Probabilistic and asymptotic methods with the Perron Frobenius's operator

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Probabilistic and asymptotic methods with the Perron

Frobenius’s operator

Guy Cirier

To cite this version:

Guy Cirier. Probabilistic and asymptotic methods with the Perron Frobenius’s operator. 2012. �hal-00691097v3�

Probabilistic and asymptotic methods with the Perron Frobenius’s operator Guy Cirier

guy.cirier@gmail.com

Résumé

1- Écart résolvant. Soit f une application polynomiale d’un compact C de Rd

dans lui-même. Une mesure P est dite invariante sous f ssi P  P_f où

Pf  P o f

1_{. P}

f est appelé l’opérateur de Perron Frobenius (PF). Soient e_fn( y ) n

eya  eyf ( a )

 /an a0 etH_n( y ) 

n_eyf ( a ) _/an

a0 où 0est point fixe def . Soit E_n l’idéal engendré par d 1 polynômesen_f( y ) 0et

ef

n1l ( y ) 0. Sous cette hypothèse et via la transformation de Laplace-Steiltjes, la solution est obtenue à partir de la distribution asymptotique des zéros réels de E_nquand n   . En effet, l’idéal E_n définit des variétés algébriques affines réelles. Si ces variétés sont 0-dimensionnelles et ont une distribution asymptotique de densité dérivable q d -dimensionnelle, la densité invariante sera de la forme:

p(a ) 1 d

(a_)d_{q(a ) /a}

1..a..ad

2-Approximation. Sous les hypothèses de la méthode du col, la distribution asymptotique des zéros des

H_n( y )(variétés algébriques affines) est image réciproque de distributions uniformes__ sur (0,1) : l  sl Im fl(a )l 1, 2, .., p

pour toutes les p  d coordonnées complexes du point critique a de (a ) sf (a )log a . __ est l’argument de _a_l , _s_l  y_l / n) . Si la hessienne de(a ) est dégénérée, comme dans le cas de linéarités partielles, nous obtenons des familles de variétés aléatoires (exemple Henon).

3- Applications aux EDO ou EDP de la forme a t  F (a ). On associe l’itération infinitésimale

f (a,) aF (a )pour lui appliquer l’opérateur de Perron Frobenius. Si F (a ) est partiellement

linéaire, alors une distribution aléatoire peut être solution asymptotique de cette équation. Les applications les plus remarquables de l’étude sont les solutions asymptotiques des équations de Lorenz, de Navier Stokes ou d’Hamilton.

4- Résonance. Les valeurs propresde la partie linéaire de f vérifient k _

1. Sous les hypothèses précédentes, le point critique a de (a ) vérifie alors la condition of résonance

1 ak1 Abstract

1- Solving deviation. Let f be a polynomial application of a compact C of Rd in C . A measure P is

invariant under f iff P  P_f where

Pf  P o f

1

. P_f is said Perron Frobenius operator (PF). Denote en_{f( y ) }n _eya _{ e}yf ( a )

 /an

a0 and H_n( y)  n

eyf ( a ) /an _a0 where0is a fixed point off . LetE_n be the ideal spanned by thed1 polynomialsen_{f( y ) 0and}

ef

n1l ( y ) 0. Under this hypothesis and via the Laplace-Steiltjes transform, we prove that the solution is given by the asymptotic distribution of the zeros of E_nwhen n   . The ideal E_n defines real algebraic affine manifolds. If these manifolds are 0-dimensional and have an asymptotic distribution with a derivable d -dimensional density q, the invariant density is:

p(a ) 1 d

(a_)d

2-Approximation. Under assumptions of the steepest descent, the asymptotic distribution of the zeros of theH_n( y ) (algebraic affine manifolds) getting the solution of (PF), is a function of iid __ on (0,1)

l  sl Im fl(a )l 1, 2, .., p

for all thep complex coordinates of the critical pointa of (a ) s f (a )log a. __ is the argument of _a_l ,

sl  lim

N yl / n . If the Hessian is degenerated, which is the case of partial linearity,

we get a family of random manifolds as in the Henson’s case.

3 – Applications to EDP, EDO. Asymptotic behaviours of ODE or PDE, as a t  F (a ), via the

Perron Frobenius’s density, are most interesting. Here, the infinitesimal iteration is

f (a,) aF (a ). But, if the Hessian is degenerated, the asymptotic design can be probabilistic.

Among applications, are asymptotic solutions of Lorenz, Navier-Stokes or Hamilton’s equations. 4- Resonance. If the eigenvaluesof the linear part of f verify k _

1. Under the previous hypothesis when n kq  , at the critical point aof (a ), we have the condition of resonance:

1 ak1 Introduction

The Perron-Frobenius‟s equation (PF) is a very difficult functional equation. Ulam and Von Neumann gave the only well-known first solution23

  for the logistic x1 4 ( x x

2

). Therefore, two important directions are used in physics: quantum chaos

 

11 and thermodynamics22

 .

This paper brings many improvements and corrections to the first preprints

 

3

 

4

 

5 . We apologize especially about some errors remaining in the previous versions of this paper.

The subject of this paper is to describe the structures of invariant probabilistic distributions when we iterate a function indefinitely. They can exist, even they are masked by others cycles or distributions. The convergence to some point or some distribution depends on zones of domination, as we match with the topographic zones of attraction of certain cycles or fixed points. This problem is not completely studied here. The reader is warned that we put forward here a very static and asymptotic conception of the physics phenomena. No transitions, no dynamics, no orbits, except perhaps for the cycles, only a probability of presence somewhere after infinite iterations in a compact set.

