and ?a; G O(2)' are

defined as the partitions

and

? + t := {(xi -t,x2 -t) ? (2/1,2/2); (^1^2) x (yi,y2) ? a;}

?a; := {(???,??2) ? (Xyi,Xy2); (^1,^2) x (2/1,2/2) ? a;}.

Let ? : O(2)' ?? O(2) be the identification mapping defined above. Then, ? is

a homeomorphism between O(2)' and O(2) such that ?(? + t) = ?(?) + t and

?(??) = ??(?) for any ? G O(2)', t G R and ? G {2n; n G Z}. Thus, O(2)' is

isomorphic to O(2) as a numeration system and will be identified with O(2).




We generalize this construction. Let A be a nonempty finite set. An element in

A is called a color. An open rectangle (xi,x2) x (2/1*2/2) in EI is called an admissible

tile if

X2 - xi = 2/1 (3)

is satisfied (see Figure 2). In another word, an admissible tile is a rectangle (xi, ?2) x

(2/1,2/2) in EI such that the lower side has the hyperbolic length 1. Let Tl be the set

of admissible tiles in H.

A colored tiling a; is a subset of TZ ? A such that

(1) R ? Rf = 0 for_any (R, a) and (R', a') in ? with (R, a) f (Ri\ a'), and

(2) Ua?A U(?,e)?w R = EL

An element in Tl ? A is called a colored tile. We denote

dom(u>) := {R; (R, a) ? ? for some a G A}.

For R G ??p?(?), there exists a unique a G A such that (R, a) G a;, which is denoted

by u)(R) and is called the color of the tile R (in a;). Let R = (??, x2) ? (2/1,2/2)? We

call 2/2/2/1 the vertical size of the tile R which is denoted by S(R). Let O(?) be the set of colored tilings with colors in A. A topology is introduced

on O(?) so that a net {??}?e? C O(?) converges to o; G O(?) if for every (R, a) G ?, there exists (Rn,an) G ?? such that

an = a for any sufficiently large ? ? I and lim p(R, Rn) = 0, ?????

where ? is the Hausdorff metric defined in (2). For an admissible tile R := (xi,x2) x (2/1,2/2), t ? R and A6i+,we denote

R + t := (xi+t,x2-\-t) ? (2/1,2/2)

XR :- (Xxi,Xx2) x (?2/?,?2/2).

Note that they are also admissible tiles.

FIG 2. Admissible tiles.



202 T. Kamae

For ? G O(?), t G R and ? G R+, we define ? + t G O(?) and ?a; G O(?) as follows:

? + t = {(R-t,a)', (R,a)eu>}

?? = {(??,a); (?, a) G a;}.

Thus, we define a continuous group action ?? + t of (?, t) G R+ ? R to ? G O(?). We construct compact metrizable subspaces of O (A) corresponding to weighted substitutions which are numeration systems. Though JJA > 2 is assumed in [7], we consider the case JjA = 1 as well.

2. Remarks on the notations

In this paper, the notations are changed in a large scale from the previous papers

[7], [8] and [9] of the author. The main changes are as follows:

(1) Here, the colored tilings are defined on the upper half plane EI, not on R2 as in

the previous papers. The multiplicative action here agree with the multiplication on

EI, while it agree with the logarithmic version of the multiplication at one coodinate

in the previous papers. Here, the tiles are open rectangles, not half open rectangles as in the previous papers.

(2) Here, we simplified the proof in [9] for the space of colored tilings coming from

weighted substitutions to be numeration systems by omitting the arguments on the

topological entropy.

(3) The roles of x-axis and y-axis for colored tilings are exchanged here and in [9] from those in [7] and [8].

(4) Here and in [9], the set of colors is denoted by A instead of S. Colors are denoted

by a, ol, a? (etc.) instead of s, s', G? (etc.).

(5) Here and in [9], the weighted substitution is denoted by (s, r) instead of (f, ?).

(6) Here and in [9], admissible tiles are denoted by R, R', Ri, R1 (etc.) instead of

S, S', Si, S* (etc.).

