Non literal tranducers and some problems of normality

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

F

RANÇOIS

B

LANCHARD

Non literal tranducers and some problems of normality

Journal de Théorie des Nombres de Bordeaux, tome

5, n

o

_{2 (1993),}

p. 303-321

<http://www.numdam.org/item?id=JTNB_1993__5_2_303_0>

© Université Bordeaux 1, 1993, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Non

literal

tranducers and

some

problems

of

normality

par

FRANÇOIS

BLANCHARD

ABSTRACT - A new _proof of Maxfield’s theorem is _{given, using} automata and results from Symbolic Dynamics. These techniques permit to prove that points that are near _normality to base

_{pk (resp. p)}

are also near

normality to base p

(resp.

pk),

and to study genericity preservation for non

Lebesgue measures when _going from one base to the other. Finally, similar results are _proved to bases the _golden mean and its _square.

0. Introduction

Consider the

_expansion

of a real number r E

_{[0, 1) =}

1r1

to the base _p.

Several arithmetical

_{manipulations}

of this

_expansion

have a transductional

cha.racter:

_{multiplication}

or division

by

an

integer;

addition of a rational

number;

changing

from base _{p to} base

_pk,

or the other _way round. What if these

_{manipulations}

are

performed

on a normal number? Do

they

_preserve

normality?

Positive answers to these

_questions

were first

_given

before transducers or automata were

_defined,

_by

_using

a result from Harmonic

Analysis, namely

Weyl’s

criterion: for instance Maxfield’s theorem

_[M]

states that

_normality

to base _p

_(i.e.

_genericity

for the uniform

_measure)

is

_equivalent

to

_normality

to base

_pk.

_Recently,

_{proofs using}

transducers and

_{ergodic techniques}

were

given

in

_[BIDT]

_concerning

addition and

_{multiplication;}

_they

allowed to

prove that near

normality

is also

preserved,

and to _say

_something

about

genericity

for measures different from

Lebesgue

measure. One _purpose of this _{paper is to prove} Maxfield’s theorem with the same

tools,

thus

obtaining

the same

refinements; another,

more

philosophical,

aim is to

underline the

_topological

character of this result.

Suppose

one has the

_expansion

of r to base

_pk,

i.e. an infinite _sequence

of

_cells,

each of them filled with one

symbol

in the

alphabet

Manuscrit _regu le 20 _{septembre 1991,} version definitive le 2 _juin 1993.

Une _partie des r6sultats contenus dans cet article ont 6t6 _expos6s au colloque "Thém.ate" ,

Any symbol a E

A has

_expansion

_7(a)

to base p; for convenience

y(a)

is

supposed

to have

_{length uniformly equal}

to

_k,

which is achieved

_by

_adding

the suitable number of 0’s in the

_beginning.

The

_rewriting

into base _p is done in three

_steps:

1- create k - 1 cells between any two of the sequence, in order to

provide

room for

writing

down the

expansion

-y(a)

instead of _a;

2- whenever one has a E A and then k -1

_empty

_cells,

write

_y(a);

one thus obtains a _sequence on

_B,

_together

with its scansion into words

of -y(A);

3-

_drop

the

_scansions,

_keeping

_only

the

_expansion

to base _p

_(scansion

seems not to matter much in this case; in fact it is

important

in the

setting

of

_doubly

infinite _sequences, where the

_question

is

_solved,

and whenever one makes use of a code with variable

length).

This construction was described in

_[BIP],

but whitout consideration of

topological

aspects,

which are essential in this article. Our

_job

consists of

_checking

that each

_step

transforms some kind of

_genericity

(first

to the uniform

_measure)

to

_{another, suitable,}

one. It is almost the same the other

way

round,

except

for an extra

_{difficulty: given}

the _sequence on

_B,

one

must first find all its

_possible

scansions in order to invert

_step

_3;

if one

wants

_genericity

to be

_preserved,

_y(A)

must be a

code,

which means _any finite concatenation of words of

_y(A)

has

_{unique decomposition}

into such words.

_Fortunately

this is the case.

This

_question

_may be

_put

in the more

_general

context of transductional

manipulations

of number

_expansions.

A transducer is an automaton with

two labels on each _arc,

_namely

an oriented

_graph

on a finite set of states

_C,

with arcs

having

one

input

label in A and

_output

label in

B*,

where A and

B are finite

alphabets.

The

expansion

of r to the base _p can be obtained from its

_expansion

to the base

_pk

_{by applying}

a 1-state transducer with arcs

having

input

label a and

_output

label

_y(a),

a E A. Transducers of different

types

may be used in the

theory

of

normality:

for instance

multiplication

by

1~ in base _p is

_{performed by}

a transducer with I-letter

_output

[BIDT];

we also introduce one in Section

_4,

when

_using

the same

techniques

for

expansions

to base the

_golden

mean. Transducers with

_output

labels

equal

either to the

_input

or to the

_empty

word can extract

_{subsequences;}

this is done in

_{[K, BrL]}

with the use of

slightly

different automata.

Another,

more

_general,

_question

is the action of such a base

change

on

sequences which are

_generic

for some non uniform invariant _{measure P.}

Sometimes the new _{sequence is} also

generic

for some other measure

J-l’,

an

significance

as

_{normality preservation;}

nevertheless our tools

provide

some

insight

into this field.

What about

_using

the same tricks when the

_expansion

is to the base

_8,

with 0

_greater

than 1 but not an

_integer?