3 Chapter 1: The resolving deviation

I- Hypothesis, notations and definitions

1--H0- Hypothesis on the polynomial application f

Let f be a polynomial application of a compact set C of Rd

in C with at least one fixed point

f (0 ) 0well isolated in C. Diameter ofC isD. f is locally invertible at 0 ._f  ( f₁, f₂, .., f_l., f_d). Then, f is C and have a finite number of reciprocal images.

2- Collection of iterations

Let C ( f ) be the collection of iterations C ( f )( f , f( 2 )_{, ..., f}( k )_...)where

f( k )  f o f( k1). We can index generically an f( k ) _{C ( f ) asf}

 with a spot «  ». We check H0 for f provided to be

invertible at 0_(see below). So, generally, all demonstrations for f are valid for all f_. 3- Notations of the fixed points and eigenvalues of the linear part at theses points

Let 0Fix ( f ): f (0 ) 0 . But, all f( k ) _{can have f}( k )_{() in C with f ()  . These } k are generally called cycles ofk order. Let Fix( f( k ))be this set and Fix ( f_) the generic union for all k. Under H0, the real eigenvalues  () of the linear part of f (f) at 0(0) are determining for the

convergence to some invariants: fixed points, cycles or distributions. We don‟t study here complex or multiplicity of . These cases are not very difficult. When exist integersr Nd

such as r _

1, there is resonance, and we present here only a trivial situation of this recurrent case. Let

R e s  _{ k N}d _ k 1, k1l  1, l 1, ...d





and R e s  _{ kN}d _k 1, k1l  1, l 1, ...d





4- General notations

If a, x, y Rd_{, xy is the scalar product; x. y the vector with coordinates}

xlyl , 1, .., d; in bold_x x₁...x_l.x_d; _dx  dx₁..dxl.dxd and h /a  

d_{h /a}

1..al.ad. II – Laplace-Steiltjes transform of the Perron Frobenius (PF) equation

1- Definitions : Perron Frobenius’s operator (PF) and its Laplace-Steiltjes transformation (LPF) Let P be an arbitrary probability measure and f an H0-application. For all Borelian set B , the transformation of P by the Perron Frobenius‟s operator (PF) is

P o f

1_{( B ) P}

f( B ). The measure is f -invariant if P ( B ) P_f( B ) for all Borelian set B . As f (resp. f_) apply a compact setC in C,

P (resp. P_) exist

 

15 . All f( k ) _{C ( f ) letsP}

f  P invariant. 2- Lemma 1

Under H0, for all f and

 p,  _f(y ) is:  _f( y )  _Rd e yf ( x )_{dP ( x )} By definition of Pf( x ) P o f 1_{( x ),}

 h( x )dPf( x )  h o f ( x )dP ( x )for all positive function hofL1_{(P )(see}

 

_{15 ), hence for e}yx

. Here , we note yinstead ofy as traditionally to lighten the writing (especially for the approximations). There is no matter because we work in a compact set. If P has a Lebesgue-Steiltjes density p, then

pf  p o f

1 _f 

1_{' is the transformation of p by (PF)} 15

 

. But this definition is not very useful, complicated and sometimes misleading. We prefer use the lemma 1. Nevertheless, if we suppose that we start the iteration process from a point x₀: either almost all points are in a neighbourhood of 0and the Lebesgue-Steiltjes density is the Dirac‟s distribution( x ), orx_n does not tends to0 with a distribution p ( x )vanishing at 0.

Now, we take the Laplace-Steiltjes transform _Lof a probability measure dP ( x ) with support contained in the compact C. This transform is naturally used to capture the mass of the fixed point0:

( y ) 0 lim    eyx_{dP ( x ). Then, we have } f( y ) 0 lim    eyx_dP

f( x ) for all f with a fixed point 0 and all probability measure P on C. Its existence is due to the compactness de C, which makes the set   ( y lim0    eyxdP_f( x )  I 0 lim _ 

 eyxdP ( x ) ) not empty: 0  convex.

3-Remark: analytical representation of ( y)

Under H0, the support of the distribution is contained in the compact C with diameter D. Hence, )

( y

 can be represented by analytic series in a poly-disk with infinite radius of convergence.

( y) _nb_nyn

converges for all probability measureP in C(b_n  Dn / n !). (More, we can take y k (n1)provided D keD k

1 / ed to keep the convergence of the series). 4-Definition: resolving equation L

With the Laplace’s transform, the PF equation gives a resolving equation L:  _f() _f( y)( y) 0 and a collection of equations 

f(

_)

f( y )

_{( y ) 0 for} f_. More, ifP is invariant under f ,

 f( y )  e

yf o f ( x )_{dP ( x ) }

f o f( y ). By recurrence: f( y)( y).

III –Main Result

1-Notations of the sets of zeros of en_{( y)}

Under H0, without resonance, let ( y) _nc_nyn _{be a solution of the resolving equation. We} notef() ncne n ( y ) 0 whereen_{( y) ( y}n _{ H} n( y)andHn( y )  n eyf ( a ) /an a  0at a 0. Let nbe the expansion of order n de  such as f(n) 0 .

IfZ (en_{)is the set of zeros of e}n_{( y), let}

Z (En) Z (e n_)

l Z (e n1k₎

1, ...d, be the set of common zeros of these polynomials. E_nis the ideal generated by these d 1 polynomials.

