A Gauss-Kuzmin theorem for the Rosen fractions

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

G

ABRIELA

I. S

EBE

A Gauss-Kuzmin theorem for the Rosen fractions

Journal de Théorie des Nombres de Bordeaux, tome

14, n

o

_{2 (2002),}

p. 667-682

<http://www.numdam.org/item?id=JTNB_2002__14_2_667_0>

© Université Bordeaux 1, 2002, tous droits réservés.

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

A

Gauss-Kuzmin theorem

for the Rosen fractions

par GABRIELA I. SEBE

RÉSUMÉ. En utilisant les extensions naturelles des

transforma-tions de Rosen, nous obtenons une

_{représentation}

de la chaîne

d’ordre infini associée à la suite des

_{quotients incomplets}

des

frac-tions de Rosen. Associé au _comportement

_ergodique

d’un certain

système

aléatoire

_homogène

à liaisons

_complètes,

ce fait nous

per-met de résoudre une version du

problème

de Gauss-Kuzmin _pour

le

_{développement}

en fraction de Rosen.

ABSTRACT.

_Using

the natural extensions for the Rosen maps, we

give

an infinite-order-chain

_{representation}

of the _sequence of the

incomplete

quotients

of the Rosen fractions.

_Together

with the

ergodic

behaviour of a certain

_homogeneous

random _system with

complete

connections, this allows us to solve a variant of Gauss-Kuzmin

_problem

for the above fraction

_expansion.

1. Introduction

The Rosen

_fractions,

introduced in

_[11],

form an infinite

family

which

generalizes

the nearest

_integer

continued fractions.

_They

are related to the so-called Hecke _groups. There is an extended literature on these

(see

the

papers by Rosen and Schmidt

_[12],

_Gr6chenig

and Haas

_[2],

Schmidt

_[13],

Haas and Series

_[3],

and Lehner

_[7],

_[8]).

It is

_{only recently}

_(see

the _papers

by

Burton,

Kraaikamp

and Schmidt

_[1]

and Nakada

_[10])

that the

_ergodic

properties

of these

_expansions

have been studied. It should be stressed that the

_ergodic

theorem does not

_yield

rates _{of convergence for}

_mixing

properties;

for this a Gauss-Kuzmin theorem is needed.

The

_{technique developped by}

Iosifescu in

_[5]

for the case of the

regular

continued fraction

_expansion

could be used for different

_types

of continued fractions. The aim of this _paper is to take _up the Gauss-Kuzmin

problem

for the Rosen fractions in this manner.

This paper is

organized

as follows.

_Using

the natural extensions for the

Rosen _maps, in Section 3 we

give

an infinite order-chain

_{representation}

of the _sequence of the

_{incomplete quotients}

of the Rosen fractions. In Section 5 we show that the random

_systems

with

_complete

connections

(RSCCs )

associated with the Rosen continued fraction

_expansion

are with

contraction and their transition

_operators

are

regular

w.r.t. the Banach

space of

Lipschitz

functions. This leads

further,

in Section

6,

to a solution

of a version of Gauss-Kuzmin

_problem

for the Rosen fractions.

Let A =

Aq

equal

2 cos 7r/q

for q

_E

{3, 4, ... }.

Fix _some such

q &#x3E;

4 and

let

_Iq

=

[-A/2,

A/2).

Then the

map Tq :

Iq

-3

_Iq

defined

_by

is called a Rosen _map.

Putting

in case

_0,

and =

0,

= 00 in case

Tq -1 (x)

=

0,

and

using

the Rosen map _Tq, one

easily

sees that _{every x} E

_~Iq

has a

unique

expansion

which is called the Rosen or

A-expansion

(ACF)

of x.

Now,

related to Hecke

_(triangle)

_group

_Gq,

_{q &#x3E; 4,}

we call x a

_{Gq-irrational}

if x has a Rosen

_expansion

of infinite

_length.