(7) Here and in [9], the terminology "primitive" for substitutions is used instead of

"mixing" in [7] and [8].

3. Weighted substitutions

A substitution s on a set A is a mapping A ?? A"1", where A+ = U?li &*. For

? G A+, we denote |?| := ? if ? G Ae, and ? with \?\ = ? is usually denoted by

?o?i ? * '?i-i with & G A. We can extend s to be a homomorphism A+ ?? A+ as

follows:

s(?):=s(??)s(??)'??s(??-?),

where ? G A^ and the right-hand side is the concatenations of tf(??)'s. We can define

s2, s3,... as the compositions of s : A+ ?? A+.

A weighted substitution (s,t) on A is a mapping A ?? A+ ? (0,1)+ such that

\s(a)\ = \t(a)\ and S*<|t(a)| T(a)i = * f?r anv ? ? A. Note that s is a substitution

on A. We define rn : A ?? (0,1)+ (n = 2,3,...) (depending on s) inductively by

Tn(a)k = r(a)irn-1(a(a)i)j

for any a G A and i, j, k with

0 < i < |s(a)|, 0 < j < \cn-l(*(a)i)\, k = ? \an-\a(a)h)\ + j. h<i




Then, (s?,t?) is also a weighted substitution for ? = 2,3, ?

A substitution s on A is called primitive if there exists a positive integer ? such

that for any a, o! G A, s?(a)? = o! holds for some i with 0 <i < \s?(a)\. For a weighted substitution (s, t) on A, we always assume that

the substitution s is primitive. (4)

We define the base set ?(s, t) as the closed, multiplicative subgroup of R+ generated

by the set

G rn(a)i ; a G ?, ? = 0,1,... and 0 < i < \s?(a)\ ?

]^ such that &n(a)i = a J '

Example 3.1. Let A = {+, -} and (s,t) be a weighted substitution such that

+ -> (+,4/9)(- l/9)(+,4/9) - - (_,4/9)(+,l/9)(-4/9),

where we express a weighted substitution (s, t) by

a -> (s(a)?, t(a)0)(s(a)?,t(a)?) ? ? ? (a G A).

Then, 4/9 G ?(s,t) since s(+)0 = + and t(+)0 = 4/9. Moreover, 1/81 G ?(s,t) since s2(+)4 = 4- and t2(+)4 = 1/81. Since 4/9 and 1/81 do not have a common

multiplicative base, we have ?(s,t) = R+. This weighted substitution is discussed

in the following sections. The repetition of this weighted substitution starting at + is shown in Figure 3 by colored tiles. The substituted word of a color is represented as the sequence of colors of the connected tiles in below in order from left. The horizontal (additive) sizes of tiles are proportional to the weights and the vertical

(multiplicative) sizes are the inverse of the weights.

Let G := ?(s, t). Then, there exists a function g : A ?? R+ such that

g(a(a)i)G = g(a)r(a)iG (5)

for any a G A and 0 < i < \s(a)\. Note that if G = R+, then we can take g = 1. In the other case, we can define g by g(ao) = 1 and g(a) := rn(ao)? for some ? and i such that s?(a?)? = a, where ?q is any fixed element in A.

Let (s,t) be a weighted substitution satisfying (4). Let G = ?(s,t). Let g satisfy (5). Let O(s, t, g)f be the set of all elements ? in O (A) such that for any

((xi,x2) x (2/1,2/2),?) ?o;, we have

(I) 2/1 ? g(a)G, and

(II) (R\ s(a)?) G ? holds for i = 0,1,..., \s(a)\ - 1, where

i?? i

R1 := (?? -l? (x2 -

x\)^2r(a)j, ??-l? (x2 - X\)^2r(a)j)

j=0 j=0

? (T(a)iyi, 2/1).

A vertical line 7 := {?} ? (?co, co) is called a separating line of a; G O(s, r, g)' if for any (R, a) G ?, R ? ? = 0. Let O(s, t, g)" be the set of all ? G O(s, r, g)' which do not have a separating line and O(s, t, g) be the closure of O(s, T,g)n. Then, the action of G x R on O(s, t,g) satisfies (jjl). We usually denote O(s, r, 1) simply by

O(s,t).