2-Theorem 1 When n :

1- Exists an integer m such as (n( y)1)

m _

E_n. With a probability 1, the distribution defined by n( y )is constant on the algebraic affine real manifolds defined by En.

2- If the manifolds are 0-dimensionnal and if ( s )is solution de the equation _f()0for the density q (a ), we can write that the invariant density of Perron-Frobenius p (a ) is p(a ) 1

d

(a_)d_{q(a ) /a}

1..a..ad

3- These results are valid for f_ C ( f ).

The demonstration needs some steps. ( x ) is the Dirac‟s distribution at 0 ;

IV- Effects of a translation on analytical expression of  _f( y ) 1- Notations of the effects of a translationa:

Observe the effects of a translationa: x u a  0 changing the origin from 0 to a in C. If

p ( x )is a distribution density of a Lebesgue- Steiltjes measure for all x 0, we have:

f ( x ) becomes f_a(u ) f (u a ) a, p ( x ) becomes p (u ) p ( x a ), p_f( x ) becomes p_f( xa ), ( y ) becomes _a( y) ( y, a ) ( y)eya, _f( y )E (eyf ( x ) ) becomes f( y, a ) E (e yf ( ua ) ), _f() becomes _f(_a)_f( y, a )( y, a ) .

5 The following proposition uses the notations: H_n( y, a )eyf ( a ) n

eyf ( a ) /anand H_n( y ) H_n( y, 0 ). All following series are uniformly convergent for all y (see Constantine

 

6 with b_n Dn

/ n !). We

shall see uniform valuations later. 2- Proposition 1

- Let ( y) _nb_nynbe the Laplace-Steiltjes transform of an arbitrary



p. Under H0, the iteration f transforms all terms yn_eya _{ }n_eya _/an _{of }

a( y)( y)e

ya _{in }n_eyf ( a ) _/an_{, and, for all f with} f (0 ) 0and



p, we have:

 _f( y, a ) _nb_nn

eyf ( a ) /an

- Invariant density is solution of the resolving equation: f()  nbn( y n _

Hn( y )) 0 . Moreover,

for aC and y, _f(_a) _nb_nn_(eya _{ e}yf ( a )_{) /a}n _{ 0. For all multi-integers q, we already} have q_ f(a) /a q _{ }q E (eya _ eyf ( ua ) ) /aq _{ } nbn nq (eya _ eyf ( a ) ) /anq _ 0 and q_ f() nbn( y nq  Hnq( y )) 0 .

- As (z ) L( p ( x ))is the Laplace‟s transform of the density p, we writep( xa ) L1((z )eza). As  _f( y, a )_{ } eyf ( x )_{dP ( x a )  e}yf ( x )_{p ( x a )dx, we have: }

f( y, a )  e yf ( x )

L1((z )eza)dx Here, ( z ) has the analytic expansion (z ) _nb_nzn_:

 _f( y, a )  eyf ( x )

L1(_nb_nzneza)dx  _nb_n(_ eyf ( x )L1(zneza)dx)

L1(zneza) is the product of done-dimensional integrals _L1(znlezal ) for

1, ..., d . Using the Bromwich‟s formula, we recognize an inverse Fourier transform: L1(znlezal ) F1(znlezal )nl( xl al) because z

nlezal _has no singularities (see Bromwich‟s

formula on Wikipedia). L1(zneza) n(( xa ) /xn n( x a ). Then:  _f( y, a ) _nb_n( eyf ( x )n( x a )dx) _nb_nn

eyf ( a ) /an

- Hence, the resolving equation is: _f(_a) ( y, a ) _f( y, a )  _nb_nn

(eya  eyf ( a )) /an  0. Then, at a  0, we have_f() _nb_n( yn  H_n( y )) 0. In _f(_a) 0 , we observe the identity:

( y) (_nb_nH_n( y, a ))ey ( f ( a )a )for a C and y, and the last result is obtained by derivation. Denote that, in combinatorial theory, these polynomials H_n( y, a )are known as Bell‟s polynomials. (Notice: H_n( y ) are polynomials inyeven if f is only analytical).

3- Remarks

- We can localise the results on a p- dimensional manifold taking a volume on the manifold instead of the Lebesgue-Steiltjes measure on Rd

.

- A diffeomorphism gin C of infinite class defined by x g (u ) induce

fg(u ) g 1 of o g (u ), hence _f(y ) becomes g1 of og( y ). - If a _kp _

Fix( f( k )) is a point of a cycle of orderk for p 1, ..., k 1 and _kp1 _{ f (} k p_{), then} ( y ) (_nb_nH_n( y,_kp))ey (k p1_ k p ) at _kp_{. So:}



_{( y )}



k _{ } p1 pk (nbnHn( y,k p )). 4- Corollary 1

Polynomials H_n( y, a ) have some recurrent matrix equations (see Constantine

 

6 ): - H_n1( y, a ) H_n( y, a ) /a H_n( y, a )yf (a ) /a

- H_n( y, a ) /y  _kk_₀n

C_nkH_nk( y, a )kf (a ) /ak.

- At a 0, the leading term of H_n( y ) is (y)n and polynomial n

H_n( y)can be seen like « unitary » in y.

V- Relation between solutions ( y )of (LPF) and( y )of (PF)

Let ( y )be the solution of (LPF) () 0 and ( y ) the transform of the solution of (PF). For conveniences, we suppose the random variable d-dimensional.