This is not obvious. In

_[Be],

A. Bertrand-Mathis

_proved

that a number is

_generic

for the

_Parry

measure to

base 0 if and

_only

if it is

_generic

for the

_Parry

measure to base

_8k,

with k an

integer,

in case 0 is a Pisot number. Here it is

_only

done for 0 =

(1 +

and k =

2;

the

proof

makes _use of the _{same set} of

results,

but the main

difficulty

is that even when 0 is

_Pisot,

_any number has several

expansions

to the base

_{0, only}

one of which is canonical

_[Fl].

Because of

_this,

several

transducers,

some of them not too _easy to build _up, have to be used in

order to obtain the canonical

_8-expansion

of a

number, starting

from its

canonical

_{82-expansion.}

This is devised as an

example

that

questions

like

preservation

of near

genericity

for the

Parry

_measure, or

preservation

of

genericity starting

from non canonical _measures, can sometimes be tackled with the use of

transducers;

but the

purely

technical

difficulty

encountered in this

_simple

case with the size of one transducer

_suggests

that new ideas would be welcome.

The results obtained also extend far

_{beyond changes}

from

_{base ~}

to

base since

_they

_may be

_applied

to codes with variable

_length;

of course

in this case

they

have

hardly

_any number-theoretic

meaning,

still another

drawback

_being

that measures with maximal

_entropy

do not

_generally

cor-respond

to each other under the tower construction.

After some

_{preliminaries,}

Section 2 contains all

_general

statements on

preservation

of

_normality

or

_genericity

that are

required

for

applications.

In Section 3

_{generalizations}

of Maxfield’s theorem for

_integer

bases are dealt

with;

in Section 4 the same is done for bases 0 =

(1

₊

B/5)/2

and

82.

This research was

_partly

done

_during

a _stay at Universidad de

_Chile,

Santiago,

in December 1990. I am indebted to Pierre Liardet and

especially

Jean-Marie Dumont for several

_stimulating

remarks and

_questions;

_also,

the work in Section 4

_parallels

some unfinished common research with him and Alain Thomas on

_{multiplication by integers}

to base the

_golden

mean

_(the

transducers

_proved

even

_bigger

than the ones in Section

4!).

1. Definitions

Let A be a finite set of

_symbols

endowed with the discrete

_topology;

A* is the set of all finite words on

_A,

the

length

of the word u is denoted

by

Jul;

a

language

on A is _any subset of A*. The sets

AZ

and

AN

of all

_doubly

and

_simply

infinite _sequences on

A,

endowed with the

product topology,

are

_compact

metric _spaces. The

shift a :

Az

(or

AN)

defined

by

is a

homeomorphism

of

(a

continuous

transformation of

_{A~ ).}

A

_subshift

on A is _any closed a-invariant subset of

(or

AN).

A subshift X is

_{unambiguously}

determined

_by

the

_language

L(X) =

ju E

A*, 3m, n

e Z or E

X,

x(m, n)

=

ul;

thus _a

language

having

some obvious suitable

_properties

defines a subshift of

Az

and a

subshift of For u E

_L(X),

denote

by

[u]

the set

_{a:(0, Jul}

-

1)

=

~}.

A transitive

_subshift

is one such that for _{any u,} v E

_{L(X )}

there exists

w E A* such that

_L(X).

A

_{subshift of finite}

_type

is defined

_by

forbidding

a finite set of

words;

a subshift is called

sofic

if

_L(X)

is

regular,

or

recognized by

a finite automaton.

A

_factor

_map is a

continuous,

_onto,

shift-commuting map

X -+

X’;

in this case X’ is a

factor

of X and X an extension of X’. A

_conjugacy

map is a one-to-one factor map: when such a _map

exists,

X and X’ are said to be

_conjugate.

A bounded-to-one factor _map is such that the number of

_preimages,

or

_lifts

of x’ E X’ is bounded

_by

some k.

Automata and transducers

An automaton ,,4 consists of: - _a finite

alphabet A,

-

a finite set of states

_C,

-

a directed

graph

on

C,

the arcs each

having

a label in A.

To a

path

in the

graph,

one associates its

label,

i.e. the word

spelled

by concatenating

the labels of its arcs

_(usually

from left to

_right,

but the natural automata for

_{multiplication by}

an

integer

work from

right

to

_left);

the set of labels is called the

_{language recognized by}

the automaton. When

using

the automaton to

_recognize

this

_language,

all states are initial and final. There is another word one _may associate to a

path:

to _{any arc,}

associate the

_couple

_(a,

_c)

of its label a and the

_origin

vertex c. Then do

the same for the

path by concatenating

all

_{corresponding}

_couples.

The

language

thus

_recognized

_may be called

_simply

the

_language

of A.

Each of these

_languages

defines a subshift:

they

are the

factor subshift

X C and the

_{lift subshift}

Y C

_(A

x

C)z

associated to ,r4. Elements of these subshifts

_correspond

to infinite

_paths

in the

_graph:

an element of X is the label of such a _one, and an element of Y is the

_pair

_consisting

of the

have

_particular

_properties

_(Y

is a subshift of finite

type; X

must be at

least

_sofic).

,~1 defines a

mapping 0

from Y to X

_{by projection}

on A of

coordinates of _{y E} Y. It is of course a factor _map. It is

convenient,

though

not

_strictly

_correct,

to call

_{also 0}

the

_projection

_map from

_(A

x

_C)*

to A*. The same definitions are

fitting

for

simply

infinite _sequences.

An automaton ,~ is said to be deterministic

_{if, given}

a E A and c E

_C,

there is at most one arc

_starting

from state c with

given

label _a;

non-ambigous

if there is at most one

_path

with

_given

label from one state to

another. Deterministic automata are

nonambigous,

but the converse is not true.