204 T. Kamae

4/9

+

1/9 4/9

+ +

Fig 3. The weighted substitution in Example 1.

Remark 3.2 ([7]). A nontrivial primitive substitution s : A ?? A+, where "nontrivial" means Sa?? ?s(a)? ^ 2, is considered as a weighted substitution in a canon- ical way. Let

M := (Jt{0 < i < \s(a)\; s(a)? = a'})a,a'eA

be the associate matrix. Let ? be the maximum eigen-value of M and ? := (?a)aeA be a positive column vector such that ?? = ??. Define weight t by

r(a)i = ?s(a)?

??a '

which is called the natural weight of s. Thus, we get a weighted substitution (s, r) which admits weight 1. We modify (s, t) if necessary in the following way. If there exists a G A with \s(a)\ = 1, so that a ?? (a', 1) is a part of (cr, r), then we

replace all the occurrences of a in the right hand side of "??" by o! and remove a from A together with the rule a ?? (a', 1) from (s,t). We continue this process until no a G A satisfies \s(a)\ = 1. After that if there exist a, a' G A such that

(s(a),t(a)) ? (s(a'),t(a')), then we identify them.

For example, the 2-adic expansion substitution 1 ?? 12, 2 ?? 12 corresponds to the weighted substitution 1 ?? (1,1/2)(1,1/2). The Thue-Morse substitution




1 -? 12, 2 -> 21 corresponds to the weighted substitution 1 -? (1, l/2)(2,1/2), 2 -?

(2,1/2)(1,1/2). The Fibonacci substitution 1 ?? 12, 2 ?> 1 corresponds to the

weighted substitution 1 -? (1,?_1)(1,?~2), where ? = (1 + Vb)/2. The weighted substitution (s,t) obtained in this way satisfies that ?(s,t) =

{??; n G ?} and that g in (5) can be defined by g(a) = ?a (a G A). Dynami- cal systems coming from substitutions are discussed by many authors (see [2], for

example). Our weighted substitutions are a generalization of them.

Let (s, t) be a weighted substitution on A satisfying (4). Let g satisfy (5). Con-

sider O(s, t, g). We call the tile Rl in (II) the ?-th child of the tile (xi,x2) x (2/1,2/2), and (xi,x2) x (2/1,2/2) the mother of Rl. Note that the vertical size S(Rl) of

Rl coincides with the inverse of the weight t(a)?. If Rj is a child of Rj+i for

j = 0,1,..., fc ? 1. Then, the tile Rq is called a fc-th descendant of the tile Rk. If ?o is the z-th tile among the set of the fc-th descendants of Rk counting as 0,1,2, ? ? ?

from left, we call ?o the (k,i)-descendant of the tile i?*. In this case, we also say that Rk is the fc-th ancestor of ?o ?

Theorem 3.3. The space O(s, t, g) is a numeration system with G = ?(s,t).

Proof. We have already proved (#1) and (fl2) in Theorem 3 of [7]. Here we prove

(#3). Let O := O(s,t, g) and ??? be the equilibrium measure. Since ??. is the unique invariant probability measure under the additive action, it is also invariant under

the multiplicative action.

By Goodman [4], it is sufficient to prove that for any X E G with ? f 1, the

transformation ? ??? ?a; on O has the metrical entropy |^?| under ???, while it

has the metrical entropy less than | log ?| under any other G-invariant probability measure. D

Lemma 3.4. Let

S := {a; G O; ? has a separating line}

S? := {a; G O; y-axis is the separating line of ?}.

Then, we have

(i) S \ S? is dissipative with respect to the G-action, so that ?/(S \ S?) = 0 for any G-invariant probability measure ? on Vi.

(ii) For any o; G S?, a; restricted to the right quarter plane (?,??) ? (?,??) and to the left quarter plane (?00,0) ? (?,??) are cyclic individually with respect to the G-action. Hence, Ga; with respect to the G-action is either cyclic or conjugate to a 2-dimensional irrational rotation with a multiplicative time parameter.