Proposition2

Under H0, ( y, a ) 2_{( y, a ) /ya  ( y( y, a )) /y}

Let ( y ) be a solution of _f() 0. In the expansion of ( y ), b_n are fixed and we can construct ( y, a ) and _f(_a) 0. But, according to proposition 1, if ( y, a ) is solution of (PF), _f(_a) 0 will be an identity for all y and a. All derivatives must vanish. In fine: (2_

( y, a ) /ya )0.

( y, a )2_{( y, a ) /ya. As( y, a ) e}ya

 eys_{dP (s ), we have the result.}

It is not difficult to rewrite the result for a random vector p-dimensional p  don a manifold. We can verify the arguments are true for every f_. In the following, no resonance gets very important.

VI- Analysis of the solution ( y ) of LPF

( y ) is now the Laplace-Steiltjes transform of a distribution with support in C and solution of LPF. We study the expansion _n( y ) limited to the order nand define the resolving deviation:

1- Definition 4: resolving deviation and the ideal E_n We call resolving deviation: en( y, a ) n

(eya eyf ( a )) /an. At the fixed point a 0,

en_{( y)e}n_{( y, 0 )y}n_H

n( y) is a polynomial of degree n.

2- Proposition 3

If (1 n_{), under H0 and }

f(n) 0 , we can construct dpolynomials n on Z (En)and

lim

n Z (En)



yf()0



.

For all analytic function ( y ), we can form the function _f()_f  where  _f( y )is obtained in substituting yn _{by H}

n( y ) in the analytic expansion of  . We perturb a solution  of f() 0 by an arbitrary polynomial _n( y ) 

k0 kn

_kyk. The coefficient of the leading term of _n( y ) is:_n  0.

The perturbed function  n( y) ( y)_n( y) becomes under f : n

f f   k0 kn kHk( y ), yet convergent, and _f()  k0 kn _kek( y). As _n  0 and 1 n

, the coefficient of the leading term of_f( n) is the one of the leading term ofen( y) if yis non-null: _n(1n

) 0. We cannot vanish this coefficient. We must seek a polynomial with a greater degree, and so on.

By induction, to vanish _f(), we must take en( y)null when n for all non-null y. Now, we take a very large fixed n to construct n by this way. Suppose yZ (e

n

) for this n Nd . Can we have _n( y ) _pnc_pyp as solution of _f(_n) 

p0 pn

c_pep( y ) 0? Response is yes: We find dsolutions, but under some other conditions.

Here, polynomial en( y )  k0 kn

k , ny

k _{is known, but vanished if yZ (e}n_{). }

f(n)has a smaller degree than n: we have d candidates _en1l ( y ) for the new leading term. Choosing one

en1l ( y ), we must identify the coefficientsc_p to compute __ln. The term of degree _n1_l is cancelled with

cn1l : (1 

n1l )c

n1l  cnn1l, n  0. Under the no-resonance hypothesis, notice that 1 

n1l  0 for 1, ..., d . But cn is arbitrary. Now, the polynomial _cn(n1le

n1l ( y ) (1 n1l )en( y )) is known.

By successive linear identifications, we shorten the degree of unknown coefficients of __ln . If we denote coefficients _c_k  c_{k , l}c_n, we have

Pn1l ( y ) cn(knck , le

k_{( y )) 0, then}

f(ln) 0. That is true for _l and we have dsolutions __ln. We take one denoted n( y ).

We examine n( y ) with a translation a of the origin: n( y ) becomes n( y, a ). As our previous computation for( y ), we have also for all a : _f(_n( y, a )) _n( y )eya _

fn( y, a ) 0. We expend _f : _f(_n( y, a )) _knc_kek_{( y, a )). Then, we can derive }

f to obtain at a  0: _f(_n( y, a )) /a_{k a0}  _mnc_mem1k_{( y )} ₀ for _k _{1, .., d} . A solution of this equation is obtained

in taking the polynomial en1k( y) 0. Hence, if y Z (en) k Z (e

n1_k

) , n( y )is solution of the both equations _f(_n) 0and_f(_n1

k) 0. Thence, thedsolutions ln are valid.

3-Some inequalities

Let  be the solution of _f() 0 and _n( y ) the expansion of order n. We have:

n( y)fn( y) 2 d yD n1 eyD / (n1)!, ( y)n( y, a ) (e ya _ 1)eyD  Dyn1eyD / (n1)!

The general term of this series is less than yD n / n !. The remaining Lagrange-Taylor‟s rest of order n of eya is less than ( y )n( y ) d yD n1 eyD / (n1)!. As f (a )  D , we have also: f( y)fn( y) d yD n1 eyD / (n1)! for  a  D , and ( y)n( y, a ) (e ya _ 1)eyD  Dyn1eyD / (n1)!.

Now, for fixed nand _n1_l , we study the solution _n, with _n(0 )1 at y 0. 4- Lemma 2

For large n, __lnconverge to  when n . Exist a number m such as (n1) m _E

n.

We take y



Z (E_n) y k (n1)



. As __ln is a Laplace-Steiltjes transform of a measure in C, we have

ln( y)lfn( y) d yD n1

eyD

/ (n1)!, and then _ln converge to  when n . We can identify lnwith n( y ) n. In such case, the coefficient of the leading term ofn( y )and the one of fn( y ) are close to 0 when n . But, _f(_n) en( y ), and the leading term of en_{( y) is 1 }n

and tend toward 1 or infinite. Either n 1has same zeros than Z (En)or y 0. Imbedding the problem in the algebraically closed field of the complex numbers, the Hilbert‟s theorem of zeros (nullstellensatz) says

(_n1)m _E n.

