Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mccme.ru/hilbert16/book2.ps Äàòà èçìåíåíèÿ: Tue Dec 23 15:16:55 2003 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 22:03:35 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: earth orbit

Hilbert's
Sixteenth
and
Related
Problems in
Dynamics,
Geometry
and
Analysis
In
honor
of
the
60-th
anniversary
of
Yulij
Sergeevich
Ilyashenko
The
Independent
University
of
Moscow
December
26-29,
2003
Multipliers
of
homoclinic
orbits
V.Afraimovich IICO{UASLP,
Mexico.
It
is
supposed
to
describe
in
this
talk
new
dierential
invariants
of
dy-
namical
systems,
the
multipliers
of
homoclinic
orbits.
It
is
one
of
the
most
basic
results
in
smooth
dynamical
systems
that
a
transversal
crossing
of
the
stable
and
unstable
manifolds
of
a
hyperbolic
xed
point
produces
a
hyperbolic
"horseshoe"
and
associated
chaotic
dy-
namics.
This
phenomenon
which
was
discovered
by
Poincare
has
been
dis-
cussed
in
many
places.
In
this
treatement
the
role
of
the
eigenvalues
of
the
linearization
of
the
map
at
the
xed
point
is
well
understood.
These
eigen-
values
control
the
rates
of
decrease
of
the
size
of
the
blocks
which
form
the
hyperbolic
set.
These
rates
in
turn
in uence
the
numerical
invariants
such
as
Hausdor
dimension,
box
dimension
and
thickness
of
the
hyperbolic
sets.
However,
in
previous
works
there
always
appear
certain
"constants"
which
remain
unexplained.
In
this
talk
we
show
that
these
constants
are
in
fact
determined
by
two
numerical
invariants
of
the
homoclinic
orbit.
They
are
indeed
invariants
with
respect
to
smooth
conjugacies.
We
call
these
invari-
ants
the
stable
and
unstable
multipliers
of
the
homoclinic.
We
show
that
these
multipliers
appear
in
the
asymptotic
formulae
for
the
thickness
and
Hausdor
dimension
of
the
invariant
sets
in
the
vicinity
of
the
homoclinic
as
the
size
of
the
vicinity
tends
to
0.
Geometrically,
the
multipliers
we
introduced
capture
the
expansion
and
contraction
eects
which
cannot
be
attributed
to
the
local
ones.To
obtain
these
multipliers
we
essentially
cancel
out
the
eects
of
linear
parts.
A
nice
feature
of
te
multipliers
is
that
they
may
be
calculateed
by
straight-
forward
computations
along
the
homoclinic
orbit.
Similar
quantities
can
be
introduced
for
heteroclinic
transversal
orbits
and
for
heteroclinic
contours.
All
results
of
the
talk
were
obtained
together
with
Todd
Young.
1

Bibliography [1]
V.S.
Afraimovich
and
T.R.Young,
Multipliers
of
homoclinic
orbits
on
surfaces
and
characteristics
of
associated
invariant
sets,
Discrete
and
Continuous
Dynamical
Systems,
vol.3
(2000),
691-704
[2]
V.S.
Afraimovich
and
Todd
Young,
Multipliers
of
heteroclinic
cycles,
Far
East
J.Dyn.
Syst.,
vol.2
(2000),
41-51.
2
Kepler's
cubes
and
nite
Lobachevsky
geometry V.
I.
Arnold
1
Dedicated
to
Yu.S.Iliashenko
Kepler
had
inscribed
5
cubes
into
the
dodecaedron
in
his
"Harmonia
Mundi"
(to
describe
the
planets
orbits
greate
axis
length
distribution).
The
12
edges
of
a
Kepler's
cube
are
some
diagonals
of
the
12
faces
of
the
dodecaedron
(one
diagonal
on
each
face).
This
cubes
are
related
to
some
5
Hamilton
subgroups
of
the
nite
modu-
lar
group
G
=
SL(2;Z5)
(of
second
order
matrices
of
determinant
1,
whose
elements
are
residues
modulo
5).
The
vertices
ot
the
Kepler`s
cubes
are
the
elements
of
order
3
in
G5;
being
also
the
vertices
of
a
dodecaedron,
forming
the
mod
5
version
of
the
Lobachevsky
plane.
Trying
to
extend
this
theory,
developed
in
[1],
to
the
more
general
mod
p
version
of
Lobachevsky
geometry
and
to
nite
modular
groups
Gp
was
led
to
the
following
theory
(p
being
a
prime
number).
Denition
1.
The
Lobachevsky
plane
is
the
group
of
the
line
aône
transformations,
u!au+b
(equipped
with
the
left-invariant
metric),
veri-
fying
the
orientation
positivity
condition,
a
=
c
2
;
c
6=
0:
This
denition
provides
the
usual
Lobachevsky
geomemtry
of
the
half-
plane
a
>
0
in
the
real
line
case.
For
the
nite
eld
Zp
case
one
should
replace
the
positivity
condition
by
its
p-version:
a
should
be
a
nontrivial
quadratic
residue,
everything
else
remaining
similar
to
the
ordinary
(real
geometry)
case.
The
replacement
of
the
calculus
inequalities
by
the
quadratic
residues
condition
might
be
useful
in
other
instances,
as
I
had
used
it
for
the
com-
lexication
goal
in
real
algebraic
geometry
[2].
Let
G
be
any
nite
group.
1
Partially
supported
by
RFBR,
grant
02-01-00655. 3

Denition
2.
The
squaring
graph
of
G
has
as
its
its
vertices
the
ele-
ments
of
G
and
its
(oriented)
edges
lead
from
each
element
g
to
g
2
:
Theorem
1.
The
squaring
graph
of
any
nite
group
consists
of
con-
nected
components,
each
of
these
components
being
one
cycle,
equipped
ho-
mogeneously
by
isomorphical
trees,
leading
to
its
point
(the
identity
element
being
a
cycle
of
length
1
and
its
component
being
a
rooted
tree).
Example
For
G5
the
graph
is
T1;1;30
G
(10A2)
G
(6A4
);
where
An
denotes
a
cycle
of
length
n
equipped
with
trees
of
1
edge
at
each
point
of
the
cycle,
T1;q1;:::
being
the
rooted
tree
with
q
r
vertices
at
the
oor
number
r. Denition
3.
A
Hamilton
group
is
a
group,
isomorphic
to
the
group
of
the
8
quaternionic
units,
f
1,
i,
j,
k
g.
Theorem
2.
The
above
tree
T
contains
5
Hamilton
subgroups.
Each
of
the
Kepler's
cubes
in
G5
consists
of
those
elements
of
order
3
in
G5;
which
preserve
one
of
the
5
Hamilton
subgroups,
while
acting
on
G5
by
conjugation. The
conjugations,
dened
by
this
subgroup's
elements,
preserve
the
cho-
sen
cube. Remark
Denoting
by
A
one
of
the
vertices
of
the
cube,
one
might
attribute
to
the
other
vertices
the
notations
A
!
;
!
2
f1;i;j;kg:
Trying
to
extend
the
Hamilton
subgroups
and
the
Kepler's
cubes
to
Gp
=
SL(2;Zp
=
Z=pZ);
I
had
obtained
the
following
results.
Theorem
3.
A
squarings
chain
of
length
r,
fAj
:
0

j

r;Aj+1
=
A
2 j
;Ar
=
1;Ar
1
6=
1g
exists
in
Gp
if
and
only
if
p

1(mod
2
r
):
Example
The
maximal
length
r
=
2
for
p
=
5
or
11,
r
=
4
for
p
=
17
or
47.
This
theory
is
related
to
the
matrix
version
of
the
small
Fermat
(and
Euler)
theorem:
tr(A
n
)

tr(A
n
'(n)
)
for
n
=
p
a
(and
for
any
order
matrices).
Here
'
denotes
the
Euler
function:
'(n)
is
the
number
of
the
residues,
relatively
prime
to
n:
Theorem
4.
The
elements
A
of
order
3
(A
3
=
1,
A
6=
1)
in
Gp
do
form
2
equally
large
conjugation
classes,
each
of
p(p