For

_{Gq-irrationals,}

there are restrictions on the set of admissible _sequences

_{of êi}

and ai. These

restrictions are determined

_by

the orbit of

A/2 [cf. [11]

and

[1]).

2. Natural extensions for the Rosen _maps

Consider

_(see

_[1])

the so-called natural extension

T for

any Rosen interval maps, which is defined as follows: for _any

fixed q &#x3E; 4,

let A =

Aq,

T(x)

be

Tq (x)

and

where we have

suppressed

the

dependence

of a =

a, and e = 61 on x. Also

notice that

T

is a transformation which on the first coordinate is

simply

the interval map while on the second coordinate is

_directly

related to the

"past"

of the first coordinate.

It has been shown in

_~1~,

that T is a

bijective

transformation of a domain

n in

R2

_except

for a set of

_Lebesgue

measure zero. We consider two cases.

2.1. Even indices. Fix _q =

2p, with p &#x3E;

2,

and let I be the interval

Iq.

Putting §j =

-T~ a~2,

with

_~o

=

-A/2,

_{we construct a}

_partition

of I

_by

considering

the intervals

Furthermore,

let

_{Kj = [0,}

_{Lj], j}

_{E {I, ...}

_{p -}

_{1 ~}

and

_Kp

=

[0, R~,

where

Lj,

1 j

p -

1,

and R

satisfy

the

system

Let S2 =

U1=1

Jk x Kk.

One has that R =

_1,

and that T is

bijective

on Q

_except

for a set of

_Lebesgue

measure zero. In

[1]

it is shown that T

preserves the

probability

measure v

(abolutely

continuous with

_respect

to

the

_Lebesgue

measure on

S2)

with

_density

~i~£,

where C is a

_normalizing

constant.

_Actually,

for q even, the constant C is

given by

2.2. Odd indices. Fix an odd _q and

recycle

notation as above. Let I

be the interval

_{Iq and Oj =}

_-T~.~/2,

with

_§o

=

-A/2.

Putting h

=

Q;3,

we

construct a

_partition

of I

by

_considering

the intervals E

_{11, - - - 2h+2},}

where

Let

_{Kj =}

_{[0, Lj], j}

2~ +

_{1 },}

and

_{K2h+2 =}

_{[0, R],}

where

_Lj,

1 j

2h +

_1,

and R

_satisfy

the

_system

Let _q = 2h ₊

3,

with h &#x3E; 1 and S1 = x

_Kj.

One has that R

and that T is

_bijective

on f2

_except

for a set of

_Lebesgue

measure zero. In

paxticulax, A

R 1. The transformation T preserves the

probability

measure v with

density

( 1 + ~ y) -~,

where

3. An infinite-order-chain

_{representation}

In this section we obtain an infinite-order-chain

representation

of the

sequence of the

incomplete quotients

of the Rosen fractions. We treat

simultaneously

the two subfamilies of Rosen _maps, those of odd indices and those of even one.

Let us consider the sets

and define

Similarly

to relations

_(2),

for all n E

N*,

let us define

which

_implies

Writing

[~i~i,~2~2?"’ ? enzn]

for the finite continued fraction

Now,

define the random variables

_{an, n}

E Z

_{= ~... , 20132, 20131,0,1,2,... }}

on Q

_by

where

TO

denotes the

_{identity map}

and

_{lii (z, y)}

=

al(x).

Clearly,

_on

f2,

Since T _preserves v, the

doubly

infinite sequence

(an)nEZ

is

strictly

sta-tionary

under v.

Let us

_put

= for all n E Z and

(x, y)

Clearly,

Projecting

v, we obtain the invariant _{measure vx} on

~- 2 , 2 ~

_{and vy}

on

[0, R].

Since the invariant

_density

_{h(x, y) =}

on St has

_projections

hx(t)

= f h(t,y)dy

and

_hy(t)

_{= f}

_h(x,t)dx

on

~-2, 2~

and

~O,R),

respec-tively,

we have

-where and denote the collection of Borel sets on

[-, ]

and

2 22J 2 2

0, RJ,

respectively.