Now, taking y complex, we give a first characterisation of the distribution on I_n as in Lucaks

 

17 or Wintner.

5- Proposition 4

With a probability 1, all real point of E_n belongs to the support of the distribution L1(_n) and tends asymptotically to the support of L1(). The distributionp_n(a ) L1(_n), defined foryZ (E_n) 0 ,

is constant on the real algebraic affine manifolfs (connected component) of E_n, null elsewhere. Let yE_n  Cd_and

ln( y )1. Consider the algebra quotientA to define the affine manifold. If

_n( y)  eyadP_n(a )1. But, by definition of the Laplace transformation, we have the same result for the real algebraic affine manifold  evadP_n(a )1. By difference: _ eva

(cos wa1)dP_n(a )0. As

cos w a1is negative continue, P_n must be purely discontinue at cos w a1 and w a 2 k ; k Z. We have P_n(a ) _kZp_kY (w a2 k),kZ : a periodic lattice distribution. HereY is the Heaviside‟s function for the discrete distribution p_kwith sum _kp_k 1.

 verifies  eyadP (a )  eyf( a )_{dP (a )}and for

n:  e ya

dP_n(a )  eyf( a )_dP

n(a ) whatever

yZ (E_n) when n . But _n 1onZ (E_n). So w f_(a ) 2 k ; k Z . Consequently,

w ( f_(a ) a ) 0 forw Im (E_n). Thena  0 and p₀ 1forvR e(E_n).

6- Remarks

- In resonant case, exist n_R 



r Nd _{1 }r



_{where the leading term ofH}

n( y )isy n_{. To} construct n, we can identify the cpfor all nR . But we shall see more interesting methods. - If the measure dP (a ) is invariant, on have the identity  eya_{dP (a )  e}yf( p )( a )_{dP (a ) for pN .}

_p( y)_p( y)( y) 0. Then, the resolving deviation f( p )_{will asymptotically vanish.} 7- Lemma 3

If n with n R e s, the zeros of en_{( y)are very close to theses of H}

n( y ). Conversely, if n withn R e s and n , the solution is y 0. We can say that the ideal E_n

 

0 is asymptotically generated by the zeros of polynomials H_n( y ) and _H_{n1l( y ) }1, 2, .., d.

If nR e s and n , we can writeen_{( y) as e}n_{( y) }n_(n_yn _n_H

n( y)). But the leading term of n_H

n( y) is 1and 

n_yn _{ 0. By the continuity of the roots, we can say that the zeros of} en_{( y)are very close to theses of H}

n( y ). Now, If n R e s



and n , en_{( y)is has its leading} term of yn_{, and e}n_{( y)is very close toy}n_{. Solution is y 0}

8-Normalisation of E_n

The number of zeros can increase with n. For instance, if k R, suppose that we have yf (ka ) such yf (ka )  when k  for a fixed couple ( y, a ) , then, by Rolle‟s theorem dn_eyf ( ka ) _{/ dk}n _{ }n_eyf ( ka ) _/an _an _has

nzeros. We normalise these zeros (think to the Hermitian polynomials) to obtain a continuum. Let y n. s (_y_l  s_ln_l ): E_n( y ) becomes Enn  E_n(n. s ). 9- Lemma 4

n( s ) 1 and Z (Enn) is bounded.

- If (_n(n.s )1)Enn, n(ks ) must converge on Z (Enn). But logarithm nis convex. For every coordinate_s_ln_l : _logn( slnl)logn( sl(nl 1)) / 2logn( sl(nl 1)) / 2. Hence:

logn(slnl)logn(sl(nl 1)) logn(sl(nl 1))logn(slnl).

n(slnl))n(sl)n((nl 1)sl) (n(sl))

nl and 

n(n. s )n(s ) n

converge only if n( s ) 1.

VII- Random d-dimensional variables

Let a d-dimensional densityq ( s )such as ( y )is a solution of _f() 0. Let ( y ) solution of the transform of PF for f with densityp ( s ). LetC is the edge of the distribution.

1- Proposition 5

If q ( s ) 0 on C, we have p(s ) (-s )q(s ) /s From Proposition 2, ( y, a ) 2_

( y, a ) /ya. Then, asq ( s ) 0 on C , we have the result. 2-Hypothesis H1 on real zeros of H_n( y) n

H1- Let s_n S_nbe such asH_n(n. s_n) 0. The repartition limit of s_n when n   has a density q ( s ) with respect to the d-dimensional Lebesgue-Steiltjes measure: in a volume dy ndds of Rd we have asymptotically dy nd_{q(s)ds zeros}

s_n. Then, nd points y_nverify H_n( y_n) 0. Let _

n( y ) 1

ndne yyn

3- Proposition 6

Under H0 and H1, if it is not masked, the Perron Frobénius’s density is asymptotically p(a ) 1 d (a_)d q(a ) /a₁..a_..a_d Let t  y / n. Under H0, H1 _n(t. n ) 1 nd ne tsn_nd_q(s n)dsn  (t )  e ts q(s )ds. VIII- Applications

1- The logistic f  (aa2_{)in dimension 1. if}

4  1 , f apply the compact set C=



f ( / 4 ), / 4



in C. Its resolving deviation is :en_f( y ) _a0  yn  H_n( y ) where

H_n( y ) H_n( y / 2 )( 2y )n _{with the Hermitian polynomials of order}

n H_n. Hence, the distribution of the zeros of H_n, denoted as t , is asymptotically the semi-circular low of Wigner: q(t )dt  k 1t2dt . q(1) q(1) 0 . Thencet  c y / with c   / 2 and  2 n 1has the semi-circular law. The distributionp₁(s )of the variable s  y / 2 n1 will verify: t  r sand: p₁(s)ds  (dq(t ) / dt )dt  kr / 2 s s(1



r2s)



ds . Then p(s) sp₁(s ) . The density of the invariant measure is a beta law(1/2,1/2) : p(s) k '/ s(1r2s) , obtained by Kawamura with the Cunts-Krieger‟s algebra, which generalise the well-known result of Ulam and Von Neumann. But this distribution will be masked if exist attractive cycles according with the values of .

