Asymptotics of Simple Branching Populations

(1)

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Asymptotics of Simple Branching Populations

Thierry Huillet, Andrzej Klopotowski, Anna Porzio

To cite this version:

Thierry Huillet, Andrzej Klopotowski, Anna Porzio. Asymptotics of Simple Branching Populations.

Journal de Physique I, EDP Sciences, 1995, 5 (9), pp.1179-1197. �10.1051/jp1:1995190�. �jpa-00247128�

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J. Phys. I élance 5

(1995)

1179-1197 BEPTEMBERI995, PAGE1i79

Classification Physics Abstracts

02.50 05.00

Asymptotics of Simple Branching Populations

Thierry

Huillet

(~), Andrzej Klopotowski (~)

and Anna Porzio

(~)

(~) ^CNRS LIMHP, Université Paris XIII, _avenue J.-B. Clément, 93430 Villetaneuse Cedex,

France

(~) ^Université Paris XIII, Institut Galilée, Département de

Mathématiques,

_avenue J.-B. Clé- ment, 93430 Villetaneuse Cedex, France

(Received

24 January 1995, received in final form 6 May1995, accepted 12 May 1995)

Résumé. On considère l'itération d'une structure déterministe arborescente selon laquelle

un ancêtre engendre _un nombre fini de descendants dont la durée de vie

(à

valeur8

entières)

est donnée. Dan8 _un prenùer temps on s'iJ~téres8e _aux trois distributions asymptotiques suivantes _:

répartition des générations, aptitude à

engendrer

des descendants et répartition selon l'âge.

Ensuite _nous développons le formalisme thermodynamique _pour mettre _en évidence le caractère

multifractal de la scission d'une _mas8e unitaire associée à cette arborescence.

Abstract. In this paper we study _a simple deterJninistic tree structure: _an initial individual generates _a limite number of oflspring, each of which has given integer valued lifetime, iterating

the _saine procedure when dying. Three asymptotic distributions of this asynchronous deterrnin-

istic branching procedure _are considered: the generation distribution, the ability of individuals to generate offspring and the _age distribution. Thermodynarnic formalism _is then developped

to reveal the Jnultifractal nature of the _Jnass splitting ^associated to _Dur process.

1. Introduction

The

subject

of this _paper is trie

study

^of _a _very

simple

deterministic tree structure,

according

to which _an _aucestor generates a

given (finite)

^number ^of

offspring,

each of which bas _a

given

lifetime with values in IN and iterates the _same

procedure. By

trie

appropriate

choice of the unit of lifetime this

assumption

seems to be _more raturai than trie

continuity (of lifetime)

and it

simplifies essentially

_our model. The fact that trie individuals have distinct lifetimes

implies

_a great ^difference ^between

asymptotical

properties of _our

population

and trie _one of

equal

lifetimes. Our

study

bas

already

been initiated in

[Ii, following

in this _some

[2-5]

in

population dynamics.

The purpose of trie

preceding

work

[Ii

was to

give

a

probabilistic

version of trie studied structure and to underline trie differences and

analogies

with _some

problems

of the

theory

of

age-dependent branching

_processes

[6,7].

^It

happens

that the deterministic structure deserves _a _more detailed

study

from trie

asymptotic point

of viewj

(3)

1180 JOURNAL DE PHi'SIQUE I N°9

this is the

object

of the present ivork. Three

asymptotic

distributions of such

populations

are therefore

considered,

relative _to the

generation distribution,

trie

ability

of individuals to

generate offspring

and the _age

distribution, respectively.

We extend trie

quoted

results

[Ii

to a

mass-splitting procedure.

Multifractal

analysis

of se'f-similar _measures has

recently emerged

as an important concept ⁱⁿ ^various ^fields

[8-12],

when _an unit _mass _can be

spread

_over _a

region

in such _a _way that its distribution varies

widely.

The

example

of Bemoulli cascade is

currently

used

[12].

In _our paper we

study

_a richer

multiplicative phenomena,

_1.e.~ _one associated _to

an

asynchronous

deterministic

branchiug

_process. Our results _can be iucludeà in trie

theory developed by

ivlichon and

Peyrière [13,14]

on self-similar structures.

By

trie

simplicity

of _our model most computaiions _are now

quite explicit.

In Section 2 _we derme _our niodel and _we recall _our results

concerning

the

generation

distribution

[Ii.

Section 3 deals with trie

ability

of individuals to generate

olfspring.

In Section 4

we consider trie

asymptotic

_age distribution. Trie

asynchronous

_mass

splitting

^is ^discussed ⁱⁿ

Section 5. We

study

the

thermodynamic

limit of the _mass

partition

function and _we show its existence and

analyticity.

We compute

explicitely

the

Legendre

transform of trie renormalized limit

partition

function aud _M;e

give

its

interpretation

_as trie dimension spectrum. The _next

section is devoted to _some

geometrical interpretation

of trie

previously

obtained facts. In trie last part ^of ^this paper we present _an

application

of trie method of

large deviations,

1-e-, ^the limit theorem for Hôlder exponents.