1)=2
elements
(provided
that
p
=
3c
+
1),
each
class
forming
the
set
of
points
either
of
a
mod
p-
Lobachevsky
plane,
or
of
its
de
Sitter
world.
The
de
Sitter
world
is
the
continuation
of
the
Klein
model
of
Lobachevsky
plane
model,
habitating
inside
a
disc
of
projective
plane.
The
continuation
4
represents
the
complementary
M
obius
band
of
the
disc,
as
it
is
explained
in
[3].
Example
For
p
=
7
the
Lobachevsky
geometry
of
the
Riemannian
sur-
face,
associated
to
the
squarings
graph
of
group
G7
is
described
in
[1]:
this
regular
polyhedron
of
genus
3
consists
of
24
heptagonal
faces,
being
the
p
=
7
brother
of
the
p
=
5
dodecahedron.
Bibliography [1]
Arnold
V.I.
Topology
and
statistics
of
formulae
of
arithmetics,
Russian
Mathematical
Surveys,
2003,
Vol.58,
4(352),pp.
3-28.
[2]
Arnold
V.I.
On
the
arrangements
of
ovals
of
real
plane
algebraic
curves,
on
involutions
of
four-dimensional
smooth
manifolds
and
on
the
arith-
metic
of
integer
quadratic
forms.
Functional
analysis
and
its
Applica-
tions,1971,
vol.5,pp
169-176.
[3]
Arnold
V.I.
Arithmatic
of
binary
quadratic
forms,
symmetry
of
their
continued
fractions
and
geometry
of
their
de
Sitter
world.
Bull.
Braz.
Math.
Soc.,
New
Series,
2003,
Vol.34(1),
pp.1-42.
5

Statistical
properties
of
the
Rauzy-Veech-Zorich
induction
map
on
the
space
of
interval
exchange
transformations Alexander
I.
Bufetov
Princeton
University
and
The
Independent
University
of
Moscow
An
induced
map
of
an
interval
exchange
transformation
on
a
smaller
interval
is
again
an
interval
exchange
transformation,
and
the
smaller
inter-
val
can
be
chosen
in
such
a
way
that
one
obtains
an
exchange
of
the
same
number
of
intervals.
One
therefore
obtains
a
renormalization
map
on
the
space
of
interval
exchange
transformations,
called
the
Rauzy-Veech-Zorich
induction.
Like
the
Gauss
map
on
the
unit
interval,
the
Rauzy-Veech-Zorich
induction
map
has
a
nite
absolutely
continuous
invariant
ergodic
measure
(a
theorem
of
Anton
Zorich);
the
square
of
the
map
is
exact
with
respect
to
that
measure.
Estimates
on
the
decay
of
correlations
are
obtained
for
the
Rauzy-Veech-Zorich
map
with
the
methods
of
Bunimovich{Sinai
and
of
Lai-Sang
Young.
6
Higher
order
Poincar
e-Pontryagin
functions
and
iterated
path
integrals
Lubomir
Gavrilov
Laboratoire
Emile
Picard,
CNRS
UMR
5580,
Universit e
Paul
Sabatier,
118,
route
de
Narbonne,
31062
Toulouse
Cedex,
France
Let
f;P;Q
2
R[x;
y]
be
real
polynomials
in
two
variables.
How
many
limit
cycles
the
perturbed
foliation
d
f
+"(Pdx+
Qdy)
=
0
(1)
can
have?
This
problem
is
usually
refered
to
as
the
weakened
16th
Hilbert
problem
(see
Hilbert
[6],
Arnold
[1,
p.313]).
Suppose
that
the
foliation
dened
on
the
real
plane
by
fd f
=
0g
has
a
period
annulus
A
=
[t2
(t)
formed
by
a
continuous
family
of
periodic
orbits
(t)

f
1
(t),
where



R
is
an
open
interval.
Take
a
segment
,
transversal
to
each
orbit
in
A
and
parameterized
by
t
=
fj

.
The
rst
return
map
P(t;
")
associated
to
this
period
annulus
and
to
(1)
is
analytic
in
t;
"
and
can
be
expressed
as
P(t;
")
=
t+
"
k
Mk(t)+
"
k+1
Mk+1(t)+
:
:
:
(2)
where
Mk(t)
6
0
is
the
kth
order
Poincar
e-Pontryagin
function.
The
max-
imal
number
of
the
zeros
of
Mk(t)
on


provides
an
upper
bound
for
the
number
of
the
limit
cycles
bifurcating
from
the
annulus
A.
In
the
case
k
=
1
the
Poincar
e-Pontryagin
function
is
an
Abelian
integral
depending
on
a
parameterM1(t)
=
R
(t)
Pdx+Qdy,
see
Pontryagin
[7].
The
higher
order
Poincar
e-Pontryagin
functions
are
computed
according
to
the
following
Fran
coise's
recursion
formula
[4].
Let
M1(t)
=






=Mk
1
(t)

0.
Then
Mk(t)
=
Z
(t)

k
7

where


1
=
Pdx+
Qdy;

m
=
rm
1
(Pdx
+Qdy);2

m

k
and
the
functions
r i
are
determined
succesively
from
the
(non-unique)
repres-
ntation

i
=
dRi
+ridf.
As
the
functions
or
dierential
forms
involved
are
analytic,
then
this
computation
can
be
carried
in
a
complex
domain.
From
now
on
we
consider
(1)
as
a
perturbed
complex
foliation
in
C
2
.
Let
l(t)
2
f
1
(t),
t
2


be
a
continuous
family
of
closed
loops
on
the
aône
complex
algebraic
curve
f
1
(t)

C
2
,
where



C
is
a
suitable
open
disc.
The
holonomy
(or
monodromy)
map
associated
to
the
family
l(t)
is
denoted
P(t;
")
and
has
the
representation
(2).
The
rst
non-zero
Poincar
e-Pontryagin
function
Mk(t),
dened
by
(2)
depends
only
on
the
free
homotopy
class
(t)
of
l(t)
[5].
For
one-forms
f1
(t)dt;
f2
(t)dt;
:
:
:
;
fr
(t)dt
on
R
dene
inductively
Z
1
0
f1
(t)dtf
2
(t)dt
:
:
:
fr
(t)dt
=
Z
1
0
(
Z
t
0
f1
()d
:
:
:
fr
1
()d)f
r
(t):
(3)
Let
S
be
a
Riemann
surface
and
!1;!2;
:
:
:
;!k
be
holomorphic
one-forms.
For
every
smooth
path
l
:
[0;
1]
!
S
we
dene
the
iterated
path
integral
[2,
3]
Z
l
!1!2
:
:
:!k
=
Z
1
0
f1
(t)dtf
2
(t)dt
:
:
:
fr
(t)dt
(4)
where
l