An automaton is said to be irreducible if its

_graph

is

_{strongly connected,}

i.e. if for _any two states _c, c~ there exists a

path

_joining

c to c’ in the

_graph.

If ,~ is

_irreducible,

subshifts Y and X are transitive.

An automaton ,r4 is said to be bounded-to-one

_(or

a transducer T bounded to-one

_for

_input)

if the factor

_{map 0}

is bounded-to-one. The

_{corresponding}

definition

_applies

to

_output.

When the automaton ,~4 is

_irreducible,

the

map 0

is bounded-to-one if and

_only

if ,A is

_nonambigous

_[B].

A transducer is an automaton in which arcs have two

_labels,

one

_input

label in

_{alphabet A,}

and one

_output

label in

B*,

where B is another

alpha-bet : thus

_output

labels _may be

_single

letters

_(in

this case the transducer

is said to be

_literal),

or words with

length

_greater

than

_1,

or the

_empty

word. A transducer may be

_thought

of as

_performing

three tasks

simul-taneously :

it

_recognizes

_input

_words,

and _output words which

_decompose

into concatenations of _output

_labels;

it rewrites on B _any

_input

word on

A. Transducers are said to be deterministic or

_nonambigous

for

_input,

or

output

when

_literal,

or

irreducible, according

to the same definitions as for

automata.

Measures

Here we state some relevant facts about invariant measures on a

_compact

metric _space X endowed with a continuous transformation

_T;

examples

are

subshifts,

or the 1-torus

~’1

endowed with

multiplication by

some

integer

p. For

proofs

see

_[DGS].

Recall the set of

_probability

measures on X is a

_compact

metric _space for the

topology

of weak _convergence, and the set

_I(X)

of T-invariant

_probability

measures on X is

always

_nonempty

and

compact.

An invariant measure _p on

(X, T)

is such that

p(E)

for _any Borel set E. The

_{(topological)}

_support

of an invariant measure _p

1;

its

_entropy

is the

_nonnegative

number

where

Cn

=

f[u], u

_E

L(X),

Jul

=

n}

(the

limit is known _to

exist).

A

subshift for which there is

_exactly

one invariant measure with maximal

entropy

is said to be

_{intrinsically ergodic.}

For _any subshift E X and

_f

E

_C(X),

define the measure

by

the formula

-A

_{point x E X}

is said to be

_generic

with

_regard

to the invariante measures

p on

_X,

or

_{simply p-generic,}

if

_Sn(x)

_converges

_weakly

_{to p}

as n _goes

to

_infinity.

As

_genericity

is an

_asymptotic

_property,

_{depending only}

on

nonnegative

coordinates of x

_{E X,}

it is

_perfectly

defined for _sequences in

A~.

When X =

A N

(resp.

and _{p is} the Bernoulli measure A with

probability

for each

_symbol

_(resp.

_Lebesgue),

a

generic x

is called

normal to the base _p.

Define

_.M(x)

as the set of measures associated _{to x,} or

limits,

for the weak _convergence of _measures, of

_{Compactness implies}

it is

always

nonempty.

Any

measure in is

_invariant;

x is

_generic

for _p

if and

_only

if

_{~1(x) _}

Given some metric

generating

the

topology

of _{weak convergence, and 6} &#x3E;

_0,

a

_{point x}

is said to be

_{(6, J.L )-generic}

if

M(x)

C

_{B(JL,6), (the}

open ball with

centre p

and radius

_b);

when _tt is the uniform measure this is

_equivalent

to

_(k,

_E)-normality

for some k and e

depending

only

on 6

_{(see [BIDT],}

Section

_3]).

Assuming § :

V 2013~ X to be a factor _map, denote

by ~ :

:

A~(K) 2013~

M(X)

the

_{corresponding}

_map for measures: = v o

0-1

(generally,

for _any factor _map from Y to

_X,

denoted

_by

a small Greek

letter,

let

the

_{corresponding}

continuous

_{shift-commuting}

_map from

_.M(Y)

to

_M(X)

be denoted

_by

the

_{corresponding capital}

Greek

_letter).

_{Map $}

is

_weakly

continuous and

_{shift-commuting.}

As a _consequence,

M(Øx) = lll(M(z))

and if y E Y is

v-generic,

0(y)

is A

lift

of measure _p on X

is a measure v on Y such that _p.

_{When 0}

is

_{bounded-to-one, $}

preserves

entropy.

There is no essential difference between the

_theory

of invariant measures on subshifts of

A~

and subshifts of Definitions are

identical,

so there

is a one-to-one

_{correspondence}

between

_invariant,

or

ergodic,

measures on a

_simply

infinite subshift and its

_doubly

infinite version.

_Properties

are

the same. For

_{instance, genericity}

_may be considered as a _property of a

doubly

infinite _sequence, but it

_depends

_only

on its restriction to

_positive

coordinates. The

_only

_(slight)

_difficulty

arises when

_trying

to find

_preimages

of some

simply

infinite _sequence under the tower

_{construction,}

because in

most cases there are several solutions. This will be dealt with in time.

2. Some

_general

statements on

genericity

preservation

Going

from base

_pk

to base _p is done in three

_steps,

as

_explained

in the

introduction. So _{is any} more

general

_coding.

After

describing

each

_step,

the

_{corresponding}

theorems on

genericity preservation

are stated

(when

previously

known)

or

proved

(when

not).

Step 1: the tower construction. In this subsection we deal

only

with the

formal construction. For instance the actual

_meaning

of the function

_f

as

regards coding

is

_{only explained}

further on.