2. Definitions

Let _us consider the

branching

model defined in reference

[Ii.

An initial individual at time _n ₌ 0 generates ^M

offspring having respective

^lifetimes (not

raudom)

Ta E IN; ¹ ^< < M. M is fixed and the _same for each individual. The parent dies _upon

giving

^birth. ^Each ^descendant iterates then the _same

procedure.

Trie

offspring

_are numbered

by

increased age; this

implies

_a natural way of

numbering

trie branches of the induced tree. Our results do trot

depend

of trie method of

numbering. Among

these

values,

let _w choose ail distinct K values

T)

E IN; ¹ <

j

<

K,

in

increasing

order and [et

~-

~~

T/,

_~+ ^~~

T(

For ail 1 < k < ~+ we let _ah be the number of

oifspring

with iiîetime

k,

if this value

figures

among the numbers

T);

¹ ^<

j

<

K,

and _zero, otherwise. One has therefore

~j

~+

ah " M.

k=i

The trivial _case _ah _" 0, 1 < k < ~±, a~~

#

0, is exduded in trie

sequel.

If _one gets ^interested ^into ^the ^number N,i, _n > 0, of individuals

living

at time _n then _one has

M

No

" 1, Nn ₌

~j (ljT,>nj

+

Nn-T, ljT,~nj ),

_n _E ~.

i=i

In the

preceding

notations

(with

_ah _" 0 for k

# Ti, ,T()

this _is also:

~+

Nn

=

~j (akljk>«j

+

akNn-kljk<nj ),

n E IN.

k=1

0/his renewal

equation

defines the generator of the

reproduction

tree. In

particular,

the self-

similarity

^{of the} tree should be

pointed

oui, since this is the

key

to the recursion formula.

(4)

N°9 ASYMPTOTICS OF SIMPLE BRANCHING POPULATIONS 1181

A closer look at this structure leads to consider _a variable

Nn(p)

which represents ^the ^number of individuals at time n of the p~~

generation,

which _means, _possessing

exactly

_p ascendants.

It satisfies trie

equations

No

(0)

₌ 1;

M

N«(p)

₌

~ jijr,>«jij~=ij

+

N«-r (v i)ijr,<«i

,

n E

il, v~(n)

< v ^<

v+(n);

1=1

with

~

if

~ integer,

P~(n)

^~~

[~]

+ l

otherwise;

fi

^if

fi integer, P~(n)

^~~

[~]

+ l otherwise.

We _suppose that

Nn(p)

₌ 0 for _p _< p~

(n)

_or _p >

p+(n).

Its

generating

function

p+(«)

ifi«(~f) ^~~

~ _~fPN«(p),

~i e jo,

ij, (i)

p=p-(«)

satisfies trie equation

M

ifi«(~f) = ~f

~ _(ijr>«j

₊

_ifi«-r, (~f)ijr,<«j

,

n e

r~, (2)

z=1

with initial condition

lfiol'f)

_" 1.

Let _us observe that

p~(n)

₌ 1 for Tj > n, which

explains

the first _term _on the

right

side of

equation (2). Moreover,

the second _term _on the

right

side of

equation (2)

is obtained since p~

(n n)

>

p-(n)

and

p+(n

_Ta) <

p+(n)

1.

In _an

equivalent

_manner,

(2)

_cari be written

~+

ifi«(~f) = ~f

~ _(ah _ijk>«j

₊

akifi«-k(~f)ijk<«i

,

n e r~.

k=1

One _may check that

p+(«)

iÎ«(1)

= Nn ₌

~j Nn(p),

_n _E _IN,

p=p-(«) and

P+(")

~fin(M~l)

=

~j _M~PNn(p)

_=1,

n E ~,

p=P~(")

which _means the conservation of _a unit _mass diifused in the _tree.

(5)

1182 JOURNAL DE

PHYSIQUE

I N°9

3.

Ability

to Generate

OfEspring

Now _we _are going ^to consider trie limit behaviour of _our

population

from the

point

of view of the

ability

to generate

oifspring.

Let Xn E lN~+ be _a vector whose l~~ component

Xn,i

represents ^the ^number ^of individuals

living

at time _n, which

produce

at least _one descendant at step n +1- 1.

It should be clear that

Xi

=

(1, 0,.

.,

0) (3)

and that the vectors Xn, n > 0,

satisfy

the

following

relation:

Xn+i

"

AXn,

_n _{2 0,}

(4)

where the _~+ _x _~+ companion matrix A is

given by:

ai 1 0. 0

a2 0 0

~ df

a~~-1 ⁰ ^0..1

a~~ ⁰ ^0. ⁰

In

particular,

the first comportent

Xn,i

represents the number of individuals

branching

at time

n.

Moreover,

the number of individuals in the

population

at time _n is

giveu by:

~+

Nn =

C~

_Xn

= ((Xn((1 ^~~ Xn

z Î, n > 0,

~j

i=i

where

C~~~(1,1,. _,1),

~+

therefore

df

~ _~«)~~

j, n ~ ~.