Now,

we are able to _prove the

following

theorem which is _very

important

in the

_sequel.

Theorem 1. For _{any x E}

_I,

we have

continued

fraction

with

_incomplete

_quotients

_ao,

_{a_1, ... ).}

Proof.

As is well known

Let us denote

by

In

the fundamental interval

for _{any arbitrarily} fixed values of

the ti

and

_{ai, i}

=

0, -1, ... ,

-n. Then we have

for _{some vn E}

_In.

Since

the

_proof

is

_complete.

Corollary.

For _{any i} E Z* we have

where a =

[fofo,

]

Proof.

For i = -1 we have

Hence the conditional

_probability

above

_equals

v-a.s. to

Fori=lwehave

Finally,

the

_proof

for i E Z*

_B

_{-1,1}

is

_analogous.

0 Remarks.

_(i)

The strict

_stationarity

of

_(an)nEZ

under v

_implies

that the conditional

_probability

does not

_depend

on n and is v-a.s.

equal

to

_{pi (a),}

with

(ii)

The process

(lnän)nEZ

is called an

_{infinite- order- chain}

in the

theory

of

dependence

with

_complete

connections

_(see

_[6],

Section

_5.5).

(iii)

Many

standard formulas for continued fractions

_{(see [9]),}

hold for the Rosen fractions.

_Letting

where -i

and ai

depend

on x E

_I,

we find that

Therefore

It follows that for _any

_{Gq-irrational}

number x E I and _any

_positive

_integer

n we have

Let us

_put

_sn = _E N*. Note that the

_{equation qn}

=

anÀqn-l

₊

enqn-2

implies

"

with _so =

0,

i.e. _sn =

[an,

ê2al];

clearly,

_{Sn E}

[0,

RJ.

Motivated

_by

Theorem

_1,

we shall consider the

family

of

probability

measures

(Va)aE[O,R]

on

BI

defined

by

their distribution functions

In

_particular

v°

is the

_Lebesgue

measure on

I,

if A = 1. For

any a E

_{[0, R]}

Let us consider the

_quadruple

By

the

_corollary

to Theorem

_1,

the _sequence is an W-valued

Markov chain on which starts at

so

= a and has the

following

transition mechanism: from state s E W the

_{only possible}

transitions are

those to states

_{(l, i)}

E

_{X B}

{(-1,1)},

and to

_~19

or to R if

_{(l, i)}

=

(-1,1)

and s E

~0, ~’ R 1~

or s E

~~ R 1,R~,

respectively,

the transition

probability being

Let

_B(W)

be the Banach _space of bounded measurable

_{complex-valued}

functions

_f

on W under the _supremum norm

III

=

Clearly,

the transition

operator

of transforms

_f

E

_B(W)

into the function defined

_by

whatever a E W.

Note that for a E

_W,

A

_{= ~ (an+1, ... )}

and n E

_N*,

This follows from Theorem 1 for all irrational a E W and

_{by continuity}

for all rational a E W . In

_particular,

it follows that the

_{Brod6n-Borel-L6vy}

formula holds under va for _any a E

W ,

that is

forxEl, nEN*.

4. The Gauss-Kuzmin

_type

_equation

because

TO

is the

_{identity map.}

Since

we can write the Gauss-Kuzmin

_type

equation

as

Assuming

that for some m E N the derivative

F§£

exists

everywhere

in

I and is

bounded,

it is _easy to see

by

induction that exists and is

bounded for all n E N*.

_{Differentiating}

the Gauss-Kuzmin

_type

_equation

we arrive at

We consider two cases.

4.1. Even indices. Let

_h(x,

_y)

_{= 1+xy~}

be the invariant

density

on

with p &#x3E;

2

_(see

Subsection

_2.1).

For

}

and

_{Kj = [0,}

_Lj]

we obtain

and,

for Thus we have where

Further,

write to

_get

for n &#x3E; m.

Clearly,

the above

equation

reduces to

with V

_being

the linear

_operator

defined on

B(I)

_by

where

Note that V is the transition

_operator

of the

_homogeneous

Markov chain

defined as

and _yo = 0.