Put Z =

A~

for some finite A. Let

_{f :}

Z - N* be a continuous map

(i.e.

for _any

_{integer n}

the set

_{z

E

_{Z/ f (z)}

=

n}

is

closed,

which

implies

of

course that

_f

is bounded

_by

some

_integer

_k).

Let us construct the tower

system

_{(Z f, r)}

over

(Z, a)

under

function f,

as in

[PS]:

define

Z f,

endowed with the restriction of the

_product

_topology,

is a _compact

metric set and r is a

homeomorphism.

Suppose p

is an invariant measure on

(Z, Q).

It is well-known that the measure defined

_by

is T-invariant on

_{Z f,}

and

_{if p}

is

_ergodic,

so is the _{map p f} is 1-to-1

and onto from

_{I( Z)}

to and Abramov’s formula . , ,

links the

_entropies

of the two measures

[A].

None of these facts

depends

on

PROPOSITION 1. The is continuous

_(hence

_{bicontinuous) for}

the

_{topodogy of}

weak _convergence

_of

measures.

Proof. Let _{pn E} tend

_weakly

_{to p}

as n - oo: this means for _any

continuous

function g

on

Z,

We wish to check that for _any continuous h on

Zj ,

~ Write h =

hi

₊

h2

_{+... +}

hk,

where

_h;

is the

_(continuous)

restriction of h to the set

_{{(z, i)}

E

_{Z f}.}

Define

hi :

Z -+ R

_by

=

hi(z,i)

whenever = 0 otherwise.

By

definition of

_hi

and

₍₁₎

one can write

Applying

the

_hypothesis

of weak _convergence to each continuous function

h

t and

_f,

the last

_expression

_goes to

_{(li(f))-l -}

_{which, by}

_(1),

is i

t

equal

to

Now here is the basic tool for all further results.

PROPOSITION 2.

_Suppose

z E

Z,

y =

(z, i~

_E

Z f.

Then one has

Pi(M(z)).

Proof.

1)

We prove first the

implication

(p

E

_{M(z)) #}

_M(y)).

As ~c is assumed to be associated to z, there is an infinite subset E of N such

that for _any continuous _g on

_Z,

tends to

_p(g)

as n tends to

_infinity

_along

E. We want to

_compute

for continuous

_h,

as N tends to

_{infinity along}

some subset E’. As

_M(y)

does not

_change

under action of _{T, it} is sufficient to

_compute

_{SN(y, h)}

for

y =

(z,1).

For z E

_Z,

define and E’

is also an infinite subset

_ofN,

and

for

_given

_z,

_fZ(~)

defines a 1-to-1 onto

For continuous h :

_{Zj -}

_R,

0 i

_k,

define continuous functions

hi :

and Z -~ Il~ as in the

proof

of

_Proposition

1. For N E

E’,

one has

By

permuting

the two

_summations,

_multiplying

and

_dividing

_by

one

_gets

Putting

= _n, since

f

is continuous and

bounded,

tends to

_1L(f)

as N tends to

_{infinity along E’,}

so that =

n/fz(n)

tends to

_(IL(f))-l.

_Apply

the

_hypothesis

to each continuous function

_hi,

and then

_(1):

2)

The converse is

proved

in a similar _way. Given v E

_I(Z¡),

there exists a

_unique

invariant measure _p on Z such that v =

Suppose

v is associated to some

(z, i)

E

_{Z f:}

for the same reason as above assume i = 1. The convergence of

SN(y,

h)

to v occurs on some subset E’. Without loss of

_generality

one _may assume E’ C

_fz(N):

since

_f

is

_bounded,

for any

p E E’

there

_{is q}

E with 0 _{p - q} k. For

_{given h}

the difference

isp(y, h) - S,(y, h)1

is bounded

_by

so that

Sq(Y)

tends

weakly

to v as well as

_Sp(y).

We claim that for _any continuous _g,

_p(g)

as n _goes to

_infinity

along

E =

fZ 1(E~).

Define continuous functions

g’

and

f’ :

Zj -

II8

by

=

g(z),

g’(z,

i)

= 0 for i _&#x3E;

1,

=

_1,

f’(z,

i)

= 0 for i _&#x3E; 1.

One has

as g’

is

_continuous,

_by

_hypothesis

the second limit in the

_product

is

_equal

to

_{v(k) _ p(g)lp(f);}

_{f. being}

defined as

_above,

as

and

_f’

is

_{continuous, by}

₍₁₎

so

11ri1nEE

g(g)-

Q.E.D.

0

A direct _consequence of

_Propositions

1 and 2 is the

_following:

COROLLARY 3. Given E &#x3E;

_0,

there exists b &#x3E; 0 such that

_M(z)

E

_B(JL,6)

implies

M (z, i)

E

_-)

ared

_conversely

_{M (z, i)}

E

_B(p¡(JL),

₆₎

_implies

M (z)

E

When E

_vanishes,

this becomes a statement on

genericity preservation.

Step

2:

_writing

the letters

This is

_by

far the

_simplest

_step.

Suppose

now the function

f

above is

_generated

in the

following

_way.

Let a be a _map from A to

_B*,

where B is some finite

_alphabet;

a can be

easily

extended into a _map from A* to

B*,

but not into a

shift-commuting

map from to

B7:

this is

_why

one has to build the tower. Define

_f

on Z

by

f (z) _

la(zo)l.

We want to

_identify

_{(Zf, 7-)}

with some convenient

_symbolic

_space

_Y,

under the

_assumption

that a is 1-to-l.