N~

=

((A"XOÎÎI

_"

ÎÎA"ÎÎJI iÎÎÎ~ÎÎ+

~=i

The

non-negative

matrix A is

primitive

and

invertible,

its characteristic

polynomial

_{x is}

x(À)

^~~ À~+

~j akÀ~+~~,

E C.

~i

The _roots de

x(À)

_are ail

distinct;

those with

non-negative

real part are ail

complex

and

conjugated

inside the unit circle except

exactly

_one, _a, which is real and _> 1; those with _a

negative

real part are

complex

or real of modulus < a [15]. ^Then o =

p(A),

where

p(A)

is the

spectral

radius of A.

It follows that

hm

NÎ

= lim

((A"(()j

⁼

p(A)

= a > 1,

n-m n-m

the~~f°~~

jjm

logNn

=

'°g°.

n-m n

(6)

4.

Asymptotic Age

Distribution

Let _us consider the _age structure of the

population.

More

precisely,

let _us introduce the vector

yT

^df

~y

y y

n n,1>

, n,z, n,~+

where

Yn,j gives

the number of individuals at time _n of

age1, supposed

_a priori bounded

by

a maximal _age _~+.

If _au iudividual of age

j

is then

supposed

to bave trie

fertility coefficient

df number of

oifspring

born of individuals of _age

j

~

number of individuals of _age

j

and if

df number of survivors among trie individuals of _age

j

~~ number of individuals of _age

j

is its suruiual

factor

from _age

j

to

j

+1, it is easy to deduce the _recurrence _on Y:

Yn+i

_"

MYn,

_n 2 1,

(5)

where

fi

₁₂

Î~+-1 Î~+

si 0 0 0

M df 0 _s2 0 0

(6)

0 0 S~~-1 ⁰

We have shown _m

[Ii

that there exists _an age distribution of the tree defined

by (5)

with the initial condition

yT_(fif~ _~)

l- _>. _, _,

where the matrix M is

given by (6)

with for

fertility

coefficients:

fg =

Î)~

E

la, Mi, j

₌ 1,.

, ~+,

(7)

£

_ah

k=j

and for survival factors:

~+

£

_ah

sj =

~@~~ E]0, Ii,

_j

= 1,.

., ~+ l.

(8)

£

_ah

h=j

Moreover,

these coefficients _are related

by

sj=1-),j=1,. _,~+-l. ₍₉₎

The asymptotic behaviour is _now

given by

the Perron-Frobenius theorem

[15,16].

More

precisely,

[et _p ^~~

(l,

_p2,

)~

and _q ^~~

(l,

_q2,

.)~

the

positive

left and

right

Perron-Frobenius

eigenvectors

of the matrix

M,

which _means, the

particular

solutions of

Mp

₌ _op,

(10)

(7)

l184 JOURNAL DE PHYSIQUE I N°9

M~q

= aq,

where _a

=

p(M)(= p(A))

is the

spectral

radius of M.

Let

~

df P

~ ((P((I ^'

~

df q

~

ÎÎqÎÎI'

be the normalized vectors so that S* ^~~

pq~

is the orthonormal

projector

_on ^the

eigenvector P*.

One checks that in the

large

time

M"

+~

a~jS*,

therefore

Yn =

M"~~Yi

~

aj~s*Yi

"

coj~p*,

where _c ^~~

q*~Yi

" M. Then

Yn

~

ÎÎYnÎÎI

^~ ^~

Let therefore

~+

ÎIM

~~

~ _llkô(k),

k=1

~+

Îllf

^~~

~j _llÎ~(k),

k=1

be the

asymptotic

_age distribution _measures.

Using equations (7), (8)

^and

(10),

the

explicit expression

of _pk is:

k-1

~l-k

^~+

l~k CYÎÎ^~

~

~l

~ ~jal>

1 < k < ~+.

l=1 1=k

An

asymptotic

_average _age

ÎM

cau be introduced:

ÎM

^~~

~

~+

kp(.

k=1

Let then

~~~~~

^~~ _~

~~~~~ Ée

~~PÎ,

E

lit,

with values in

]0, mi.

Trie cumulant function

KM(À)

^~~

logLm(À)

is _concave and

analytic

on lit. Moreover,

K[(0)

= TM.

Let

1 10, 0

p2 01, 0

jdf

p~~-1 ⁰ ^0, ¹

~~

0 0.

(8)

N°9 ASYMPTOTICS OF SIMPLE BRANCHING POPULATIONS l185

The matrix

ÀÎ

_is

non-negative

and

primitive,

_so it has _a simple real dominant

eigenvalue

which coincides with its

spectral

radius denoted

by

_où.

Also,

_où > 1. It should be clear that

KM(logaù)"1°giiPii.

and it is therefore

possible

to introduce the number

DM ^~~

1°g

_[[P[[i

TM iog

_a _ù

~1°'

_l.

Let _us consider the

propagation

system ^{of trie}

weights:

$~T

(1

~

~),

0 '. '

9n+1= ÀÎ9n,

n > 0.

The numbers

Mn ^~~

((Qn il,

_n _E

i~,

represent then trie

weights

of trie

population

at time _n.