2- First approach of the Julia’s iteration

We prevent the readers that this example doesn‟t pretend characterise the Julia‟s sets. Many important works have been made. Here, we don‟t worry with cycles or resonances, or relations of domination. Let f (z ) z2 _{1 / 4 }2_{be the iteration de Julia in C}

 

_{14 . It has 2 fixed points: a}

0 1 / 2and

a '₀ 1 / 2 . We denote here with « bold » characters the vectors of R2 for the complex numbers a  (a, b ),   (,)and a'₀  (a '₀, b '₀) 1 / 2 .

Consider the function f ( z ) at the fixed point a₀; we can write with z  a + a₀:

f (a2b22 a₀a2b₀b, 2 ab2 a₀b2b₀a ). We can suppose that it applies a compact C in C by convergent reciprocal iterations. We have the function: yf ( a )=2 yL ( a ) yQ ( a ).

Let ₀  a2 0 b

2

0 ,  x

2_{ y}2 _{,  ( x ) / 2 and   (  x ) / 2 . We denote the}

orthogonal matrices: ₀O₀  a0 b0 b₀ a₀         _O y x y y x       O          

- The linear part is yL (a ) x(a₀ab₀b ) y(a₀bb₀a ) ₀yO₀a where O₀ is orthogonal. We suppose that the eigenvalues of the linear part a₀ and its conjugated a₀ have a module greater than 1. - The quadratic part is orthogonal yQ (a ) xa2 _{ xb}2 _{ 2 yab aO}

ya. Oy has a characteristic equation x2 _y2_{ }2 _{0 with 2 eigenvalues, one positive , and the other negative  .}

If t

Odesign the transposed matrix of O, denoting y O sanda  Ou with u = (u, v) and s = (r, s), then ya su. yf ( a )=2₀ s tOO₀Ou +u2 v2 2 r (a₀ub₀v)2 s(a₀vb₀u ) +u2v2 Hence : yf (a )(2 ra₀2tb₀)u u2 (2ta₀2 rb₀)vv2. In the orthogonal eigenspaces, the application splits into two independent iterations: (2 ra₀2 sb₀)u u2 and (2ta₀2 rb₀)v v2. AsO is orthogonal  r2 _{ s}2 _.

The resolving deviation gives two Hermitian polynomials: H_n((2 ra₀  2 sb₀) / i 2)(i 2)n andH_n((2 sa₀  2 rb₀) / 2)( 2)n_{. The first is always positive. The zeros of the second follow} asymptotically a semi-circular Law. Thence, (2 s₁a₀  2 r₁b₀) / 2 (with r₁  r / 2 n1 ands₁  s / 2 n1) follows a semi-circular Law. And s₁ 2 n1 sgives the distribution.

We change coordinates fora'₀ without modify the distribution. But, they may be masked (dominated) locally by every cycles or distributions of f_.

Chapter 2: Approximation of the zeros I- Notations, rappels and hypotheses

1-Leading idea

We search the asymptotic distribution of the real zeros of H_n( y ) when y  with n. As Plancherel and Rotach

 

20 did for the Hermitian polynomials, we represent H_n( y )in the field of the complex numbers with the Cauchy‟s d-dimensional integral :



H_n1( y ) n1_eyf ( a ) _/an1

a0  c

—



_

eyf ( a )_da

an whereis a closed poly-disk around the fixed point 0 of f , a Cd . Note

n1 (n₁1, n₂ 1, ...n_d 1). c is a finite non-null constant.

The steepest descent‟s method, due to Riemann-Debbie, gives us an approximation of this integral wheny . If yis a zero of H_n1( y ), we will hope that the approximation will give this zero. The method needs many assumptions. Ifyf (a )is polynomial in a, we can see Pham

 

19 or Delabaere

7

 

for a good formulation, but only if the Hessian of yf (a ) is definite. Definition 1

We call Plancherel-Rotach’s function (PRF): (a ) yf (a ) n log a where log a is the main determination of the complex logarithm. Then _H_n1( y) c—_ e( a )_da

2-Basic notations

yRd tends toward the infinite with n Ndand we writeyn. s where _y_l  n_ls_l . All the_n_l   . We shall see soon that all the _n_l must be equal. Now we can write (a ) n. (sf (a )log a )

3-Recalls on the steepest descent’s method and assumptions

The approximation of H_n1( y ) by the steepest descent‟s method is the finite sum of contributions

_cQ_c(a ) of all the critical points of (a ). Among these contributions, some of them are exponentially negligible. So, we keep only that one of which the real part is the most important. The approximation can be written (see

 

7 ):

H_n1(ns )Q_n(a ) c ' e( a )

(1g(a, n, s ) / n ')

Where the functiong is bounded when n ' depends onn and n '  with n. c ' is never null. A point ais critical point of (a ) if we have (a ) /a  0. That can be written aln. sf (a ) /al  nl  0. Then a(a ) /a is polynomial with the same degree thanf .