Let X = be the set of all

points

x E

BZ

for which there exists a

sequence t

E

_{0,1}~

such

_{that ti}

_{= tj}

= 1

implies

x(i, j - 1)

_E

(a(A))*;

t is called a scansion of x into words of and

4&#x3E;’ being

the natural

projections

from

_(B

x

{0,1})~

on

B~

and

(0, 1)~’~,

define a subshift Y on the

_alphabet

B x

_{0,1}

_by

the

_following

condition:

for

any y E

_Y,

_0(y)

E

_X,

and

_0’(y)

is a scansion

_of

_0(y)

into words

of

a(A).

For a E

_A,

write

_a(a)

=

al(a)a2(a)...

a’a(a),(a),

a;(a)

E B. Define the

This definition consists of

_replacing

each letter zn in z

by

the

corresponding

word

_a(z),

for which

_{Z f}

_provides

convenient room

owing

to the choice of

f,

while

_keeping

trace of the

_beginning

of all words

_{a(zi). ~}

is

_obviously

continuous and

_onto,

and

_by

₍₂₎

it

_changes

T _{to or;} it is 1-to-I if and

only

if a

_is,

which means that in this

_{case ~}

is a

_conjugacy.

Now let us examine the action

of ~

(and

~-1)

on

_points

that are

generic,

or almost

_generic,

for some

_given

measure.

That ~

preserves

genericity

and almost

genericity

is a

straightforward

consequence of its

being

a factor _map.

PROPOSITION 4. Let 7r be a

factor

_map

_from

the

_compact

metric _space Y to the

_compact

metric space X. For any - &#x3E; 0 there exists b &#x3E; 0 such that

_for

_any invariant measure v on

_Y,

_{y E}

_Y,

_if

_M(y)

E

_{B(v, b),}

then

M

The

_proof

is

_elementary

and left to the reader. As a

consequence g

preserves

genericity and, assuming

c~ to be

_1-to-1,

so does

~-1.

Step 3:

_dropping

or restoring the scansion

Dropping

the scansion

_only

means

_applying

the factor _map

0; by

Propo-sition 4 this _{map preserves}

_genericity.

But the converse is far from

being

always

true: some extra conditions are needed. Here is the

general ergodic

statement we are

_going

to

_apply:

it is an _{easy consequence} of

Proposition

3.3 in

_[BIDT].

PROPOSITION 5.

_{Suppose 0 :}

Y ~ X is a

_factor

_{map, A}

is an invariant measure on X such that there is a

_unique

invariant measure v on Y with

~(v) _

A. _{For any s} &#x3E;

_0,

there is b &#x3E; 0 such that

_if

C

_B(A,b)

_for

some x E

_X,

then

_for

_{any y E} Y with

_0(y)

= _x,

M(Y)

_C

B(V,ê).

Of course one wants sufficient conditions _{on p} for

_uniqueness

of its lift

v. The _next, still

general,

remark settles the uniform case:

Remark 6. When Y and X are the extension and factor subshift of a

nonambiguous

irreducible

_automaton,

_{and /-t}

is the

_unique

measure with

maximal

_entropy

on

_X,

then its

_unique

lift on Y is the measure v with maximal

_entropy.

This is a mere _consequence of intrinsic

ergodicity

of sofic

A more

general

sufficient condition for an invariant measure to have a

unique

invariant

_preimage

under some

nonambiguous

automaton is

_given

in

[BIDT,

Propositions

2.7 and

_4.9].

And _now, how can we use Remark 6 when

introducing

all

_possible

scansions of x E X? This is done

_{by introducing}

an

important

further combinatorial

_assumption

on a:

a(A)

must be a code.

DEFINITION. r C B* is said to be a code

_if

implies

rra = n and _x~ =

y=, 1 i n.

Many

finite and infinite codes are described in

[BeP];

the ones used in Section 3 and Section 4 are

elementary.

When r =

a(A)

is a

code,

one

usually

wishes a to define a 1-to-1

mor-phism

from A* into

_B*,

so as to make it

_possible

to tell what element of A* any word u E r* is the

_{image of;}

for this to be true one must also assume a to be 1-to-l.

The

_petal

automaton

_AA

of a

language

A =

zs)

has _state

set

_fF}

U xt E

_A}.

There are arcs

(i)

from 6’ to

_(x;,1)

with label

_xi(l),

x; E

A;

(ii)

from

_{(x;, n)}

to

_{(xt, n}

+

₁₎

with label

_xi(n

+

₁₎

for n ₊ 1

_{Ixi 1;}

(iii)

from

_{(xi, IXil- 1)}

to 6’ with label

This automaton is

_obviously

irreducible. When E is the

_only

initial and final

state,

,A~

recognizes

A*;

when all states are initial and

final,

it

recognizes

the set

_F(A*)

of

_"pieces"

of words of A*.

PROPOSITION 7

_[BP].

The

_{petal automaton of}

a

(finie)

language

A on B

is irreducible. It is

_{nonambiguous if}

and

_{only if}

A is a code.

In

_fact,

if A is not a

code,

there exists no

_nonambiguous

automaton

permitting

to obtain the scansions of words

_belonging

to

_F(A*).

Here is now the theorem we have been

_aiming

_at;

it

just

collects our

previous

results.

PROPOSITION 8.

_{Let p}

be an invariante measure on

_Z,

a : A 2013~

B*,

let

f

be a continuous _map Z to

_{~’, ~}

the two

_factor

_maps

_{defined above,}

and

_finally

let b &#x3E; 0. There exists 6’ &#x3E; 0 such that

_if

z is a

(6’, p)-generic

for

A = lll o

_Conversely

assume a is 1-to-1 and

_a(A)

is a

_code,

and reduced to a

_unique

invariant measure v.