Let _us consider _now the variable

9n(p)

which represents the

weight

of trie individuals at time _n of the p~~

generation,

that

is, possessing exactly

_p ancestors. It satisfies the

equation

Vo (o)

= i;

9n(p)

M

=

~j ljr>«jlj~=ij

+

Qn-T, (P 1)ljT,<«j

, n E

~,

_p / 1.

i=i

Its

generating

^function

~

ifi«(~f) ^~~

~É~ ~fPil«(v),

_~f E 10,

il,

p=p-(«) with

v"(n)

^~~ _n,

~,

if

~ integer,

v~(n)

^~~ ^~ ^~

[fi]

+ 1,

otherwise,

satisfies the

equation

~+

ifi«l~f) = ~f

~j (vkijk>«i

+

vkifi«-k(~f)iik<«i

, n e

i~,

k=i

with the initial condition

trio(~f) =

One _may check that

p+(«)

ifin(i)

= M~

=

~ it«(p),

n e i~,

p=p-(n)

(9)

aud

p+(«)

ifi«(ÎÎPÎÎi~)

=

~ ÎÎPÎÎi~iI«(P)

₌ 1, n E

Il,

p=p-(«)

which _means the unit _mass conservation.

A direct extension of results of

[Ii gives:

THEOREM 1:

If DM(n)

^~~

~£ Îi ((p(([~"~~~logm~

((p(([~"~~~, n E ~, where

pn(1)

denotes trie number

of

ancestors

of

trie i~~ indiuidual _ai lime _n, then trie limit _SM ^~~ lima-m

DM(n) always

ezists and _one bas trie

equality:

_SM ₌

DM

EXAMPLE. For the Fibonacci tree:

M=2, ~+=2,

ai"a2=1,

one has _ri

" 1, _r2 _"

),

where _a

= 1.618....

Moreover,

_((r((~

= a so that

TA

"

@

_@

1,381964....

One has _pi

"1,

_p2 _"

),

where _o =1.618...

Moreover,

_((p((~

=

Q

_so that

ÎM

"

fl@

_*1, _230718.... _This

_gives

2a + 1

log Q

~~

2(a

₊ ₁₎

logâm

^'

where âM is trie

largest

real solution of the

equation

z~~ ₊

z~~

= 1.

2a

5. Mass

Splitting

We turn _now to the

description

^of

singular

_mass distributions _on the interval [0,

Ii, exploiting

the

peculiarities

of the

branching

tree structure defined in the

preceding

sections. In this set-up,

we compute ^the

corresponding

partition function and _we show that it is

exactly

renormalizable.

Next, _we

develop

_some

thermodynamic

formalism, in

particular

_we show that the free energy function is

analytic.

Besicovitch has considered _a

repartition

of the unit _mass in the interval [0,

Ii.

His method

firstly

consists to distribute _a unit _mass in the interval [0,

Ii

in such

jw~y

^that ^the ^mass ^xi ^> ⁰

is

uniformly

distributed _on the interval ^~

,

~

jj~

[, 0 < < M

-1, ~j

xi = 1. The

following

M

,_~

steps ^iterate this

procedure

for each sub-interval. ^~

The

asymptotic

distributions _are then concentrated _on

everywhere

dense sets in [0,

Ii,

whose Hausdorlf dimension is

equal

to the entropy [17]:

M-1

d ~~

~j

Xi

log

~ Xi z=0

One _can consider the

following

situation

adapted

to our case: each segment ^~

,

~ ~ Nn Nn

~[;

0 < < Nn -1, at step n receives trie _mass

M~P«(~),

l < <

Nn, uniformly distributed,

(10)

where

pn(1)

denotes trie number of ancestors of the i~~ individual

(with

_respect to the natural

ordering

^defined

before)

at time _n.

This method

naturally

leads _to consider trie entropy

DG(n)

^~~

Îi f M~P«(~) logN~ M~P"(~),

_n E IN.

We have

proved

^that:

THEOREM 2

Ill:

The limit

DG

^~~ lim

DG(n) always

ezists and _one bas trie

following equality:

M

logm

1

logm

~'~

£ ka~ loga Î loga ^~''

iiki~+

where

fldf £

k~*

~*df

^ah

k, k

i<k<~+

£

_ah

i<k<~~

EXAMPLE

(eONTINUED).

^For ^the ^Fibonacci tree _one finds

DG

_" ^~ ^~°~^~

"

0.9603...,

^where

3

loga

a = 1.618... is trie

golden

number.

Moreover, Î

=

~.

2

Assume _now _a unit _mass is

being split

into M

branches,

_ah of which _are

being

of

length k;

M

k =

1,.

^~+ Each branch receives the _mass _xi _>

0,

1= 1,

2,.

,

M; £

_xi

" 1. The

splitting

i=i

is then iterated _as before. A

particular

_case of this model when _xi

=

ù,

=

1,2,...,M,

has

already

been considered above. Now _we shall consider the situation when trie _masses xi >

0,

₌

1,2,. .,M,

_are not

necessarily equal.