4- Hypotheses: general position

The critical point a giving the maximum of e( a )

is assumed be in general position

 

7 :

Definition 2

H1- For all fixed s, it is isolated, unique, (except its complex conjugate) with finite distance. It is called Morse‘s point, if it is not degenerated.

H2- A sufficient condition to obtain this maximum is that the Hessian (which is Hermitian) of  (a ) is definite negative at a. Critical points not degenerated are isolated.

Unfortunately, in a lot of problems, the Hessian is not definite negative (see V). H3- n. sf tends toward  when n in order to define the Laplace’s integral. But, if the iteration is in a compact set, this hypothesis will be redundant.

II- Asymptotic equivalence relation of the real zeros of H_n1( y ) when n 

We first identify the equation of the zeros of the approximation, then we study under what conditions the distribution of the approximation is asymptotically very near of the zeros of H_n( y ). The variety of the situations is so rich that we examine here only the most common ones.

As we are only interesting by the real zeros of H_n1( y ), we can neglect in the representation Q (a ) all the factors  0: that may be functions, constants, etc. Notice that g / n ' 0 when n  and c ' 0 in Q (a )c ' e( a )(1g(a, n. s,1 / n ') / n '). Then e( a )is the one and only one factor making

Q (a ) null. We write this equivalence H_n1( y )

0e

( a )_{. Then}

n ' has no importance when n . 1-Lemma 1

Almost everywhere (when the Hessian is not degenerated): H_n1( y )

0Jn( y ) sin(Im(a )) - First, if H_n1( y ) 0, all contributions e( a )

must be null at the critical point a  a ( y ). As the polynomial a /a  0 has real coefficients, if ais a complex solution, the conjugate a is solution.

But the contour gives a negative contribution for a, and the sum is e

( a ) _euuuuu( a )r



0Jn( y).

- Conversely, if J_n(a ) 0, we can imagine that another contribution Q₁(a ) takes the place ofQ. Hence, it can existy and Q₁ such as H_n1( y)  Q₁(a )  0when Q (a ) 0. But, it is impossible

because the two contributions are continuous:

Let a ' be a point in a small neighbourhood of awhere Q is the dominating contribution and suppose that exists Q₁ dominated. At a ', Q₁(a ')Q (a ')  / 3 because Q is dominating. Then

H_n1( y ') Q (a '). As (a,a ') determines a unique couple (y,y ') : H_n1( y )  H_n1( y ) H_n1( y ')  H_n1( y '). In this inequality, we can substitute H_n1by Q or

Q₁each time they are dominating: H_n1( y)  Q₁(a )Q (a ')  Q (a ')  Q₁(a )Q (a ')  / 3, but

Q1(a )Q (a ')  Q1(a )Q1(a ')  Q1(a ')Q (a ') . The continuity of Q1 induces the contradiction.

2- Come back to the ideal E_n

Recall that Z (E_n) be the set of common zeros of the polynomials en( y)0and en1l( y)0,1, ...d . Enis ideal of these polynomials. We have seen that, if n with n R e s, the zeros of en( y)are very close to those of H_n( y ). But, we can study H_n( y )even though nR e s. Let Z ( H_n) the set of common zeros of H_n( y )instead of en_{( y).}

Let Z (J_n) be the set of the common zeros of J_n( y )for n-1 and _n 11_l , _1, 2, .., d.

III – Main theorem

Let pbe the fixed number of complex coordinates of a critical point aof PRF. 1- Theorem 2

If the critical pointaZ ( J_n)is in general position, then:

1 - (a ) depends only on one common index of derivation_n_l  n. We can write_y_l  ns_l and the PRF is n(a ) n (sf (a ) log a)with _a  _la_l . The coordinates are defined by the system :

alsf (a ) /al 1  p1, 2, .., d . As f is polynomial, solutions are algebraic affine manifolds.

Moreover, the resolving deviationen_{( y) must be computed only for common derivation n. Thus,} let    f be the determinant of the linear part of f . We have

If  1 the iteration converges to 0;

If  1 the iteration, if it is not masked, can converge to a distribution defined byZ (J_n)

2 - The imaginary parts of the complex coordinates of asatisfy a uniform code of the real zeros ofH_n( y ), which are algebraic affine manifolds for all complex coordinates _1, 2, .., p. These

sl Im fl(a )l  kl / nwith the codekl (1, ..., n 1). Between two zeros (manifolds Mf ) at time

n1there is one zero at timen: then we have a Rolle’s foliation. When_n_l  

if

l  lim

n kl / n for  1, .., p , then  is asymptotically a random uniform vector on the unit cube

0,1

 

p

.Then, The d  pdimensional many folds are indexed by a random vector of

 

0,1 p 

 

0 and define the random equation __l  s_l Im f_l(a )_{l }1, 2, .., p

Remark : It seems possible to extend the results to the rational fractions and analytical functions. 2- Proof: Analysis of the zeros ofJ_n( y )

Letp  0be the fixed number of complex coordinates of the critical point. We must find the zeros of sin(Im(a )). Hence n. (Im sf (a ))) k . -Proof of point 1

Suppose we found a complex critical point a ( y ) such as J_n( y ) 0. As sin k Im(a ) 0for all integerk N , ais invariant if we multiply y by any integer k. Hence, k(a )gives the same critical point athan k '(a ). We can write _k(a ) k '_ln_l(s_l f_l(a ) log a_l)