If it

is the

_unique

measure on Z

corresponding

to then

_for

_any

_{(b, A)-generic}

x E

_X,

if x

= ~ o ~(z, i), z

is

_(b’,

Proof.

_Starting

from _p on Z it is

_enough

to

_apply

the classical

_properties

of towers and the direct

_part

of

_Corollary

_3,

and

_Proposition

_4,

in order to

obtain the direct result.

_Conversely,

there is a

_unique

invariant measure v on ~’ such that =

A’; apply

_Proposition

_{7 to} Y and

_X,

which

exactly

coincide with the extension and factor subshifts of

_petal

automaton

of code

_a(A);

then

_Proposition

5 to

_A,

and

_finally

_Proposition

4 and the

converse

_part

of

_Corollary

3. 0

Remark 9. For number-theoretic

_{applications,}

one wants one-sided infinite

sequences. In order to obtain the same results in this _case,

just

remark that

(a)

given

an

invariant p

on

AN,

there is a

unique

invariant measure

p’

on

A~

such that its restriction to the

_a-algebra

of future events is

equal

to _p.

(b)

any

point x

in some subshift of

A~

can be extended into at least one

point

x’ in the

corresponding

subshift of

A~ .

(c) x

is

_p-generic

if and

_only

if x’ is

_{p’ -generic.}

Thus _any

_question

about

_genericity

in

AN

can be carried over to

Az

and

solved

_there;

all former statements _may be

_applied

in

A~ .

3.

_Refining

and

_extending

Maxfield’s theorem

Let us denote a measure on

1r1,

invariant

by multiplication by

_p,

by

a

starred Greek

_letter,

for instance and the almost

_always

_unique

corre-sponding

measure on

(0; .. ,

_{,p -}

1}

by

the same Greek letter without a star: _}1.

The next

_proposition

states that

_b-normality

to the base _p is

_equivalent

to

b’-normality

to

_the

base

_{pk for}

some b’. The first

_part,

about

preservation

of

normality,

is the well-known Maxfield theorem

_[M];

another

_ergodic

_proof

can be deduced from the main result in

[BrL].

The remainder is

mostly

original, though

J.-M. Dumont

_[D]

has an

unpublished

combinatorial

proof

of the converse

_part,

which _may also be deduced from

[C].

However,

the

present

proof

establishes

_clearly,

in the line of

_[BIDT],

that such results are within _easy reach of

_{Ergodic Theory,}

combined with

_elementary

_techniques

of the

_Theory

of Automata.

PROPOSITION 10. For _any two

_integers

_p &#x3E;

_2,

k

&#x3E;_

2,

and r E

_(0,1),

r is normal to the base

_{p if}

and

_{only if}

it is normal to the base

_pk.

_Moreover,

for 6

&#x3E;

_0,

there exists b’ &#x3E; 0 such that

_if

r is b’-normal to the base _p

_(resp.

pk),

then r is b-normal to the base

_pk

_{(resp. p).}

Proof. Put A =

f 0,...

ip k -

1 ~,

B =

{0,’"

p -

1}.

Since the

Lebesgue

measure

corresponds

to

_only

one measure on

AN

or we can work

directly

on

_expansions;

and

_according

to Remark 9 all

_propositions

on

genericity

of

_doubly

infinite _{sequences may} be

_applied

to _sequences in

AN

or To

prove the statement we have to

_perform

three tasks:

- first to see how the former construction

(tower

and factor

_map)

can be

applied

to deduce the

_expansion

of r in base _p from its

_expansion

in base

pk

(and conversely);

- then

to _prove that the measure

corresponding

to the uniform

mea-sure on

AN

under this construction is the uniform measure on

BN,

and

conversely;

-

finally,

to check the

hypotheses

of

Proposition

7 are fulfilled.

1)

In order to obtain the

_expansion

of r to base _p from its

_expansion

to

base

_pk,

one has to

_replace

each

_symbol

a

belonging

to A =

{0,’’’

by

a word

_y(a)

with

_{length k}

on

alphabet

B =

~0, ~ ~ ~ , p -

1}: y(a)

is

the

_expansion

of a in base _p, with a suitable number of 0’s added in the

beginning

when _necessary.

_{Obviously y}

is 1-to-1 onto between A and

_Bk;

moreover

y(A)

is

evidently

a

prefix code,

therefore 0

is the canonical factor map of a deterministic automaton.

2)

Call _p the uniform measure on Z =

AN,

A’ the uniform measure on

X =

BN .

We establish that and is the

uni-que element of

~-1 (~’).

Starting from p

on

AN

one has

f

= k

everywhere,

so

by

Abramov’s formula = =

log p

=

h(A’).

As

they

are associated to bounded-to-one factor

_{maps, -=}

and $ _preserve

_entropy:

since

a is

_{1-to-1, =-}

is a

_conjugacy;

since

y(A)

is a

code, ~

is bounded-to-one. As

A = lll is the ultimate

_image

of _p on

_X,

h’(A’)

=

h(A);

but _as A’

is the

_unique

measure on X with

_entropy

logp,

one has A’ = a.

Conversely,

there is a

_unique

invariant measure v on Y such that A’:

_{since 0}

is the canonical factor _map of a

nonambiguous

automaton it is

bounded-to-one,

thus A’

_implies

_{h(v) -}

_{h(A’) =}

_{log p;}

as the automaton is irreducible Y is

_{intrinsically ergodic,}

and v is

_unique.