We want to derive _some statistical limit

properties

^of trie _mass

repartition

within each branch _as _n _- _cc.

At step _{n, it is} ^here also of interest to compute the

partition

function of the _mass distribution

along

the

cylinders. Usually

_m trie literature _on infinite trees, ^the ^word

"cylinder

_1" denotes the subtree of ail the descendants of

partide

in the infinite _tree. In _our model it _can be

identified,

in trie

equivalent

_mariner, to trie set of the ancestors of

partide1.

To this purpose let _us introduce the _concave

partition

function

[loi

_:

~«j~) flf~j)>,

_~

~

», (ii)

imi

where _~1(1) denotes the _mass attached to trie

cylinder

_1;

=

1,

2,.

,

Nn.

Let _us remark that ~fin(À) ^is also:

p+(«) _Nn(p)

~fin(À) ^~~

£ £

_~1(1)~, _E

_lit,

p=p-(«) z"1

so

that,

_~fin(À)

dearly

reduces to trie

analogous

partition function

given by (1),

when

xi "

ù, 1=1,2,.

,

M.

(11)

As

before,

~fin(0) =

Nn

_is the number conservation

equation

and ~fin(1) ₌ ^{1 is} ^trie mass

conservation equation.

It follows

clearly

from the

preceding

^that:

LEMMA 1:

For each _E lit _we bave trie

equality

ifi«(À) =

ÎÎX«iÀ)ÎÎI,

where

Xn(À)

is trie solution

of:

Xn+i(À)

"

A>Xn(À)> Xo(À)

_"

11>°; °)~,

Xo(À)

₌

(1,0,.. 0)~,

with

ai(À)

1 0..0

a2(À)

0 1..0

AA~~

,

(12) a~~-i(À)

0 0..1

a~~(À)

⁰ ^0.

ak(À)~~ (

xl,

d(i)=ki=i

where trie last _sum is

performed

_Oder ail branches

of depth d(1)

= k

of

trie generator.

It should _now be recalled

that,

due to the

primitivity

of

AA,

the matrix AA bas _a

unique

maximal absolute

eigenvalue,

_say

a(À)

> 0, ^of

algebraic (and

hence

geometric) multiplicity

one.

a(À)

satisfies the characteristic equation ^of

(8)

_as _an

eigenvalue

of AA,

namely:

~j

~l ^~)~k

" i, ~ M.

(13)

ÎÎ ^l~

d(1)=k

Moreover,

_a _: lit

-

lit+

_is

strictly decreasing

and _goes _to _zero _as _- _+cc [15].

Also,

due to the structure of AA and from the

preceding

sections:

o(0)

= a > 1 and

a(1)

= 1.

From this

situation,

one can conclude:

THEOREM 31

Let _us

dejine:

Fn(À) Î

logN~

~fin(À), E

lit,

_n E IN.

Trie

pointwise "thermodynamic

limit"

lim

Fn(À)

^~~

F(À)

=

log~ a(À),

E

fil, (14)

is _concave and

analytic.

Moreouer

F(0)

₌ -1,

F(1)

= 0.

(12)

Proof:

From Lemma 1,

ifin(À) =

ÎÎXnIÀ)ÎÎI

=

ÎÎAIÎÎjIj,

therefore

lim

log~

_~fin(À) ₌ lim

log~

ii

Ai

ii

)j

=

log~ a(À),

E lit.

F(À)

is thus _concave _as _a

simple

limit of _concave functions.

The

analyticity

results from trie fact that

a(À)

is itself

analytic,

_as it _was

kiudly brought

to our attention

by

Michon.

Indeed,

[et

P(À,x) ~~det(AA XI),

E

lit,

_x E lit.

By

the Perron-Frobenius theorem

P(À, a(À))

₌ 0 for ail E lit.

Suppose

that

P(Ào, a(Ào))

= 0 for fixed Ào. Since

o(Ào)

is _a

simple

_root of P:

$(Ào, ^a(Ào)) ^#

^0,

therefore

by

the theorem of

implicit analytic

functions there exist _an

analytic

function _r _: ]Ào a, Ào +

a[-

fil and

fl

> 0 such that ]Ào a, Ào +

a[x]a

_> _(Ào)

fl, a(Ào

+

fl[n(P

=

0)

is the

graph

of

r(À).

Moreover, letting

_xi < z2 < <

o(to)

denote ail real roots of

P,

trie

continuity

of _roots of

an

equation depending continuously

on a parameter reads _as follows:

For _e _> 0 there exists ~ such that, if _Ào

Î<

1~ and if

P(À,z)

= 0, then

z

E]zi

_e, _~i +

e[u]z2

_{e, z2} +

e[u. u]a(Ào)

_e,

a(Ào)

+

e[.

If _e is well

chosen,

then those intervals _are

disjoint

and for _<

1~, the greatest root is

r(À).

This _proves that

r(À)

=

a(À)

at _some

neigbourhood

of Ào.