But, if yZ ( J_n), all change of _n_l 1 in _n_l gives the same point y for all _:

then_a_lyf (a ) /a_l  n_l  0becomes _(a_l  a_l)yf (a a ) /a_l  n_l 1 0. Dividing y by a common_n m in(n_l), we obtain by difference





(al( y / n )f (a ) /al) /a



a 1 / nand a c / n. Then, the integrand is quite constant:

k(a ) k 'l(nl 1)(sl fl(a ) log al) k ' (a a ) k 'l(sl fl(a a )log al  al) (k  k ')(a ) k 'l(sl fl(a ) log al)  C te / n

Then, with k k '1all the_n_l must be equal: _n_l  nfor_1, 2, .., d. -Lemma 2

If _k_{c l}  k_l / n , the zeros of J_n(a ) must verify the condition

lnl(sl Im fl(a )l kcl)0

s Im f (a ) contains only sinus, the arcs of which are linear combinations of multiple integers of  ,

then s Im f (a ) is periodic: Im(a( 2 k '))Im(a())2 k ' . Hence, all

N Im(a (2 k ')) k  0 are solutions. Writing

k  lnkc_l where kc l  kl / nl and yc l  sln, the solutions can be written as

nl(sl Im fl(a )  kcl) 0 where p is the number of complex

coordinates. Let _k_l runs over 1, 2,.., n1, then the solution compounds n1 distinct points for each complex coordinates.

In particular,   0 such asIm(a (0 )) 0 gives the fixed point and a non-null solution l(sl Im fl(a ) kl)0. But we don‟t know if there is other 0 (,)

pand under what

conditions such a solution exists. AsaZ ( J_n), we examine now a complex coordinate_a_l , when nl 1 becomes nl , the others k l remaining constant.

- Proof of point 2

If yZ ( J_n), theny remains invariant when _n_l 1 becomes _n_l . The equation of the complex _a_l ,

alyf (a ) /al  nl becomes alyf (a ) /al  nl 1 0, the other equations don‟t move. The critical point a becomes a a. If we normalise the equations by the common n, we have again : 



(alyf (a ) /al) /a



a  and a c. Now, the equation

_jn_j(s_j Im f_j(a )_j  k_c

j) 0 becomes :jn 'j(sjIm fj(a ')'j k 'cj) 0

k '_cj  k_cj if _j  l and _k '_{c l}  n_lk_cl / (n_l 1)for the range of _k_{c l} + a new point : _n_l / (n_l 1) By difference we obtain;

S (aa ) /aasl Im fl(a a ) k 'cl  k 'cl   0

where

nlkcl is the point when the range change from (1, n1)to (1, nl)

Examine the quantityS (a a ) /aa. As S (a ) /a 0at the critical point a, then S (a a ) /a  2_{S (a a ) /a}2 _{a . We obtain:}

Im S (aa ) /aa c 'a c ' c / n For all

nlkcl running over(1, n), sl Im fl(a )l  kc l  0 whenn .

This is true for all complex coordinates and , when n is large, we have the result.

We obtain a uniform code by k_c of coordinates of real zeros ofeC ( a )_{sin S (a ). Note that a n1 -th} zero is between two n-th zeros. If 0 is the unique solution of Im (sf (a ))  0, the order

n1manifolds can‟t intersect order n manifolds. Then we have a Rolle‟s foliation Suppose now that n 

We have already

s limn y/ n,_ limn k for 1, .., p ;

Let  a random vector with uniform repartition on the unit cube equation_s_Im f_(a )_ k_ / n_) tend to_s_Im f_(a )__

3- Examples

- One-dimensional case

a_c depends only on s y / n :a_cf (a_c) /a1 / s 0 and the equation of zeros is (s Im f (ac)) /  . thenq(s )ds  Pr ob 1 zéro



(s, s ds )



 Im f (ac)ds /

- Case of a cycle

Let a cycle : f( p )_{(0 )0with f}( k )_{(0 )}

k  0 pour k 1, 2, .., p1. A point k of this cycle is fixed point for the application f( p )_{as f}( np )_(

k)k. If exists a critical point ack of (a )sf( p )_{(a )log a we can have a probability distribution. But this distribution will move at} each iteration and be cyclical as in a Markov process.

- Hermitian case : f  aa2 / 2 :  cH_n1() cn1_e(aa2/ 2 ) _/an1 a0  — e(aa2/ 2 )_da an The PR function

 

11 gives for t   / 4 n  cos : H_n1()

0 sin



 sin(2) / 2



when   under the condition 2

 4 n  0 . The density of zeros of t is the semi-circular low : p(t )dt  (1 cos(2))d / (2 /) 1t2dt .

Demonstration that seems more explicative than the theorem of Sturm. - r-Hermitian case: f  a ar _{/ r .}

We write the PR function (z ) zzr

/ rt log z and the critical point is solution of the

Lambert‟s

 

9 equation of z '(z )zzr t 0.

If r  3 and if we use the Fourier‟s transform, the characteristic function is the Airy‟s function. - Quadratic case in Rd

: f (a ) aQ (a ) : Q (a ) is quadratic. We can diagonalize yQ with

transformation orthogonal aO_yu . In this basis, the volume is invariant, thenn log a  n log u. The function (a ) is

(u ) yOyul  (lul

2 _{n log u}

l), the eigenvalues l are solutions of yQ (a )I  0. The semi circular low is generated by each coordinate having a __l  0. Moreover, the distributions are independent in orthogonal eigenspaces. The equations __l  0 give zones where the number p of complex coordinates remains constant.