!i
=
fi
(t)dt.
This
does
not
depend
on
the
parametrization
of
l.
Every
non-constant
polynomial
f
denes
a
smooth
bration
f
:
C
2
nf
1
(S)!
C
nS
where
S
is
a
nite
set.
Let
!
be
a
polynomial
one-form
in
C
2
.
It
de-
nes
a
geometric
section
of
the
at
cohomology
bundle
associated
to
the
above
bration.
Denote
by
!
0
=
d! d f
the
covariant
derivative
of
this
section
and
consider
the
free
associative
dierential
ring
(R;
d d f
)
with
integer
coef-
cients
generated
by
!.
The
elements
of
R
are
integer
linear
combinations
of
words
!
(i1
)
!
(i2
)
:
:
:!
(is
)
.
Consider
nally
the
linear
dierential
operator
L
=
!
d d f
:
R
!
R.
We
have
for
instance
L!
=
!! 0
,
L
2
!
=
!(! 0
)
2
+!
2
!
00
etc.
Note
that
L
k
1
!
is
a
linear
combinations
of
words
of
lenght
k.
After
this
preparation
we
may
reformulate
the
Fran
coise's
recursion
formulae
as
follows
8
Theorem
1.
The
k-th
order
Poincar
e-Pontryagin
function
Mk
dened
by
(2)
is
given
by
an
iterated
path
integral
of
lenght
k
Mk(t)
=
(
1)
k
1
Z
(t)
L
k
1
!
:
This
allows
to
prove
on
its
hand
Theorem
2.
The
kth
order
Poincar
e-Pontryagin
function
Mk
satises
an
equation
of
Fuchs
type.
Bibliography [1]
V.I.
Arnold,
Geometrical
methods
in
the
theory
of
ordinary
dierential
equations,
Grundlehr.
Math.
Wiss.,
vol.
250,
Springer-Verlag,
New
York
(1988).
[2]
K.-T.
Chen,
Algebras
of
iterated
path
integrals
and
fundamental
groups,
Trans.
AMS
156
(1971)
359-379.
[3]
K.-T.
Chen,
Iterated
Path
Integrals,
Bull.
AMS
83
(1977)
831-879.
[4]
J.-P.
Fran
coise,
Successive
derivatives
of
a
rst
return
map,
application
to
the
study
of
quadratic
vector
elds,
Ergod.
Theory
and
Dyn.
Syst.
16
(1996),
87{96.
[5]
L.
Gavrilov,
I.D.
Iliev,
The
displacement
map
associated
to
polynomial
unfoldings
of
planar
vector
elds,
arXiv:math.DS/0305301
(2003).
[6]
D.
Hilbert,
Mathematische
probleme,
Gesammelte
Abhandlungen
III,
Springer-Verlag,
Berlin
(1935),
pp.
403{479.
[7]
L.S.
Pontryagin,

Uber
Autoschwingungssysteme,
die
den
Hamiltonis-
chen
nahe
liegen,
Phys.
Z.
Sowjetunion
6
(1934),
25{28;
On
dynamics
systems
close
to
Hamiltonian
systems,
Zh.
Eksp.
Teor.
Fiz.
4
(1934)
234-238,
in
russian.
9

Internal
1:3
resonance
revisited
N.K.Gavrilov Nizhny
Novgorod
For
a
two
parameter
unfolding
_ z
=
e
i'
z
+(e
i'
1) z
z
2
 z;
z
=
x+iy;
 z
=
x
iy;
being
a
principal
approximating
family
for
the
study
of
a
system
near
a
pe-
riodic
orbit
with
the
multiplier
exp[2i=3],
we
present
an
explicit
formula
for
a
bifurcation
curve
in
the
parameter
plane
(;')
corresponding
to
the
the
existence
of
heteroclinic
contour.
We
also
prove
the
uniqueness
of
a
limit
cycle
without
appealing
to
any
small
parameter
methods.
10
Yu.S.Ilyashenko's
contributions
to
the
theory
of
holomorphic
foliations
and
uniformization Alexey
Glutsyuk ENS-Lyon
An
approach
to
the
Hilbert
16-th
problem
initiated
by
Petrovskii
and
Landis
and
continued
by
Yu.S.Ilyashenko
was
to
study
the
foliation
by
com-
plex
phase
curves
of
a
polynomial
vector
eld.
In
1957
Petrovskii
and
Landis
had
made
an
attempt
to
prove
that
the
number
of
limit
cycles
of
a
poly-
nomial
vector
eld
is
bounded
by
a
cubic
polynomial
in
the
degree.
They
have
shown
that
it
is
no
greater
than
the
maximal
number
(called
Petrovskii-
Landis
genus)
of
homologically-independent
cycles
whose
projections
to
a
given
complex
line
are
homotopic
to
disjoint
circles.
Their
key
statement
said
that
the
latter
genus
is
uniformly
bounded
by
a
function
in
the
degree.
It
was
found
that
Petrovskii-Landis
arguments
do
not
give
a
proof.
In
one
of
his
rst
papers
[7]
Yu.S.Ilyashenko
had
proved
that
the
latter
key
statement
was
false:
the
Petrovskii-Landis
genus
of
a
vector
eld
of
a
given
degree
can
be
arbitrarily
large,
and
the
number
of
complex
limit
cycles
can
be
countable. While
developing
a
complex-analytic
approach
to
the
Hilbert
16-th
prob-
lem,
Yu.S.Ilyashenko
had
obtained
distinguished
results
in
the
theory
of
holomorphic
foliations.
In
late
60-ths
-
beginning
of
70-ths
he
investigated
generic
properties
of
polynomial
vector
elds
in
complex
plane.
He
proved
in
[10]
that
a
typical
polynomial
vector
eld
has
at
most
countable
number
of
homologically
independent
complex
limit
cycles
and
is
rigid
with
respect
to
the
orbital
equivalence.
In
the
same
paper
he
had
proved
density
of
complex
phase
curves
for
certain
class
of
polynomial
vector
elds,
which
generalized
a
previous
result
by
Khudai-Verenov
[6].
Density
of
phase
curves
of
a
generic
complex
polynomial
vector
eld
was
proved
by
his
student
B.M
uller
[15]
in
dimension
two
and
recently
by
F.Loray
and
J.Rebelo
[14]
in
any
dimen-
sion.
In
his
joint
work
with
A.S.Pyartli
[11]
Ilyashenko
proved
that
the
11