Remark 1:

A _more detailed

study

of F shows that when

- -cc,

F(À)

behaves like the fine of equation y =

tmaxÀ,

where furax ^~~

log~

_x~, _x~ ^~~ inf _~ri

~~ l<l<M

d(1)=~~

Also when _- +cc,

F(À)

behaves like the line of

equation

_y ₌

tmmÀ,

where

tmm ^~~

~à _log~

K*, K* ~~

Sllp Xl l<1<M

d(1)=~+

We _are _now in the

position

to introduce trie

Legendre

^transform ^of

F(À):

f(t)

^~~ inf

(Àt F(À)),

_{t E}

_[tmm,

_furax].

(15)

The function

f(t)

is _concave and

analytic

_on [tmjn, furax].

Moreover,

it is involutive:

F(À)

₌ inf

(Àt fit)),

_E fil.

tElt~;n,t~axj

f(t)

also reads

f(t)

= ~*

(t)t F(~*(t)),

t ~ jt~~~,

t~~~j,

with À* defined

by

$ _(À*(t))

= t.

(13)

1190 JOURNAL DE PHYSIQUE I N°9

Jtt)

i

F'a)

0 tmin F'(1) 1 F'(0) tmax _t

Fig. 1. The Legendre transform of the free _energy function.

It has the usual bell

shape [loi

_see

Figure

1. It follows from the above that:

f(F'(0))

=

-F(0)

₌ ',

f(F/ji))

m

Fiji) F(i)

₌

F/(i) flD(i),

and from Theorem 1 and equation

(14)

that

~+

£

^a~~

£ log~

_K1

k=1 1<1<M

F'(0)

= ~~

~~~~~~

> l,

£ kaka~k

k=1

£

M _7tl _ÎOg~_7tl

F'(1)

=

~/"~

^< ^l.

£~ £

_~l

k=1 1<1<M

d(1)=k

We _are not able at this step ^to ^derive more properties of both

f(t)

and

F(À)

_as defined

by

equations

(13)

and

(15),

in particular, their explicit expression. Nevertheless, _some additional

information _cari be extracted.

To do

this,

_we [et:

t*(À)

=

F'(À)

(14)

be the _inverse of

À*(t),

_so that _one _can derme the function

/lÀ) fl _f(t*(À))

=

ÀF'IA) FIA),

E M.

l16)

Let _us define _a one-parameter

family

of

probability

_measures:

~1>(1) ^~~

~~~~~

,

l <1 <

Nn,

E

lit,

É

Jt(1)>

z=1

with _~li

Ii)

= ~1(1), ^<1< Nn.

The

imprecision function

_[18]:

1«j~) fl _Fjj~)

=

f ~>ji)

_iog~v~

~ji),

~ _{~ J~~,}

ami

is _a _measure of the lack of information of _an observer

who, ignoring

the true distribution

~1>(1) decides to affect his _own _guess ~1(1). Also

Nn

Î~(À)

^~~

~j Îl(~) Î°~Nn

ÎIÀ(~), ~ É~,

z=1

is the dual

imprecision function, reversing

the rotes of _~1(1) and ~1>(1).

Another

interesting

function is trie

Renyi's

entropy of of ~1(1):

Dn(À) Î _Fn(À), _#

_1.

~~~~~~~'

for ail À

#

1 ~~~

~"~~~ fi ùF(À)

^~~

_D(à) ₍₁₇₎

~"~~~m#Dli)

^~~

^F'il).

^~~

Let _us derme trie Shannon's entropy of ^of _~1>

Ii)

Nn

Sn(À)

^~~

~j

~l

Ail

_lOgN~ _~LÀ

ii ),

E lit.

z=1

An immediate extension of Theorem 1 is:

THEOREM 4:

For each _{E lit} trie

following

convergence froids:

~~z(À)~Î(À),

with

M

/(À)

^~~

Î~

~~~~~~~~~_~~~~~

Îi

k

L ~l(À)

k=1 1<1<M

d(1)=k

(15)

and

~

~ri(À) ^~~

/~

,

l <1< M.

£

^~>

i i=1

Letting

Kullback's

information gain

be

G«(~) fl1«(~) _s«(~)

=

( ~>(i)

_iog~v

4,

~ ~

m,

ami

" ~l>(1)

and its dual expression:

Gj(~) fl _çj~) _s«ji)

=

( ~ji)

_iog~v~

fl,

_~

~

m,

after _an _easy computation, _one has

Sn(À)

=

Dn(À) )Gn(À), _#

_1.

₍₁₉₎

As _n _- _cc,

by

^Theorem

3, il?)

and

(19)

we have

G«(À)fiGlÀ).

By (16), il?), (19),

GIÀ)

₌

(/(À) D(À))j

=

(i À)F'(À)

+

F(À)

for ail

#

and

/(1)

=

D(1),

due _to

G(1)

= 0,

G'(1)

= 0.

