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Dynamics of Cremona transformations

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84 pages
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


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Dynamics
of
Cremona
Arnaud
transformations
Beauville
Universite´deNice
Pisa,
October
Arnaud Beauville
2008
Dynamics of Cremona transformations
The dynamical degree
fBir(P2)f= (f0f1f2) homogeneous polynomials of degreed
Arnaud Beauville
Dynamics of Cremona transformations
The dynamical degree
fBir(P2)f= (f0f1f2) homogeneous polynomials of degreed
d:= deg(f) :
f
:H2(P2)H2(P2) is multiplication byd.
Arnaud Beauville
Dynamics of Cremona transformations
The dynamical degree
fBir(P2)
f= (f0f1f2) homogeneous polynomials of degreed
d:= deg(f) :
f:H2(P2)H2(P2 multiplication by) isd.
ForfgBir(P2), deg(fg)deg(f) deg(g)
Arnaud Beauville
=
Dynamics of Cremona transformations
The dynamical degree
fBir(P2)f= (f0f1f2
d:= deg(f) :
) homogeneous polynomials of degreed
f:H2(P2)H2(P2 multiplication by) isd.
ForfgBir(P2), deg(fg)deg(f) deg(g)
=
lim (degfn)1n=λ(f) :=dynamical degreeoff n→∞
Arnaud Beauville
Dynamics of Cremona transformations
The dynamical degree
fBir(P2)f= (f0f1f2
d:= deg(f) :
) homogeneous polynomials of degreed
f:H2(P2)H2(P2 multiplication by) isd.
ForfgBir(P2), deg(fg)deg(f) deg(g)
=
lim (degfn)1(f) :=dynamical degreeoff n=λ n→∞
1λ(f)deg(f);λ(f) = deg(f) iffis generic, λ(f) = 1 iffperiodic orfPGL(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 conjugatesµihave|µi| ≤1.
Arnaud Beauville
Dynamics of Cremona transformations