monodromy
group
at
innity
of
a
generic
polynomial
vector
eld
is
free.
In
late
60-ths
Ilyashenko
initiated
the
study
of
uniformization
of
folia-
tion
by
complex
phase
curves.
The
classical
Poincar
e-K
obe
uniformization
theorem
says
that
each
simply-connected
Riemann
surface
is
conformally
equivalent
to
either
C
,
or
C
,
or
the
unit
disc.
The
uniformization
of
arbi-
trary
(not
necessarily
simply-connected)
Riemann
surface
is
its
parametri-
sation
by
the
canonical
coordinate
of
its
universal
cover
(which
has
one
of
the
three
previous
types).
As
it
was
proved
by
the
speaker
[3],
[4],
Lins
Neto
[13],
and
in
a
particular
case
by
Candel
and
Gomez-Mont
[2],
for
a
generic
polynomial
vector
eld
all
the
universal
covers
of
the
complex
phase
curves
are
discs. In
1972
Ilyashenko
had
proved
that
the
union
of
universal
covers
of
leaves
intersecting
a
cross-section
(taken
with
marked
points
on
the
cross-section)
is
a
Stein
manifold
[8],
[12].
The
classical
Bers'
simultaneous
uniformization
theorem
[1]
says
that
for
any
holomorphic
bration
by
compact
Riemann
surfaces
the
bers
admit
a
uniformization
holomorphic
in
the
parameter
of
the
transversal
cross-
section.
In
early
70-ths
Ilyashenko
had
obtained
an
important
result
in
the
theory
of
Kleinian
groups
[9]
that
extended
Bers'
theorem
to
holomorphic
families
(parametrized
by
a
disc)
of
compact
surfaces
with
a
unique
singular
ber
having
only
nondegenerate
double
singular
points.
(For
general
alge-
braic
families
of
compact
Riemann
surfaces
with
singularities
the
previous
statement
is
false
[5].)
Bibliography [1]
Bers,
L.
Simultaneous
uniformization.
Bull.
Amer.
Math.
Soc.
66
1960
94{97.
[2]
Candel,
A.;
Gomez-Mont,
X.
Uniformization
of
the
leaves
of
a
rational
vector
eld.
Ann.
Inst.
Fourier
(Grenoble)
45
(1995),
no.
4,
1123{1133.
[3]
Glutsyuk,
A.A.
Hyperbolicity
of
phase
curves
of
a
general
polynomial
vector
eld
in
C
n
.
(Russian)
Funktsional.
Anal.
i
Prilozhen.
28
(1994),
no.
2,
1{11,
95;
translation
in
Funct.
Anal.
Appl.
28
(1994),
no.
2,
77{84
[4]
Glutsyuk,
A.A.
Hyperbolicity
of
the
leaves
of
a
generic
one-dimensional
holomorphic
foliation
on
a
nonsingular
projective
algebraic
variety.
(Russian)
-
Tr.
Mat.
Inst.
Steklova
213
(1997),
Dier.
Uravn.
s
Veshch-
12
estv.
i
Kompleks.
Vrem.,
90{111;
translation
in
Proc.
Steklov
Inst.
Math.
1996,
no.
2
(213),
83{103.
[5]
Glutsyuk,
A.A.
Nonuniformizable
skew
cylinders.
A
counterexample
to
the
simultaneous
uniformization
problem.
C.
R.
Acad.
Sci.
Paris
Sr.
I
Math.
332
(2001),
no.
3,
209{214.
[6]
Huda
-Verenov,
M.G.
A
property
of
the
solutions
of
a
dierential
equa-
tion.
(Russian)
-
Mat.
Sb.
(N.S.)
56
(98)
1962
301{308.
[7]
Ilyashenko,
Yu.
S.
An
example
of
equations
dw=dz
=
Pn
(z;
w)=Qn
(z;
w)
having
a
countable
number
of
limit
cycles
and
arbitrarily
high
Petrovski
-Landis
genus.
(Russian)
Mat.
Sb.
(N.S.)
80
(122)
1969
388{404.
Translated
in
Math.
USSR-Sb.
9
(1969),
365{378.
[8]
Ilyashenko,
Yu.S.
Foliations
by
analytic
curves.
-
Mat.
Sb.
(N.S.)
88(130)
(1972),
558{577.
Translated
in
Math.
USSR-Sb.
17
(1972)
[9]
Ilyashenko,
Yu.S.
Nondegenerate
B-groups.
(Russian)
-
Dokl.
Akad.
Nauk
SSSR
208
(1973),
1020{1022.
Translated
in
Soviet
Math.
Dokl.
14
(1973),
207{210.
[10]
Ilyashenko,
Yu.S.
Topology
of
phase
portraits
of
analytic
dierential
equations
on
a
complex
projective
plane.
(Russian)
Trudy
Sem.
Petro-
vsk.
No.
4,
(1978),
83{136.
Translated
in
Selecta
Math.
Sovietica
5
(1986),
no.
2,
141{199.
[11]
Ilyashenko,
Yu.S.;
Pyartli,
A.S.
The
monodromy
group
at
innity
of
a
generic
polynomial
vector
eld
on
the
complex
projective
plane.
-
Russian
J.
Math.
Phys.
2
(1994),
no.
3,
275{315.
[12]
Ilyashenko,
Yu.S.
Covering
manifolds
for
analytic
families
of
leaves
of
foliations
by
analytic
curves.
-
Topol.
Methods
Nonlinear
Anal.
11
(1998),
no.
2,
361{373.
[13]
Lins
Neto,
A.
Simultaneous
uniformization
for
the
leaves
of
projective
foliations
by
curves.
-
Bol.
Soc.
Brasil.
Mat.
(N.S.)
25
(1994),
no.
2,
181{206.
[14]
Loray,
F.;
Rebelo,
J.
Minimal,
rigid
foliations
by
curves
on
CP
n
.
-
J.
Eur.
Math.
Soc.
(JEMS)
5
(2003),
no.
2,
147{201.
13

[15]
Mjuller,
B.
The
density
of
the
solutions
of
a
certain
equation
in
CP
2
.
-
Mat.
Sb.
(N.S.)
98(140)
(1975),
no.
3(11),
363{377,
495.
Translated
in
Math.
USSR-Sb.
27
(1975),
no.
3,
325{338.
14
How
often
a
dieomorphism
has
innitely
many
sinks?
Anton
Gorodetski
Independent
University
of
Moscow,
Caltech
This
question
is
known
as
R.Thom's
Conjecture
[T].
S.Newhouse
dis-
proved
this
conjecture
[N1],
[N2].
It
turns
out
that
in
the
space
of
C
r
smooth
dieomorphisms
Di
r
(M)
of
a
compact
surface
M
there
is
an
open
set
U
such
that
a
Baire
generic
dieomorphism
f
2
U
has
innite
many
co-
existing
sinks.
But
some
results
indicates
that
these
dieomorphisms
have
zero
measure
[LY].
Our
(with
V.Kaloshin)
work
in
process
is
devoted
to
this
phenomena.
Innite
number
of
sinks
for
two
dimensional
dieomorphisms
appears
near
homoclinic
tangencies.
Let
V
be
a
small
neighborhood
of
a
homoclinic
contour.
A
sink
is
V
-localized
if
its
trajectory
is
inside
of
V
.
It
has
a
complexity
s
if
it
turns
around
V
exactly
s
times.
Preliminary
results
are
the
following. Theorem
1.
For
any
integer
s
there
exists
a
neighborhood
V
=
V
(s)
of
a
homoclinic
contour
such
that
a
prevalent
unfolding
of
a
dieomorphism
has
only
nite
number
of
V
-localized
sinks
of
complexity
s.
It
is
not
trivial
to
dene
a
notion
of
prevalence,
see
[K],
[HSY].
Theorem
2.
Given

>
0
and
an
integer
s,
there
exists
a
neighborhood
V
=
V
(s;)
such
that
a
prevalent
unfolding
f
of
the
dieomorphism
as
above
has
the
following
property.
For
some
constant
C
>
0
and
any
V
-
localized
periodic
point
p
of
complexity
not
greater
than
s,
f
n
(p)
=
p;
jTrDf
n
(p)j
>
C
(1
)n
:
The
last
Theorem
is
in
some
sense
parallel
to
results
of
Palis
and
Takens
[PT]
and
Palis
and
Yoccoz
[PY].
They
prove
stronger
hyperbolic
properties
15

in
more
restrictive
case.
The
approach
we
use
involves
Newton
Interpolation
Polynomials
and
Discretization
method,
derived
by
V.Kaloshin
in
his
previous
works
[K1],
[KH1],
[KH2]. Bibliography [HSY]
Hunt
B.,
Sauer
T.,
Yorke
J.,
Prevalence:
a
translation-invariant
"al-
most
every"
on
innite-dimensional
spaces,
Bull.
Amer.
Math.
Soc.
(N.S.)
27
(1992),
no.
2,
217{238;
[KH1]
Kaloshin
V.,
Hunt
B.,
A
stretched
exponential
bound
on
the
rate
of
growth
of
the
number
of
periodic
points
for
prevalent
dieomorphisms.
I
&.
Electron.
Res.
Announc.
Amer.
Math.
Soc.
7
(2001),
17{27
&
28{36
(electronic);
[KH2]
Kaloshin
V.,
Hunt
B.,
A
stretched
exponential
bound
on
the
rate
of
growth
of
the
number
of
periodic
points
for
prevalent
dieomorphisms.
I.
preprint,
86pp;
[K]
Kaloshin
V.,
Some
prevalent
properties
of
smooth
dynamical
systems,
Proc.
Steklov
Inst.
Math.
1996,
no.
2
(213),
115{140;
[K1]
Kaloshin
V.,
Ph.D.
thesis,
Princeton
University,
2001;
[LY]
Tedeschini-Lalli
L.,
Yorke
J.
How
often
do
simple
dynamical
processes
have
innitely
many
coexisting
sinks?
Comm.
Math.
Phys.
106,
(1986),
no.
4,
635{657;
[N1]
Newhouse
S.,
Dieomorphisms
with
innitely
many
sinks.
Topology
13,
(1974),
9{18;
[N2]
Newhouse
S.,
The
abundance
of
wild
hyperbolic
sets
and
nonsmooth
stable
sets
of
dieomorphisms,
Publ.
Math.
I.H.E.S.,
50,
(1979),
101{
151;
[PT]
Palis
J.,
Takens
F.,
Hyperbolicity
and
sensitive
chaotic
dynamics
at
homoclinic
bifurcations.
Cambridge
University
Press,
1993;
[PY]
Palis
J.,
Yoccoz
J.-C.,
Fers
cheval
non
uniformment
hyperboliques
engendrs
par
une
bifurcation
homocline
et
densit
nulle
des
attracteurs.
(French)
[Non-uniformly
hyperbolic
horseshoes
generated
by
homo-
clinic
bifurcations
and
zero
density
of
attractors]
C.
R.
Acad.
Sci.
Paris
Sr.
I
Math.
333
(2001),
no.
9,
867{871; 16
[T]
Thom
R.,
Stabilite
structurelle
et
morphogenese.
(French)
Essai
d'une
theorie
generale
des
modules.
Mathematical
Physics
Monograph
Series.
W.
A.
Benjamin,
Inc.,
Reading,
Mass.,
1972;
17