As

_{~-1 (v)}

also has

entropy

logp,

by

Abramov’s formula the

_{corresponding}

measure on Z has

3)

The direct

_part

of

_Proposition

8

_only

_requires

that

_f

be

_continuous,

which is evident. For the reverse

_part,

it is sufficient to

_{remark 7}

is 1-to-l

and also a

prefix

_code,

so

_that,

by

_Proposition

7 and Remark

_6,

_0-1(p)

is

a

singleton. D

Now let us

drop

uniform _measures, and _suppose r E

71

is

_generic

for

some

xpk-invariant

measure

~*

(in

order to avoid _any trouble when

_passing

to the

_expansions,

_{assume is}

_nonatomic).

_By

_{using Proposition}

7 it is easy to _prove r is also

generic

for some

corresponding xp-invariant

A*.

But,

supposing

one inverts the roles of xp and

xpk,

when is the

corresponding

statement true ?

An answer to this

_question

can be obtained in

particular

_cases,

_using

some results in

[BIDT].

For instance:

PROPOSITION 11.

_Suppose

the measure A* on

1f1

is the

_image

of

an

ergodic

Markov measure A with

topological

_{support BN.}

Then there is a

unique

measure v on Y such that

_A;

thus there is a measure

_~,*

on

1f1

such that

_{if x is}

_generic

_for

A* under

_{xp, it is generic for J1 *}

under

_xpk.

Proof. The

_hypothesis

_{implies A}

is

_unique,

and

_by

_[BIDT,

_Propositions

2.7

and

_4.9]

that v is also

_unique,

thus

_defining

_unique

_{measures /-I} on

A N

and

p*

on

1ft.

_By

_Proposition

8 one

_gets

the

_expected

results on

preservation

of

_{genericity. D}

4.

_Analogous

results for

_expansions

with

_respect

to the

_golden

mean

Given a real number

~3 &#x3E; 1;

not in

_N,

a _sequence

(st,

i &#x3E;

₀₎

of

_symbols

in A =

{0,1," ’ ,

[,8]}

is said _to be _an

expansion

of

_{r E}

1r1

_to the base

,8

if

its valuation

_V(s)

checks

When

_{3

is not an

_integer

there are _many

possible

expansions

of _any real number in to the base

_{3.

One of these

_expansions,

the

_greatest

for

lexi-cographical order, plays

a

special

role and _may be called the canonical one. The matter of

_{{3-expansions}

has been

_given

some attention in the

literature,

and an

_analogue

to Maxfield’s theorem was

proved

by

A. Bertrand-Mathis

example

how to use transducers and

_{ergodic techniques}

to

_get

the same

re-sult, by

the _way

_obtaining

the same refinements as for

expansions

to

integer

bases in Section 3.

Consider the case

_{3

- 9 =

(1

₊

B/5)/2:

then the canonical

expansion

of r is the

only

one in which the word 11 never _occurs; the set of all such

expansions

is the subshift of finite

_type

So

defined

_by

this exclusion rule. The canonical

_expansions

to the

_{base (}

=

02

are _sequences in which there

is at least one occurrence of 0 between two occurrences of 2.

_They

form

a sofic

_system

_{By erasing}

the _output labels in the

graph

of 7í

below,

one obtains an automaton

_recognizing

_S,.

On the sets

_So

and there

is a

unique

measure

having

maximal

_entropy

(equal

to

logo

and

log20

respectively),

which is called the

_Parry

measure; its role

corresponds

to

that of the uniform measure with

regard

to

integer

bases.

Given r E

1r1,

and its canonical

_{expansion s’}

to the base

_(,

we want to

compute

its canonical

_expansion

s to the base 0. This has to be done

by

combining

the action of two transducers: on account on their

_acting

in

_opposite

_directions,

it is

_impossible

to _merge them into

_just

one

[F1].

The first _one,

_{Tl, performs}

two tasks:

_creating

one

_empty

cell between any two

symbols

of

s’,

belonging

to

alphabet

f 0, 1, 2},

and

then, using

the

additional _room,

_rewriting

these

_symbols

on

alphabet

(0, 1) ;

the _sequence

s" thus obtained has valuation r and is on

alphabet

A =

{0,1,’*’

~~3j~

but

_generally

does not

_satisfy

the constraint that 11 must never occur. The second transducer

_T

is

_literal;

it eliminates all blocks 11 in

_s’,

thus

yielding

s. A

simpler

transducer

might

be used in order to do

just this,

as

in

_[Fl];

_but,

in order to use the various results of Section

_2,

we want

_T

to be

_{nonambiguous,}

which means its

input

subshift must be

_equal

to the

output

subshift

_{of Tl;}

this makes it more

complicated.

This also illustrates the kind of difficulties one runs into when

_using

transducers in the field of

{3-expansions.

Transduces

This transducer acts from left to

_right;

the

_input

words

_(on

_alphabet

f 0, 1,

21)

are

_exactly

those

_belonging

to

_S,.

_Any

_symbol

in

_{f 0, 1,}

₂₁

is

replaced by

a two-letter word

depending

on the state the transducer is in.

Initially

a cell

_containing

a 0 is created between _any two

_symbols

in

_s’;

then

any

symbol

2 is subtracted and

replaced by

_adding

two

_1’s,

one in the first cell to the left and one in the second cell to the

_right,

_according

to

_identity

2

_{= e + e-2}

_{(thus V(s")}

=

Yes)).

The cell to the left

formerly

contained a

a new

_2,

and this must be eliminated in its turn

_(this

is done

_by

the

_loop

with labels

_1/10

on state

_b).

Its

_graph

_appears in

_Figure

1.

Figure

1

The

_input

subshift

_Xl

is

_exactly

_S~.

The

_output

subshift

_X1

has two

relevant features:

13

never _occurs, and two occurrences of 11 must be

separated by

at least one occurrence of

02.

_Figure

2 below

_pictures

an automaton

_recognizing

it.