Clearly,

(yn )n&#x3E;o

satisfies the recursion

equation

and _yn E I for all indices n. Let us consider the

quadruple

where v : I x X - I is

_{given by}

v (x, (l, i))

=

vii (x)

and

Q :

I x X -

[0, 1]

is

_given

_by

_{Q (x, (l, i))}

=

for all

_{(1, i)}

E X

_and,

_Hx

= 1 _on

[0, ~),

_we obtain

Since it follows that

Hence the

_right-hand

side of the above

_equation

_equals

Next,

to prove that we note that

Hence

For the other cases, similar

proofs hold,

and we leave them to the reader. Thus we have shown that the _sequence is an I-valued Markov chain on which start at _{yo -} 0 and has the

_following

transition

mechanism: from state s E

I,

the

_{only possible}

transitions are those to

states

_m,

_{(l, i)}

E X

_{B {(-1,1), (1,1)~,}

and

_{to 2}

or

to ’

if l =

~1,

i = 1 and s _E

~- 2 , ~ -

A)

_{or s E}

( ~ - a, 2 ) ,

respectively,

the transition

probability

being

qls(s).

4.2. Odd indices. We have the same

_goal

as in the

previous

subsection.

Let _q = 2h ₊

3, with h &#x3E; 1,

and

recycle

notation as above. Since

h(x, y) =

c

is the invariant

_density

_y on fl =

Kj

(see

Subsection

j

(

2.2),

it follows that for t E E

_{{ 1, ... ,}

2 h +

_1}

and

_{Kj =}

_[0,

while for t E

_J2h+2

=

[0,

2 ~

and

K2h+2 =

[0, RI,

_{we get}

Thus

where

Further,

write

_{f n (x) -}

_{EC7Fn(x), x}

E

_I,

to

_get

m, where V is the linear

_operator

defined on

_by

with _qli and _vlz,

_{(1, i)}

E X defined as in the

_previous

subsection. As in the even _case, we _{may prove} that

qii(x)

= 1 _on I.

Also,

it _appears that

V is the transition

_operator

of the

_homogeneous

Markov chain on

-5. Characteristic

_properties

of the transition

_operators

U and V Here we restrict our attention to the transition

_operators

U and V in-troduced in Sections 3 and 4. In this section we deal with the

properties

of

U and V on function _spaces different from

B(W)

and

B (I ) .

In connection with the

_operators

U and

_V,

we note the

_following

prop-erties.

_First,

if we define

and

then we have

U’U’f

=

_{U°° f}

for all

f

E

_B(W)

and

_V’V’f

=

V°° f

for all

_f

E

_B(I)

and n E N*.

_Second,

let

_{L(W) (respectively L(I))}

be the Banach space of all bounded

complex-valued

Lipschitz

functions on W

(respectively I )

under the usual norm

IIIIIL

=

I f

₊

s ( f ),

where

Then U

_{(respectively V)}

sends

_boundedly

_L(W)

_{(respectively L(I))}

into itself.

_Moreover,

we have the

following

results.

Theorem 2. The random

_system

with

_complete

connections

_(RSCC)

associated with the

_{aCF-expansion}

is with contraction and its transition

operator

U is

_regular

w.r.t

_L(W).

Proof.

We have for all

_(l,

_i)

E X

Hence the

_requirements

of the definition of an RSCC with contraction are met with k = 1

(see

Definition 3.1.15 in

[6]).

By

Theorem 3.1.16

in

_[6],

it follows that the Markov chain

_(s:)n&#x3E;O

associated with the RSCC

{(W,W),

(X, X),

u,

P}

is a Doeblin-Fortet chain. Hence

_by

Definition 3.2.1

in

_[6],

the Markov chain

_(sn)n&#x3E;o

is

_compact,

because its _{state space} is a

compact

metric _space

_{(W, d) _ (~0, R~, d),}

with

_{d(x, y) _ ~x - y I,}

_{dx, y E W}

and its transition

_operator

is a Doeblin-Fortet

_operator.