Multiple
Time
Scale
Systems:
an
Update
John
Guckenheimer
Mathematics
Department,
Cornell
University
The
theory
of
multiple
time
scale
dynamical
systems
can
be
traced
at
least
as
far
back
as
the
work
of
van
der
Pol
during
the
1920's.
The
subject
has
undergone
extensive
mathematical
development
since
that
time,
from
three
distinct
approaches.

Asymptotic
analysis
of
these
systems
was
developed
systematically
in
Russia,
with
Pontryagin
an
early
leader,

Nonstandard
analysis
of
these
systems
was
developed
systematically
in
France
with
Georges
Reeb
in
Strasbourg
as
a
visionary
leader.

Geometric
methods
were
introduced
into
the
subject
in
the
United
States
and
Russia,
with
the
work
of
Fenichel
in
the
1970's
an
early
examplar.
The
subject
remains
in
an
unsatisfactory
state
with
these
dierent
languages
that
are
diôcult
to
translate
into
one
another
and
with
the
inherent
com-
plexity
of
the
phenomena
the
subject
investigates.
Periodic
eorts
to
review
the
subject
and
organize
the
major
results
have
been
made.
Arnold,
Afra-
jmovich,
Ilyashenko
and
Shil'nikov
wrote
a
geometric
overview
in
a
paper
that
appeared
in
English
in
the
Springer-Verlag
Encyclopaedia
of
Mathe-
matical
Sciences.
This
lecture
will
survey
some
of
the
developments
that
have
occurred
since
this
work,
with
a
focus
on
some
of
my
own
recent
con-
tributions.
In
particular,
it
will
discuss
computational
aspects
of
multiple
time
scale
systems,
using
the
forced
van
der
Pol
system
as
an
example.
The
origins
of
the
hyperbolic
theory
of
dynamical
systems
can
be
traced
to
the
work
of
Cartwright
and
Littlewood
on
this
example,
and
we
shall
illustrate
how
modern
methods
and
computations
clarify
the
phenomena
they
studied.
18
Instabilities
of
totally
elliptic
points
of
symplectic
maps
in
dimension
four.
Vadim
Kaloshin Caltech
We
show
that
generically
certain
totally
elliptic
points
of
symplectic
maps
in
dimension
four
are
unstable.
Proof
is
based
on
variational
principle
developed
by
J.Mather.
This
is
a
joint
work
with
J.Mather
and
E.
Valdinoci.
19

Normal
forms,
geometric
structures
and
rigidity
Anatole
Katok
We
discuss
how
local
and
semi-local
normal
form
theory
for
smooth
maps,
dierential
equations
and
group
actions
can
be
used
for
proving
global
dierentiable
classication
in
various
cases,
including
various
rigidity
prop-
erties.
Here
is
a
list
of
topis
and
key
words:

Resonanace
and
subresonanace
relations.

Non-stationary
normal
forms
for
contractions
and
applications
to
rigid-
ity.

Critical
regularity
of
invariant
geometric
structures
for
hyperbolic
dy-
namical
systems:
nonresonanace
and
resonance
cases.

Anosov
cocycle
and
its
generalizations.

Invariant
connections
and
higher{order
structures.
Bibliography [1]
S.
Hurder,
A.Katok
Dierentiability,
rigidity
and
Godbillon-Vey
classes
for
Anosov
ows,
Publ.
Math.
IHES,
vol,
72,
(1990),
5-61.
[2]
L.
Flaminio
and
A.Katok,
Rigidity
of
symplectic
Anosov
dieomor-
phisms
on
low
dimensional
tori,
Erg.
Theory
and
Dynam.
Systems,
vol.
11,
(1991),
427-440.
[3]
A.Katok,
Hyperbolic
measures
and
commuting
maps
in
low
dimension,
Disc.
and
Cont.
Dyn.
Sys,
vol.
2,
(1996),
397-411.
[4]
M.Guysinsky,
A.
Katok,
Normal
forms
and
invariant
geometric
struc-
tures
for
dynamical
systems
with
invariant
contracting
foliations,
Math-
ematical
Research
Letters,
vol.
5,
(1998),
149-163.
20
Circle
dieomorphisms
with
breaks.
Konstantin
Khanin
21

An
example
of
non-coincidence
of
minimal
and
statistical
attractors
V.Kleptsyn
Moscow
State
University,
Independent
University
of
Moscow,

Ecole
Normale
Sup
erieure
de
Lyon.
E-mail:
kleptsyn@mccme.ru
1.
Introduction.
There
are
many
dierent
denitions
of
what
one
should
call
an
attracting
set
for
a
dynamical
system,
such
as
the
maximal
attractor,
the
nonwandering
set,
the
Birkho
center,
the
Milnor's
attractor,
the
statistical
attractor
and
the
minimal
attractor.
The
latter
notion
was
introduced
by
A.Gorodetski
and
Yu.Ilyashenko,
see
[1].
The
aim
of
this
work
is
to
present
an
example
of
a
dynamical
system
for
which
the
two
last
denitions
give
dierent
attracting
sets
(an
example
which
have
been
found
only
short
time
ago).
2.
Denitions.
Let
us
consider
a
smooth
dynamical
system
(X;
g
t
),
and
let
L
be
the
Lebesgue
measure
on
X
(normalized
to
L(X)
=
1).
Denition
1.
An
open
set
U
is
called
Stat-inessential,
if
for
L{almost
every
point
x
2
X
1 T
 

tj
0

t

T;
g
t
(x)
2
U
 
!0;
as
T
!1;
where
jZj
denotes
the
Lebesgue
measure
of
the
set
Z

R.
Denition
2.
An
open
set
U
is
called
Min-inessential,
if
1 T
Z
T
0
(g
t 
L)(U)
dt!
0;
as
T
!1:
Denition
3.
A
statistical
(resp.
minimal)
attractor
is
dened
as
a
complement
to
the
union
of
all
Stat-inessential
(resp.
Min-inessential)
open
sets.
We
denote
it
as
AMin
(resp.
AStat
).
22
3.
Main
result.
Consider
a
planar
vector
eld
v.
Suppose
that
it
has
two
singular
points,
a
saddlenode
A
and
a
saddle
B.
Also,
suppose
that
an
outgoing
separatrix
of
A
coincides
with
an
incoming
separatrix
of
B
and
that
an
outgoing
separatrix
of
B
coincides
with
an
incoming
separatrix
of
A
(see
g.
11.1): A
B
Figure
12.1:
Modied
Bowen's
example
Let
AB
(resp.
BA)
be
the
separatrix,
going
from
A
to
B
(resp.
from
B
to
A),
let
=
fA;Bg
[
AB
[
BA,
and
let
M
be
a
domain,
bounded
by
.
Then
there
exists
a
small
\inner
neighborhood"
V
of
the
loop
(an
intersection
of
some
neighborhood
of
with
M)
such
that
every
trajectory
with
initial
conditions
in
V
stays
in
V
forever
and
tends
to
as
time
tends
to
innity. Let
us
consider
the
dynamical
system
(X;
g
t
),
whereX
=
V
,
and
g
t
=
g
t v
is
the
ow,
generated
by
v.
We
will
call
this
system
\modied
Bowen's
example"
in
an
analogy
with
the
Bowen's
example
(which
is
constructed
in
the
same
way,
but
both
singular
points
are
saddles).
Theorem
1.
For
the
modied
Bowen's
example,
AMin
=
fAg;
AStat
=
fA;Bg. 4.
Sketch
of
the
proof.
It
is
clear
that
every
point
x
2
X
passes
almost
all
the
time
in
arbitrarily
small
neighborhoods
of
A
and
B,
thus,
AMin