COROLLARY Il

The limit entropy

of

trie _mass distribution

along

_ouf

branching

tree

always

ezists and:

Nn

lim

~j

~1(1)

logN~

~1(1) =

D(1).

n-m z=1

6. Geometrical

Interpretation

The

preceding

considerations suggest to introduce _more

general

computations relative to the discrete _measures

/l

~~

É

akôjkj,

/l* ^~~

É

alôjkj.

k=i k=i

Let

L(À)

^~~

/ _e~~~d~l*

=

f e~~~al,

_E

_fil,

M ~-~

(16)

K~$)

i

dG <DG ~

,ia=r'(oi

iz$=DGK'(0)

iii =daK'(o)

~oe

Fig. ^2. The geoJnetrical interpretation of ~G.

with values _m

]0, mi,

^be ^trie ^real

^Laplace

^transform ^of ~1*.

Then trie cumulant function

K(À)

^~~

logL(À)

is _concave and

analytic

_on lit.

Moreover,

_one has

K(loge)

₌

logm

and

K'(0) ^=1.

This

gives

_an

interesting geometrical

interpretation ^of the entropy ^dimension ^and "shows"

that DG

E]0,1[.

It is quite natural to introduce _now the

Legendre

transform of trie function

K(À).

Let:

s(t)

^~~ inf

(Àt K(À)),

t E

[~-, ~+].

>EM

The function

s(t)

is _concave and

analytic

on

[~-, _~+]. Moreover,

it is involutive:

K(À)

= sup

(Àt s(t)),

_E lit.

tel- _>~+l

s(t)

also reads

S(~) " À~(~)~

KIÀ~(~)),

_~

l~-, ~+l,

with À* defined

by

~

) _(À*(t))

= t.

If t* denotes trie inverse of À*, one can m an

equivalent

_mariner work _ou trie entropy

S(À) ÎÎ _s(t*(À))

=

ÀK'(À) K(À);

E lit.

Therefore

E(À)

^~~

K'(À)

_can be

interpreted

_as _an internai euergy and

F(À)

^~~

)K(À)

^as ^a ^free

energy,

beiug

the _inverse of the temperature.

(17)

In_a

Fig. 3. The internai _euergy.

The evaluation of these

quantities

at

point

₌

loge gives

F(loge)

^~~

£K(log o)

=

DGT,

°g °

E(log o) É _K'(log _o)

~~

dGfl,

with dG <

DG

and

S(log o)

=

Î(dG

_DG

_log

o < 0.

7. Statistics of trie Hôlder

Exponents

In this

section,

we

give

^first trie heuristics of trie multifractal

study

^{of trie} _mass distribution

along

the _tree defined in Section 2. A

rigorous approach

ta this

problem

is then

given

which

essentially

is _an

application

of the

large

deviation theorem for _a non-Markov renewal _process that _we exhibit.

At step n of the above

branching procedure,

Nn

+~ o" individuals _are present ⁱⁿ ^trie population. Let 1(n) ^E

(1,.

,

Nn)

denote one of them. The _mass associated ta 1(n) ^is ~1(1(~)). ^The quantity

~~ ~~ _~~

~~l"

ÎÎ ^ÎÎÎ~

is called the _coarse Hôlder exportent

of1(~) [12].

For each value of

tz~~~, ^it ^is of interest ta evaluate the

number,

_say

N~(ti~~~),

^of individuals

taking approximatively

this _coarse Holder exponent, 1-e-,

Nn(tz~

^~~

#(i

E

(1,.

,

Nn);

^~°~^~~~~

+~ tj~

).

"

1°g

Nn "

(18)

Let _us fix _n E IN and suppose that there exists trie

functions logNn(tj~~~

~~~~l"~~ ^'~

logNn

^'

~~~

ciizj«~)

~

i°iijiin~)

⁼ ~ji~~~~~ 1,

~~~~

vnltzj«~

^~~

"j~~i«)1

m~ ~ynC(~.j~~

)),

~

(the"probability"

ta take the value

tz~~~ for _some

individual).

Indeed, if this _were trie _case,

using

trie definition of the partition function

Ill),

_one gets,

regrouping

all the individuals at step n with the _same exponent tq~~ ^and

recalling Nn

+~ o",

~fi~( À) +~

~

cy~"~~~'ln) ~~~~'ln)_~~, _fi É~,

~In)

where trie last _sum is

performed

_over ail individuals

having

distinct Holder exponents. ^For n

large

the effective contribution to this _sum is

given by

trie term which mmimizes

Àtz~~~

f(ti~~~

^).

We _may therefore expect

~~

^~°~~

^~"~~~

~

tli[

^~~~zin)

^fitzjn~)),

^~ ^~

^»,

for

large

_n, as

required.

So _we define

F(À) ~~inf(Àt f(t)),

_E lit.

The _measure carried

by

those individuals whose _coarse Holder exponent is

tj~~~ is:

~~~iin)

~~

N«(tj~~~

)Nn

^~'l")

+~

~y-«(t.~~~ f(t,~~~⁾⁾

In the _case the limit exists, the subset of those individuals for which

i~n t~~~~ ^m

Dji)

=

Fiji

Ii.e.,

the

unique

value which minimizes t

fit))

_carnes all the _measure, 1-e-, ^lim ~1(tq~~ = 1.