To _prove the

_regularity

of U w.r.t.

_L(W),

let us define

_recursively

_wn+1 =

(w"

+

2)-1, n

E

_N,

with _wo = _w. A criterion of

_regularity

is

expressed

in Theorem 3.2.13 in

_[6],

in terms of the

_supports

_En(w)

of the

_n-step

transition

_probability

functions

_{P~(w, ~), n}

E N*

_where,

with the usual

notation,

Clearly

Therefore,

Lemma 3.2.14 in

_[6]

and an

induc-tion

_argument

lead to the conclusion that wn E E N*. But

Now,

the

_regularity

of U w.r.t.

_L(W)

follows from Theorem 3.2.13 in

_[6]

and the

_proof

is

_complete.

0

Theorem 3. The random

_system

with

_complete

connections

_(RSCC)

is with contraction and its translation

_operator

V is

_regular

w.r.t.

_L(I).

Proof.

The

_proof

_parallels

that one of Theorem 2. We have for all

(l, i)

E X

We deduce that the Markov chain associated with the RSCC

(X, X),

v,

Q}

is a Doeblin-Fortet

chain.

_Moreover,

the Markov

chain is

_compact

and its transition

_operator

is a Doeblin-Fortet

operator.

To _prove the

_regularity

of V w.r.t.

_L(I)

we

_proceed

as in the

preceding

proof.

Here the transition

_probability

function is

A similar

_argument

leads to the

_regularity

of V. 0 Remark. Theorem l’ in

_[4]

shows that there exist

_positive

constants

Kl, K2, ,~

1 and 0 1 such that

6. A Gauss-Kuzmin

_type

_problem

The results obtained allow to a solution of a Gauss-Kuzmin

_type

problem.

The solution

_presented

here is based on the

ergodic

behaviour of the RSCC

associated with the transition

_operator

V.

It should be mentioned that it is

_only

very

recently

that there has been any

investigation

of the metrical

properties

of the Rosen fractions

(Nakada

(in

~10~)

started

_{investigations}

of metrical

_properties

of the Rosen

_fractions,

but

_only

with even

q’s) .

_Thus,

we _may

_emphasize

that our solution obtained

in an

_elementary

manner is

_{surprisingly simple}

and could be used in similar

contexts.

(see

Section

_3).

By equation

(4),

we have

By

elementary

computations

we obtain

and

_If:1

_2,

_8(1:)

3 for all

_possible

values of

_sn.

Now,

we note that on account of the last remark in Section

5,

we

actually

proved

the

_following

Gauss-Kuzmin

_type

theorem.

Theorem 4

_(Solution

of the Gauss-Kuzmin

_problem).

For all a E W there exist two

_positive

constants K and 0 1 such that

there p denotes the invariant measure _vx

for

T on

x3I.

Remark. Theorem 4 reduces for a = 0 and A = 1 _{to a} version of

Gauss-Kuzmin

_type

theorem for the nearest

_integer

continued

_fraction,

_{where p°}

represents

the

_Lebesgue

measure on

[-1/2, 1/2]

_{2 2} ’

It should be

_{emphasized that,}

to our

knowledge,

Theorem 4 is the first

Gauss-Kuzmin result

_proved

for the Rosen fractions.

Acknowledgements.

I would like to thank Marius Iosifescu for many

stimulating

discussions and Cor

_Kraaikamp

for useful information

concern-ing

the

_present

_subject.

I also wish to thank an _anonymous referee for a

number of

_suggestions

and comments which allowed to

_improve

the content

and

_presentation

of the paper.

References

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emphasis on the _{Markoff spectrum (Provo, UT, 1991), 227-238,} Lecture Notes in Pure and

Appl. Math., 147, Dekker, New _York, 1993. Gabriela I. SEBE

University Politehnica of Bucharest

Department of Mathematics I

Splaiul Independentei 313 77206 Bucharest

Romania