(iii) S? is a finite union of minimal and equicontinuous sets with respect to the G-action. In fact, there is a mapping from the set of pairs a G A and i with

0<z<z+l< \s(a)\ onto the set of minimal sets in S??

Proof, (i) If the line ? = u is the separating line of ? G O, then ? = Xu is the

separating line of ?a;. Hence, S \ S? is dissipative.

(ii) Let o; G S?? Denote by a;"1" the restriction of ? to the right quarter plane

(0,00) ? (0,00), while by ?~ the restriction of a; to the left quarter plane (?00,0) ?

(?,??). Let (Rf)i^z be the sequence of tiles in ?sp?(?) such that Rf intersects with the upper half lines of ? = ?0, and Rf is a child of Rf+i for any z G ?

(db respectively). Let af :? u(Rf) be the colors of Rf (? respectively). Define

mappings s? from A to A by s+(a) = s(a)? and s_(a) = s(a)|<t(a)|_?. Since

0?(af) = af-i (i ? Z) (? respectively), the sequence (af)iez is periodic, which



206 T. Kamae

also implies that the vertical sizes S(Rf) of Rf, which coincide with the inverses

of the weights r(a?+i)?, are also periodic in z G ? with the period, say r* which

is the minimum period of (af)iez (? respectively). Then, ?+ := rr (aj)^* is

the minimum (multiplicative) cycle of a;"1", while ?~ := tG (a$ )7\-( _.. is the

minimum (multiplicative) cycle of ?~, that is, ??? = a;* holds for ? = ?* and ?*

is the minimum among ? > 1 with this property (? respectively).

Therefore, ? is cyclic with respect to the G-action if ?+ and ?~ have a common

multiplicative base. In this case, the minimum cycle of ? is the minimum positive number ? such that ? = (?+)? = (X~)m holds for some positive integers ?,p?.

Otherwise, the G-action to Go; is conjugate to an 2-dimensional irrational rotation

with a multiplicative time parameter.

(iii) We use the notations in the proof of (zz). Take any pair (a, i) with a G A and

0<z<z + l< |s(a)|. Take any a/ G O having a tile R G ??t?(?') with o/(?) = a

such that the z/-axis passes in between the z-th child of R and the ? + 1-th child of

?. Let ?(a, i) be the set of limit points of ?a/ as ? G G tends to co. Note that this

does not depend on the choice of a/. Then, ?(a,?) is a closed G-invariant subset

of S0. Moreover, since the sequence (s"(s(a)?),s+(s(a)?+?))?=0,?,2,... enter into a

cycle after some time, ip(a,i) is minimal and equicontinuous with respect to the

G-action.

To prove that the mapping ip is onto, take any ? G S?? There exists ?? G

O(s, t, g)" which converges to ? as ? ?? co. We may assume that there exists a

pair (a, i) such that for any ? = 1,2,???, there exists R G ??p?(??) with a = o/n(?) such that the y-axis separetes the z-th child of R and the z + 1-th child of R. Then, ? G f(a, i), which proves that f is a mapping from the set of pairs (a, i) with a G A

and 0<z<z + l< \s(a)\ onto the set of minimal sets in S0 with respect to the

G-action. D

Example 3.5. Let ? with 0 < ? < 1 satisfy that logp/log(l - p) is irrational.

Let (s,t) be a weighted substitution on A = {1} such that 1 ?? (l,p)(l,l - p).

Then, ?(s,t) = R+ holds. Let O = O(s, t). In this case, elements in S0 are not

periodic, but almost periodic as shown in Figure 4. Then, the dynamical system

(S0, ? (? G R+)) is isomorphic to ((R/Z)2, ?? (X G R+)) with

Tx(x,y) = (x + logA/log(l/p), y -r logA/log(l/(l - p))).

Lemma 3.6. It holds that h?n(X) = \ ̂ ?| for any X G G. Let ?f 1 and ? be any other X-invariant probability measure on O, then h?(X) < \ logA|.

