Dynamics of Cremona transformations

Arnaud Beauville

Description

Dynamics of Cremona transformations Arnaud Beauville Universite de Nice Pisa, October 2008 Arnaud Beauville Dynamics of Cremona transformations

• lim n?∞


• cremona transformations


• universite de nice

Sujets

• Beauville
• Universidad de Niza Sophia Antipolis

Informations

• Publié par: pefav
• Nombre de lectures: 36
• Langue: English

Extrait

Dynamics

of

Cremona

Arnaud

transformations

Beauville

Universite´deNice

Pisa,

October

Arnaud Beauville

2008

Dynamics of Cremona transformations

The dynamical degree

f∈Bir(P 2 )f= (f 0 f 1 f 2 ) homogeneous polynomials of degreed

Arnaud Beauville

Dynamics of Cremona transformations

The dynamical degree

f∈Bir(P 2 )f= (f 0 f 1 f 2 ) homogeneous polynomials of degreed

d:= deg(f) :

f

∗ :H 2 (P 2 )→H 2 (P 2 ) is multiplication byd.

Arnaud Beauville

Dynamics of Cremona transformations

The dynamical degree

f∈Bir(P 2 )

f= (f 0 f 1 f 2 ) homogeneous polynomials of degreed

d:= deg(f) :

f ∗ :H 2 (P 2 )→H 2 (P 2  multiplication by) isd.

Forfg∈Bir(P 2 ), deg(fg)≤deg(f) deg(g)

Arnaud Beauville

=⇒

Dynamics of Cremona transformations

The dynamical degree

f∈Bir(P 2 )f= (f 0 f 1 f 2

d:= deg(f) :

) homogeneous polynomials of degreed

f ∗ :H 2 (P 2 )→H 2 (P 2  multiplication by) isd.

Forfg∈Bir(P 2 ), deg(fg)≤deg(f) deg(g)

=⇒

lim (degf n ) 1 n =λ(f) :=dynamical degreeoff n→∞

Arnaud Beauville

Dynamics of Cremona transformations

The dynamical degree

f∈Bir(P 2 )f= (f 0 f 1 f 2

d:= deg(f) :

) homogeneous polynomials of degreed

f ∗ :H 2 (P 2 )→H 2 (P 2  multiplication by) isd.

Forfg∈Bir(P 2 ), deg(fg)≤deg(f) deg(g)

=⇒

lim (degf n ) 1 (f) :=dynamical degreeoff n =λ n→∞

1≤λ(f)≤deg(f);λ(f) = deg(f) iffis generic, λ(f) = 1 iffperiodic orf∈PGL(3).

Arnaud Beauville

Dynamics of Cremona transformations

Theorem

1

f

(Diller-Favre)

∈Bir(P

2 ):

Arnaud Beauville

Dynamics of Cremona transformations

Diller-Favre 2

2

λ(f) is an algebraic integer; its conjugatesµ i have|µ i | ≤1.

Arnaud Beauville

Dynamics of Cremona transformations