AStat

fA;Bg.
Consider
for
a
point
x
2
X
two
sequences
T
A n
and
T
B n
of
time
intervals,
passed
in
some
(xed)
neighborhoods
of
A
and
B
respectively.
For
simplicity,
we
restrict
to
the
case
where
the
saddle
can
be
written
as
_ x
=
x;
_ y
=
y,
the
saddlenode
as
_ x
=
x
2
;
_ y
=
y
(the
general
case
is
similar,
but
a
bit
more
diôcult
technically).
Direct
computation
gives
the
following
(asymptotic)
behavior
of
this
sequence:
T
B n

T
A n
;
T
A n+1

e
T
B n
:
So,
for
each
individual
point,
there
exists
a
sequence
of
times
Tn
which
tends
to
innity,
such
that
for
every
Tn
the
total
amounts
of
time,
passed
near
A
and
near
B,
are
almost
equal.
Thus,
AStat
=
fA;Bg.
23

On
the
other
hand,
it
can
be
shown
that
for
any
two
points
x;
y
2
V
either
after
some
moment

they
will
never
appear
near
B
simultaneously,
or
they
belong
to
the
same
trajectory.
It
can
be
seen
from
comparing
the
sequences
T;T;
e
T
;
e
T
;
e
e
T
;
e
e
T
;
:
:
:
and
T
0
;T
0
;
e
T
0
;
e
T
0
;
e
e
T
0
;
e
e
T
0
;
:
:
:
of
the
times,
passed
near
A
and
B.
Thus,
the
Lebesgue
measure
will
con-
centrate
(as
time
goes
to
innity)
near
the
point
A:
g
t 
L
!ôA;
as
t!1:
Thus,
the
minimal
attractor
is
AMin
=
fAg.
5.
Remarks. Remark
1.
The
modied
Bowen's
example
is
an
example
of
codimen-
sion
3
in
planar
vector
elds.
The
denitions
of
inessentiality
may
be
rewritten
in
a
following
way.
Consider
for
an
open
set
U
the
family

U T
(x)
of
time-averages
of
the
char-
acteristic
function

U
of
U:

U T
(x)
=
1 T
Z
T
0

U
(g
t
(x))dt:
Remark
2.
The
set
U
is
Stat-inessential
(resp.
Min-inessential)
if
and
only
if

U T
tends
to
0
in
measure
(resp.
almost
surely)
w.r.t.
L.
Remark
3.
Considering
family
of
functions

U T
for
U
=
U"(B),
we
obtain
a
dynamical
realization
of
the
Riesz's
example.
Acknowledgements The
author
is
deeply
grateful
to
A.Gorodetski,
Yu.Ilyashenko,
and
E.Ghys
for
useful
discussions. Bibliography [1]
A.Gorodetski,
Yu.Ilyashenko,
Minimal
and
strange
attractors,
Interna-
tional
Journal
of
Bifurcation
and
Chaos,
1996,
v.
6,
N
6,
1177{1183.
24
Integrable
linearizable
and
normalizable
saddles
in
quadratic
systems
in
C
2
P.
Marde
si c
We
present
some
results
from
a
joint
work
with
C.
Christopher
and
C.
Rousseau. We
study
for
the
family
of
polynomial
vector
elds
given
by
_ x
=
x+
X i+j=2
c
ij
x
i
y
j
_ y
=
y
+
X i+j=2
dij
x
i
y
j
;

>
0;
c
ij
;
dij
2
C
A
vector
eld
is
linearizable
if
by
an
analytic
change
of
coordinates
(without
multiplication
by
a
function)
it
can
be
transformed
to
a
linear
vector
eld.
A
vector
eld
is
integrable
if
it
can
be
tranformed
to
a
multiple
of
a
linear
vector
eld.
A
resonant
vector
eld
i.e.

2
Q
+
is
normalizable
(orbitally
normalizable)
if
by
an
analytic
change
of
coordinates
it
can
be
transformed
to
a
vector
eld
containing
only
resonant
monomials
(up
to
a
multiplicative
factor).
In
the
parameter
space
R
+

C
6
,
denote
L,
I,
N
and
ON
the
sets
of
the
parameter
values
corresponding
to
linearizable,
integrable,
normalizable
and
orbitally
normalizable
vector
elds
and
for
xed

2
R
+
let
L,
I
,
N
and
ON
denote
their
restriction
given
by
the
prescribed
value
of
.
The
following
inclusions
are
obvious L

I
\
\
N
ON
and
analogously
for
their
restrictions
to

xed.
25

What
can
be
said
about
the
sets
L,
I,
N
and
ON
and
about
their
restrictions
for

xed?
The
dependence
of
these
sets
on

is
higly
discontinuous.
For
instance
if

is
a
Bruno
number,
then
L
=
C
6
,
whereas
if

is
rational,
then
the
sets
L,
I
are
algebraic
sets.
For

irrational
non-Bruno
number
these
sets
are
very
complicated.
We
study
two
mechanismes
for
linearizability
which
work
well
in
the
family:
the
Darboux
method
and
the
blow
down
to
a
node
or
saddle-node.
In
particular
these
two
methods
suôce
for
proving
linearizability
of
all
strata
in
L2.
Our
calculations
showed
some
strange-looking
features:
holes
in
the
lin-
earizability
strata
and
their
accumulation,
strata
stopping
for
some
value
of

etc.
We
give
a
geometric
explanation
of
some
of
these
phenomena
in
terms
of
the
unfolding
of
Ecalle-Voronin
moduli.
We
study
also
the
relationship
between
integrability
and
linearizability.
We
show
that
for

2
R
+
,
I
=
L
if
and
only
if

is
not
a
Cremer
number.
We
consider
more
generally
analytic
saddles.
We
prove
the
equivalence
between
the
following
three
conditions:
-
linearizability
of
an
integrable
saddle
-
relative
exactness
of
a
certain
time
form

-
vanishing
of
the
integrals
of
the
form

along
all
asymptotic
cycles
in
the
leaves
of
the
saddle.
Analogous
results
for
saddle-nodes
have
been
obtained
by
L.
Teyssier.
We
discuss
also
some
generalizations.
Bibliography [1]
Yu.
S.
Il'yashenko,
In
the
Theory
of
Normal
Forms
of
Analytic
Dier-
ential
Equations,
Divergence
is
a
Rule
and
Convergence
the
Exception
When
Bryuno
Conditions
are
Violated,
Vestnik
Moskovskogo
Univer-
siteta.
Matematika,
Vol
36
2
(1981),
10-16.
[2]
Yu.
S.
Il'yashenko;
A.
S.
Pyartli,
Materialization
of
Resonances
and
Di-
vergence
of
Normalizing
Series
for
Polynomial
Dierential
Equations.
Trudy
Sem.
Petrovsk.
No.
8
(1982),
111{127.
[3]
C.
Christopher;
P.
Marde
si c;
C.
Rousseau,
Normalizable,
Integrable,
and
Linearizable
Saddle
Points
for
Complex
Quadratic
Systems
in
C
2
.
J.
Dynam.
Control
Systems
9
(2003),
no.
3,
311{363
26
[4]
C.
Christopher;
P.
Marde si c;
C.
Rousseau,
Normalizability
Synchronic-
ity
and
Relative
Exactness
for
Vector
Fields
in
C
2
.
to
appear
in
J.
Dynam.
Control
Systems
[5]
L.
Teyssier,
Equation
Homologique
et
Classication
Analytique
des
Germes
de
Champs
de
Vecteurs
Holomorphes
de
Type
Noeud-Col,
Th
ese
de
l'Universit e
de
Rennes
I,
(2003)
27