Proof. To prove the lemma, it is sufficient to prove the statements for ? > 1. Take

any G-invariant probability measure ? on O which attains the topological entropy of the multiplication by ?? G G with ?? > 1, that is, hv(Xi) = log ??. We assume

also that the G-action to O is ergodic with respect to v. Then by Lemma 3.4, either

?(S?) = 1 or ?(O \ S) = 1. In the former case, h?(X) = 0 holds for any Xe G since

the G-action on S? is equicontinuous by Lemma 3.4, which contradicts with the

assumption. Thus, we have ?/(O \ S) = 1.

For ? G O, let ?o(o;) G ??p?(?) be such that ?o (a;) = (^1,^2) x (2/1,2/2) with

Xi < 0 < x2 and 2/1 < 1 < 2/2? Take a0 G A such that

?/({? G O; ?(??(?)) = a0}) > 0.




vertical sizes

1/(1-p)

vertical sizes

i/p

IF

Fig 4. An element in S? in Example 2.

Take oo '?= max{6 < 1; b G g(ao)G} (see (5)). Let

O? := { ? G O; the set {? G G; ?a;(??(?a;)) = a0}

is unbounded at 0 and co simultaneously }

O? := { ? G O?; ?o (?) = (xi,x2) x (2/1,2/2)

with yi = 60 and o>(?o(o;)) = a? }.

For a; G O?, let ??(a;) be the smallest X ? G with ? > 1 such that ?a; G O?? Define

a mapping ? : O? ?? O? by ?(?) := ??(a;)a;. For fc = 0,1,2, ? ? ? and i1 = 0,1, ? ? ? , |s*(a0)| - 1, let

P(k,i) := { ? G O0; ?0(?)~1 ??(?0(?)?)

is the (fc, z)-descendant of ?o(o;)}

(see Figure 5) and let

V:={P(k,i); fc = l,2,??? , 0<z<|afe(ao)| }

be a measurable partition of O0. Note that ??(a;) = Tk(a0)~l if o; G P(fc, z). Since ?^(O?) = 1 by the ergodicity and

O?= jj (J AP(fc,z),

P(k,i)eV 1<?<t*=(a0)-1 xeG

there exists a unique ?-invariant probability measure v$ on O? such that for any Borei set ? C O, we have

/boTfc(ao)t_1 iy0(X'1BnP(k,i))dX/X



208 T. Kamae ?

ao

6?

a?

Fig d. ? e P(3,4) with ?0(?) = 13/3

with

if G = R+ and

C(v) := S -?oETkMi vo(P(k,i)) < co

P(kyi)?P

v(B) = C(v)-1 S S ??-^????))

with

P(k,i)ev ^G 60<?<60t*(a0). X

<?("):= S (-logrfc(ao)?/log/3)i/o(P(M))<oo

?(*:,?)???

if G = {/?"; ? ? ?} with /? > 1.

Since

S rfc(ao)? = land ? m(P(fc,t)) = l,

p(fc,t)ep p(fc,?)?P

we have

i/JP) := - JZ logn,(P(M))-i*(P(M))

p(k,i)ep

< - S logrfc(ao)?-ô(P(fc,?))

P(M)?T>

by the convexity of - log ?. The equality in (8) holds if and only if

H>(P(fc, i)) = Tk(a0)i (VP(fc, t) ? ?).

(6)

(7)

(8)

(9)




By (6), (7), and (8), we have

H,0(V) = - S log^(P(M))-"o(P(M))<co.

p(k,i)er

For any o;, a/ G O? such that A/c(o;) and ?* (a/) belong to the same element in

V for fc = 0,1,2, ? ? ?, the horizontal position of ?o(o;), say (xi,x2), coincides with

that of ?o(o/). Therefore, ? and a/ restricted to (ari,x2) x (0, &o) coincide. In the

same way, if Afc(o;) and Afc(o/) belong to the same element in V for any fc G ?, then ?o := ?o (a;) = ?o (a/) holds and all the ancestors of ?o in ? and a/ coincide

as well as their colors. Therefore, ? and a/ restricted to the region covered by the

ancestors of ?o coincide. Hence, if ? or a/ does not have the separating lines, then

? = ?' holds.