If Nn(tz~~~ âttains îts ^maximum ât ^the

value(s)

n-m

tz~~~ then lim tij~~ ^"

F'(0)

n-m

for which

f(F'(0))

=

-F(0)

=

D(0)

= 1

is maximum.

Therefore, D(0)

= 1 is the dimension of the support of this _measure

[9,12].

Let _us discuss _now the

rigorous approach

to the above heuristics.

Let T be _a random variable with

probability

distribution

~+

l~

~~

~j (

_~k

k=1

(19)

Let

AXT

be _a random variable with

probability

distribution

~] ~ (ôi-jag~,j=(~ôj-jag~~j.

ÎÎ iÎ iÎ

d(i)=k

Thus, given [T

=

k], AXT

_expresses _a ^random

equidistributed

choice of _one _among the _ah values

log~rii

1 < 1< M of

depth d(1)

= k.

AXT

will be considered _as _a random increment of _a _process defined _as follows. We _s±art

by

the

following equation

Xn ₌

AXTIy>q

+

(În-T

₊

AXT)ljT<nji

n 2 0,

(20)

where trie

equality

_is ^considered in law and

in-k _{given [T}

= k <

ni

has the _same

probability

distribution _as jKn-k. In this

equation

the

only required hypothesis

_on Xn _is that after _some

regeneration

time T the future of this _process _is

supposed

to be _a statistical copy of the initial process. Let _us check _now that this _process is well defined in laàv.

Lot

~n(À) Ee~"

=

(lbn(À);

n > 0.

It follows from

(20)

that

~+

~n(À)

"

j ~j(~k(À)1[k>n]

⁺

~n-k(À)ak(À)1[k<n] Ii

~l ~ Ù.

k=1

Also, letting

4l(s,

_À)

fi

e~~"4ln(À)

n=o

one gets

~ É

~k(À)li ~~~~l 4l(s,

_À) ₌ ^~"~

~~

E

llt~,

_s _>

logaN(À), (21) s(1- ù £ e-Skak(À))

k=i

where

oN(À)

is the the

largest

_zero of trie denominator of

(21),

_1-e-,

~+

(

_~~_~~~

oN

À)k

~'

which is _a normalized version of

(13).

It is well known that iim

jé~j~))+

ma

p~j~).

Letting

_nnw

FN (À) ^~~

log

_oN

(À),

and

fN(t)

its

Legendre transform,

it follows from Theorem II.6.1. of reference [19] ^that

li ~l~

_'°~OE~~n >

tl

_" _exp

f~(t~

= ~ f(t)-1

(20)

provided

t >

F'(0),

with

f

defined

by iii

and

Xn

^~~

log Mn. Alternatively,

lim

#(1

5 5

Nn; )~ _~/~

/

t)1

= af~~~.

"-" °g

n

References

iii

Huillet T. and Klopotowski A., J. Appt. Prob. 31

(1994)

333-347.

[2] ^Leslie P-H-, Biometrika 33

(1945)

183-212.

[3] ^Leslie P-H-, Biometrlka 35

(1948)

213-245.

[4] ^Lewis E-G-, Sankhyà 6

(1942)

93-96.

[5] Pollard J-H-, Biometrika 53

(1966)

397-415.

[6] Harris Th.E., The Theory of Branching Processes

(Spnnger-Verlag,

Berlin, Gottingen, Heidel- berg,

1963).

(7] Mode C.J., Multitype Branching Processes

(American

Elsevier Publ. Camp., New York,

1971).

[8] Collet P., Lebowitz J-L- and Porzio A., J. Statist. Phys. 47

(1987)

609-644.

[9] Falconer K., Fractal Geometry. Mathematical Foundations and Applications

(John

Wiley lc Sons, Inc., Chichester, New York, Brisbane, Toronto, Singapore,

1990).

[10] Halsey T.C. et ai., Phys. Rev. A 33

(1986)

l141-1150.

[Ill

Mandelbrot B-B-, The Fractal Geometry of Nature,

(W.H.

Freeman lc Go., San Francisco,

1982).

[12] Peitgen H-O- et ai., Chaos and Fractals. New Frontiers of Science

(Springer-Verlag,

Berlin, Gottingen, Heidelberg,

1992).

[13] ^Michou G., "Arbre, Cantor, DiJnension", Thèse d'habilitation, Université de Bourgogne

(Novem-

bre1989) _pp. 1-126.

[14] Michon G. and Peyrière J., "TherJnodynamique des ensembles de Cantor autosiJnilaires", Preprint, Université de Bourgogne (1992) _pp. 1-28.

[15] Horn R-A- and Johnson C.R., Matrix Analysis

(CaJnbridge

University Press,

1985).

[16] GantJnacher F-R-, Théorie des matrices

(Dunod,

Paris,

1966).

[17] Billinsgley P., Ergodic Theory and Information

(John

Wiley lc Sons, Inc., New York, London, Sydney, 1967).

[18] ^HaJnmad P., Information, systèmes et distributions

(Cujas,

1987).

[19] ^Ellis R-S-, Entropy, Large Deviations and Statistical Mechamcs

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New York, Berlin, Gottingen, Heidelberg,