On
the
analytic
solvability
of
the
center-focus
problem
N.
B.
Medvedeva
Chelyabinsk
State
University
Chelyabinsk,
Russia
The
singular
point
of
the
vector
eld
in
the
plane
is
called
monodromic,
if
the
Poincare
rst
return
function
(monodromy
transformation)
is
dened
for
it.
Tthe
monodromic
singular
point
of
the
analytuc
vector
eld
can
be
either
a
center
or
a
focus.
The
set
M
of
all
the
monodromic
germs
can
be
represented
as
an
union
of
semialgebraic
sets
:
M
=
[M.
All
the
germs
from
oneM
have
the
same
process
of
desingularization
connected
with
the
Newton
diagrams.
The
following
theorem
is
proved
Theorem
1.
The
center-focus
problem
is
analytically
solvable
in
each
class
M.
28
Topology
of
plane
real
algebraic
curves:
some
solved
and
open
problems S.
Orevkov
How
the
branches
of
a
plane
real
algebraic
curve
can
be
arranged
on
the
projective
plane?
This
is
the
rst
half
of
the
rst
part
of
Hilbert's
16th
problem
(the
second
half
is
about
spatial
surfaces).
This
question
just
indi-
cated
a
large
area
for
investigation.
However
several
concrete
questions
and
conjectures
served
as
repers
in
this
area.
For
example,
in
the
16th
problem,
Hilbert
himself
posed
an
explicite
question:
can
a
6th
degree
curve
con-
sist
of
11
ovals,
none
of
them
being
inside
another
(answered
by
Petrovski)
and
more
generally,
what
arrangements
of
ovals
are
possible
for
a
6th
de-
gree
curve
(answered
by
Gudkov).
Other
questions
and
conjectures
which
motivated
the
development
were
formulated
by
Ragsdale,
Gudkov,
Arnold,
Rohlin,
Viro,
Kharlamov
et
als.
For
instance,
the
famous
Gudkov's
con-
jecture
about
the
congruence
modulo
8
maybe
caused
the
Arnold-Rohlin
revolution
in
the
area.
In
my
talk
I
am
going
to
survey
the
matter
of
art
for
the
old
problems
and
to
formulate
some
new
(or
old
but
forgotten)
problems
which
seem
to
me
important.
29

Rigidity
theorems
for
generic
holomorphic
germs
of
dicritic
foliations
and
vector
elds
in
(C
2
;
0)
L.
Ortiz-Bobadilla
It
is
well
known,
that
the
analytic
and
formal
classication
of
holomor-
phic
vector
elds
at
generic
singular
points
coincides
(Poincare).
The
failure
of
the
genericity
assumptions
(in
the
case
of
saddle
resonant
and
saddle
node
singularities,
for
example)
leads
to
a
relatively
simple
formal
classication
which
diers
from
the
analytic
one:
the
analytic
classication
has
functional
moduli
(Brjuno,
Voronin,
Grintchy,
Mescheryakova,
Teyssier).
The
same
happens
for
orbital
equivalence
(analytic
and
formal)
of
holomorphic
vector
elds
(Ilyashenko,
Martinet,
Ramis
),
and
for
mappings
(Ecalle,
Ilyashenko,
Voronin).
In
the
cases
of
higher
codimension
the
remarkable
presence
of
\rigidity"
takes
place:
it
consists
on
the
coincidence
of
formal
and
analytic
classication,
while
already
the
formal
classication
is,
generally
speaking,
boundless
and,
as
rule,
has
functional
moduli.
The
rst
rigidity
theorem
(where
rigidity
is
understood
as
above)
was
proved
for
the
analytic
classication
of
nonsolvable
nitely-generated
groups
of
germs
of
holomorphisms
on
the
complex
line
(Cerveau,
Moussu,
Ramis,
Elizarov,
Ilyashenko,
Shcherbakov,Voronin).
The
rigidity
phenomena
in
the
classication
of
foliations
given
by
holomorphic
germs
of
vector
elds,
known
as
orbital
rigidity,
was
found
for
degenerate
singular
points
having
nilpotent
Jordan
cell
at
this
linearization
(Cerveau,
Moussu,
Elizarov,
Ilyashenko,
Shchervakov,
Voronin).
A
rigidity
theorem
for
foliations
in
the
class
(next
by
complexity),
n
,
consisting
of
germs
having
vanishing
n-jet
at
its
singular
point,
was
proved
(for
generic
non
dicritic
germs)
by
S.
Voronin.
Later,
an
analogous
result
for
the
rigidity
of
vector
elds
was
obtained
(Ortiz,
Rosales,
Voronin).
Finally,
orbital
rigidity
for
families
of
germs
with
xed
geometry
(under
weak
genericity
assumptions)
was
obtained
by
Le
Floch
and
can
be
obtained
from
the
results
given
by
J.F.
Mattei
and
E.
Salem.
In
the
present
exposition
a
rigidity
theorem
(in
the
case
of
foliations
and
vector
elds)
is
30
obtained
for
generic
dicritic
germs
in
the
class
n
(joint
work
with
S.Voronin
and
E.
Rosales).
31

Algebraic
Integrability
of
functional
equations
and
dynamical
systems
with
discrete
time
dened
by
birational
mappings
K.V.
Rerikh
Bogoliubov
Lab.
of
Theoretical
Physics,
JINR
e-mail:
rerikh@thsun1.jinr.ru http://thsun1.jinr.ru/rerikh
The
talk
is
based
on
the
papers
[8],
[9]
that
are
a
development
of
the
algebraic-geometrical
approach
to
integrability
of
functional
equations
(
see
[7])
of
the
form
below
and
is
devoted
to
an
algebraic
integrability
(AI)
of
functional
equations
dened
by
birational
mappings
(BFEs)
for
functions
y(w)
:
C!C
N
in
one
complex
variable
w
of
the
form
y(w
+1)
=
Fn(y(w));
y(w)
:
C!C
N
;
w
2
C;
Fn
2
Bir(C
N
):
(1)
For
w
=
m
2
Z
the
BFEs
above
is
a
dynamical
system
with
a
discrete
time
(DDS)
or
cascade
(see
[2]).
Here
the
map
Fn
:
y
7!
y
0
=
Fn(y)
=
fi
(y) fN+1(y)
;
i
=
(1;
2;
:
:
:
;N),
fi
(y)
for
8
i
are
polynomials
in
y,
degFn(y)
=
max
N+1 i=1
fdeg(fi
(y))g
=
n;
is
a
given
birational
one
of
the
group
of
all
automorphisms
of
C
N
!C
N
with
coeôcients
from
C
[10]-[13].
The
BFEs
(1)
can
also
be
derived
at
author's
discretization
of
a
standard
autonomous
dierential
equation
of
rst
order
with
the
vector
eld
[1]
connected
with
Fn(y(w))
by
a
corresponding
manner.
We
obtain
a
general
solution
of
these
BFEs
for
n
=
1
and
any
N
but
also
the
one
for
8(n;N)
at
Fn

UmôF1ôU
1
m
where
Um
2
BirC
N
.
We
discuss
the
dynamics
of
the
map
(1)
at
N
=
2;8n
following
to
[7]
and
such
important
characteristic
of
integrable
mappings
as
the
Arnold
complexity
C
n A;1
2
(k)

dn
(k)
def =
deg
k n
(see
Denition
of
the
one
in
[3],
[4]
and
[5],
[6]
).
We
derive
for
the
one
the
linear
autonomous
dierence
32
equation
with
constant
coeôcients.
A
spectrum
of
characteristic
equation
for
this
dierence
equation
denes
a
polynomial
or
exponential
growth
of
the
Arnold
complexity
that
is
testing
of
algebraic
or
nonalgebraic
integrability.
The
AI
of
the
BFEs
for
n