It is

_important

to notice

_7í

_performs

the three

_operations

described in

Section 2 at the same time. In order to

_{apply previous}

results it _may be

thought

of as the

composition

of two transducers. Put u =

_{00, v}

=

_01,

_w =

10,

and

_replace

_output words in

_{7í by}

these new

symbols:

one thus obtains a

literal,

deterministic tranducer

T’

with the same

graph,

but with _output

alphabet

lu,

v,

w}.

The second one,

T",

has

only

one state and

_replaces

letters

_by

their value on

alphabet {0,1}.

It is a

typical

tower

transducer, performing

steps

1, 2

and 3 of the Introduction.

Remark. it is

_important

to notice is a code.

Transducer T :

This one acts from

_right

to left. Before

_constructing

it one should build up an automaton

_recognizing

the output subshift of

_Tl,

_X’,

from

_right

to

left

_(this

makes it easier to check that the

_input

subshift of

_T

is

_equal

to the

output

subshift of

_Tl).

One

_{such, deterministic,}

automaton is

_represented

in

_Figure

2.

Now this is how

_T

acts.

_Everytime

the word 11 occurs in the

input

sequence

s",

the two l’s are

replaced by

0’s and a 1 is created to the

left,

according

to the

_identity

1 =

8-1

₊

8-2

(which

ensures

V(s’) = V(s") =

V (s )) .

As the

_output

subshift of

_{T ,}

never contains

13

or

12 012

this never creates

_2’s,

or words

1 n, n

&#x3E;

_2;

when a new word 11 is created to

the left of the one

just

eliminated it must be dealt with in its turn: this is achieved

_by

the

_loop

on

_{states g}

and h in the

graph.

Its

graph

is featured

out in

_Figure

3.

-Figure

3

Finally,

72

is

_nonambiguous

for

_input

and

_output.

Let

_{and It,}

be the

_Parry

measures on

1r1

_{corresponding}

to 8 and

_(.

There

_only

remains to

_apply

results of Section 2 to the transducers

_{T’, T",}

and 72

to _prove the

_following

PROPOSITION 12. For

_{any b}

&#x3E;

_0,

there exists b’ &#x3E; 0 such that the two

following properties

are

equivalent :

(1)

r E is

_b)-generic

under

_{multiplication}

_by

0.

(2)

r is

_{b’)-generic}

under

_{multiplication by (.}

Proof. It is

_closely

related to the

_proof

of

_Proposition

10. First one

reformulates the

_problem

into its

_{symbolic equivalent}

_{on S9}

and Then one has to

_identify

measures _IL8

_and

as

_{corresponding}

to each other under action of the

_{transducers 7i}

and

T :

this is an _easy

job

with the

use of

_entropy.

_Finally

_apply

_Proposition

8 to

_T"

_(this

is

_possible

since

(u, v, w)

is a

_code)

and

Propositions

4 and 6 to

_T’

_{and 72}

to reach the conclusion. 0

Remarks.

1- Recent work

_by

C.

_Frougny

_[F2]

and

_by

C.

_Frougny

and B.

_Solomyak

[F3]

suggests

this

_proof

_may be

_generalised

to all Pisot numbers r such that

1 has a finite

decreasing

canonical

_expansion

to base T

[F2].

2- The same set of results

_suggests

there are other measures for which

genericity

is

_preserved.

REFERENCES

[A]

L. M. _ABRAMOV, The _{entropy of} a derived automorphism, Amer. Math. Soc. Transl. 49

_(1965),

162-166.

[BP]

J. _BERSTEL, D. PERRIN, Theory of codes, London, Academic Press, 1985.

[Be]

A. _{BERTRAND-MATHIS, Développements} en base 03B8 et _répartition modulo 1

de la suite

_(x03B8n),

Bull. Soc. math. Fr. 114

_(1986),

271-324.

[BIP]

F. _BLANCHARD, D. PERRIN, Relèvements d’une mesure _{ergodique par} un

codage, Z. Wahrsheinlichkeist. v. Gebiete 54

_(1980),

303-311.

[BIDT] F. _BLANCHARD, J.-M. _DUMONT, A. _THOMAS, Generic sequences, trans-ducers and _{multiplication of} normal numbers, Israel J. Math. 80 _(1992), 257-287.

[BrL]

A. _BROGLIO, P. _LIARDET, Predictions with _{automata, Symbolic Dynamics} and its _{Applications, AMS, providence, RI,} P. Walters _{ed., Contemporary} Math

135, 1992.

[C]

J. W. S. _CASSELS, On a _{paper of} Niven and _Zuckerman, Pacific J. Math. 3

(1953),

555-557.

[D]

J.-M. _DUMONT, Private communication.

[F1]

C. _{FROUGNY, Representations of} numbers and finite automata, Math. _Systems

Theory 25

_(1992),

37-60.

[F2]

C. _FROUGNY, How to write _{integers in non-integer base,} Lecture Notes in

Computer Science _(Proceedings of _{Latin ’92) 583,} 154-164.

[F3]

C. _FROUGNY, B. _SOLOMYAK, Finite _{Beta-expansions, Ergod.} Th. &#x26; _Dynam.

Syst. 12 _(1992), 713-723.

[K] T. _{KAMAE, Subsequences of} normal sequences, Israel J. Math. 16 (1973),

121-149.

[M]

J. E. MAXFIELD, Normal _k-tuples, Pacif. J. Math. 3

_(1953),

189-196.

[PS]

K. _PETERSEN, L. _SHAPIRO, Induced _flows, Trans. Amer. Math. Soc. 177

(1973), 375-390.

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