Since ?/(S) = 0, we have ??(S ? O?) = 0. Hence, the above argument implies that V is a generator of the system (O?, ?, ?). Thus, /??0(?) = hUQ(A,V). It follows

from (8) that

K0(A) = K0(A,V)

<??0(?)

<- S logrfc(ao)iô(P(fc,z)). (10)

p(k,i)ev

The equality in the above that

?^(?) = - S logTk(ao)i'Vo(P(k,i))

P(k,i)ev

holds if and only if (AnV)n?.z is an independent sequence with respect to i/o satis-

fying (9). Since

MAi)/logAi = ??)

?Oo ???<??)

?,?(?)

-Ep(fc,i)eplog^fc(?o)i ? vo(P(k,?))

'

hv(Xi) < ^?? follows from (10), while the equality holds if and only if (AnV)n?:Z is an independent sequence with respect to i/o satisfying (9). Let this probability measure be ?. Then, it is not difficult to prove that ? is invariant under the additive

action. Hence, the uniqueness of such measure ([7]) proves ? = ?^, which completes the proof of Lemma 3.6 and Theorem 3.3. D

The following Theorem 3.7 follows from a known result about the spectrum of unitary operators corresponding to the affine action (Lemma 11.6 of [13], for

example).

Theorem 3.7 ([10]). Let O be a numeration system with G = R+, that is, with

the multiplicative R+-action. Then, the additive action on the probability space O

with respect to ??\ has a pure Lebesgue spectrum.



210 T. Kamae

4. ?-function

Here, we listed only the results on the ?-functions. For the proof, refer [10]. Let O := O(s,t,a) satisfying (4) and (5). For a G C, we define the associated

matrices on the suffix set A ? A as follows:

Ma := ? ? TÂ (?)

?a,+ := (?s(?)?=a< r(a)J)a|Q/?A

ia,- := (la(a)k(a)|_1=a' T(a)k(?)|-l)a^,? ?'?A

Let ? be the set of closed orbits of O with respect to the action of G. That is, ? is the family of subsets ? of O such that ? = Gu for some o; G O with ?? = ? for

some X ? G with ? > 1. We call ? as above a multiplicative cycle ?? ?. The minimum

multiplicative cycle of ? is denoted by c(?). Note that c(?) exists since ?? f ? for

any a; G O and XeG with 1 < ? < min{r(a)~1; a G A, 0 < z < |r(a)|}. We say that ? G ? has a separating line if o; G ? has a separating line. Note that

in this case, the separating line is necessarily the y-axis and is in common among a; G ?. Denote by ?? the set of ? G ? with the separating line.

Define the ?-function of G-action to O by

&(*):= H(l-c(t)-*r\ (12)

where the infinite product converges for any a G C with ?1(a) > 1. It is extended

to the whole complex plane by the analytic extension.

Theorem 4.1. We have

det(/-Ma,+)det(/-MQ)_) ??

=-det(/-MQ)- ?S?(a)'

where

CE0(a):= ??1"^)"0)"1 ????

is a finite product with respect to ? e ??.

Theorem 4.2. (i) ??(a) f 0 i}Tl(a) f 0.

(ii) In the region Tl(a) f 0, a is a pole of ??(a) with multiplicity k if and only

if it is a zero of det(I ? Ma) with multiplicity fc for any fc = 1,2, ?

(iii) 1 is a simple pole of ??(a).

Theorem 4.3. For O = O(s,t/,#), if ? (s,t) = {??; ? G ?} with ? > 1, then

there exist polynomials p,q G Z[z] such that ??(a) = p(X?t)/q(Xa)- Conversely,

if ??(a) = p(XQ)/q(Xa) holds for some polynomials p,q G Z[z] and X > 1, then

?(s,t) = {Xkn; n G ?} for some positive integer fc.

Theorem 4.4. If ?(s,t) = {Xn; n G ?}, then X is an algebraic number.

Acknowledgment

The author thanks his old friend, Prof. Mike Keane for his useful discussions and

encouragements to develop this research for more than 10 years.




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