2;
N
=
2
is
dened
through
an
ex-
istence
of
the
linear
system
of
algebraic
curves
[12]
(LSAC)
(z(w))
:
Pk+1 j=1
Cj j
(z(w)
(y
i
(w)
=
z i
(w)=z3
(w);
z
2
CP
2
)
of
degree
,
dimen-
sion
k

1,
of
genus
p

0,
being
invariant
subspace
relatively
to
the
birational
map
n
:
z
7!
z
0

(z)
=
z
n 3
f(zi
=z3
).
As
result
of
this
deni-
tion
we
derive
k
rst
integrals
in
extended
phase
space
C
2
C
in
terms
of
(k
+
1)
basic
functions
of
the
LSAC
(in
Jordan
basis).
These
integrals
are
rational
functions
in
y(w);w;
(
i k+1
)
w
;
i
2
(1;






;
k)
but
i
are
dened
by
the
spectrum
of
the
matrix
D
(in
Jordan
basis):
(z
0
(w))

Dô (z(w)),
and
z
0
(w)
def =
z(w
+1)
def =
nz(w).
We
established
necessary
and
suôcient
conditions
of
the
existence
of
the
LSAC
as
well
as
the
construction
of
the
one
(LSAC)
in
explicit
form.
In
[14]
we
made
rst
step
to
the
integrating
of
the
functional
equations
(1)
for
8(n;N).
Bibliography [1]
V.I.
Arnold,
Yu.S.
Il'yashenko,
Ordinary
Dierential
Equations,
in
Dy-
namical
Systems,
Vol.
1,
eds.
D.V.
Anosov
and
V.I.
Arnold
(Ency-
clopaedia
Math.
Sciences,
Vol.
1,
Springer,
Berlin,
1988).
[2]
D.V.
Anosov
et
al.,
Smooth
Dynamical
Systems,
in
Dynamical
Sys-
tems,
Vol.
1,
eds.
D.V.
Anosov
and
V.I.
Arnold
(Encyclopaedia
Math.
Sciences,
Vol.
1,
Springer,
Berlin,
1988).
[3]
V.I.
Arnold,
Dynamics
of
the
complexity
of
intersections,
Bol.
Soc.
Bras.
Mat.
21
(1990)
1{10.
[4]
V.I.
Arnold,
Dynamics
of
intersections,
in:
Proceedings
of
a
Conference
in
Honour
of
J.
Moser,
eds.
P.
Rabinowitz
and
E.
Zehnder
(Academic
Press,
New
York,
1990)
pp.
77{84.
[5]
A.P.
Veselov,
Integrable
mappings,
Russian
Math.
Surveys
46,
no.
5
(1991)
1{51.
[6]
A.P.
Veselov,
Growth
and
Integrability
in
the
Dynamics
of
Mappings,
Commun.
Math.
Phys.
145
(1992)
181{193.
33

[7]
K.V.
Rerikh,
Algebraic-geometry
approach
to
integrability
of
birational
plane
mappings.
Integrable
birational
quadratic
reversible
mappings.
I,
J.
of
Geometry
and
Physics
24
(1998)
265-290.
[8]
K.V.
Rerikh,
Integrability
of
functional
equations
dened
by
byrational
mappings.
(General
theory
of
integration
of
birational
cascades
and
cascade- ows.)II, Submitted
to
Journal
"
Mathematical
Physics,
Analysis
and
Geome-
try".
[9]
K.V.
Rerikh,
Algebraic
integrability
of
functional
equations
dened
by
birational
mappings.
I,
Submitted
to
"Moscow
Mathematical
Journal".
[10]
I.R.
Shafarevich,
Basic
Algebraic
Geometry
(Springer,
Berlin,
1977).
[11]
H.
Hudson,
Cremona
Transformations
in
Plane
and
Space
(Cambridge
University
Press,
Cambridge,
1927).
[12]
Selected
Topics
in
Algebraic
Geometry,
Bulletin
of
the
National
Re-
search
Council,
part
I,
(1928)
Number
63,
part
II,
(1934)
Number
96,
Report
of
the
Committee
on
Rational
Trnsformations.
[13]
V.A.
Iskovskikh
and
M.
Reid,
Foreword
to
Hudson's
book
"Cremona
transformations",
Preprint
(Cambridge
University
Press,
Cambridge,
1991).
(Cambridge
University
Math.
Library)
[14]
K.V.
Rerikh,
General
approach
to
integration
of
reversible
dynamical
systems,
dened
by
mappings
from
Cremona
group
Cr(P
n k
)
of
bira-
tional
transformations,
"Mathematical
Notes"
,
v.
68
(
2000),
594-601.
34
The
cyclicity
of
singular
loops
of
reversible
quadratic
Hamiltonian
under
quadratic
perturbations
Robert
Roussarie
As
it
is
well-known,
the
principal
diôculty
in
the
study
of
a
planar
vector
eld
family
is
to
understand
how
in
the
family,
the
polycycles
bifurcate
into
limit
cycles.
The
reason
is
that
a
polycycle
bifurcation
is
in
general
a
highly
non-analytic
phenomenon.
A
case
of
particular
importance
is
when
one
considers
limit
cycles
created
in
unfolding
an
Hamiltonian
polycycle.
Introducing
a
small
parameter
"
the
dual
unfolding
can
be
written

=
dH
+"  
+o(")
where

=
(
 ;
"):
The
integral
function
unfolding
I(h;
 )
=
R
fH=hg
 
is
called
Abelian
integral
(unfolding)
of
the
given
vector
eld
unfolding.
Limit
cycles
bifurcating
from
an
annulus
lled
by
regular
Hamiltonian
cycles,
are
in
general
directly
controlled
by
the
zeros
of
the
associated
Abelian
integral
unfolding.
Then,
a
general
question
is
to
understand
in
what
measure
the
Abelian
integral
unfolding
also
control
the
limit
cycles
bifurcating
from
a
Hamiltonian
polycycle.
In
the
present
talk,
one
restricts
the
investigation
to
quadratic
polyno-
mial
vector
elds.
In
this
context,
the
Hamiltonian
function
H
is
a
cubic
polynomial
and
  
is
a
family
of
quadratic
1-forms.
The
exact
number
of
zeros
of
Abelian
integrals
corresponding
to
the
integration
of
quadratic
1-
forms
along
cycles
of
cubic
Hamiltonians
was
recently
obtained,
even
for
the
degenerate
cases.
These
Abelian
integrals
have
no
more
than
2
zeros
in
their
whole
domain
of
denition,
which
may
be
the
union
of
two
intervals.
One
can
deduce
from
this
result
the
number
and
the
bifurcation
diagram
of
limit
cycles
which
bifurcate
from
any
region,
union
of
regular
annuli
of
Hamiltonian
cycles
in
the
space
of
quadratic
vector
elds.
Moreover,
one
can
extend
this
study
at
the
whole
plane,
in
the
case
of
generic
quadratic
Hamiltonian
vector
elds.
For
such
a
generic
Hamiltonian
vector
eld,
the
35

boundary
of
region
lled
by
the
regular
Hamiltonian
cycles
contains
just
non-degenerate
centers
or
homoclinic
saddle
connections,
and
in
these
two
cases
it
is
known
that
all
the
bifurcating
limit
cycles
are
related
to
zeros
of
the
associated
Abelian
integral.
For
non-generic
Hamiltonian
vector
elds,
the
problem
seemsmuch
more
diôcult
because
one
has
to
look
at
the
limit
cycles
bifurcating
from
polycy-
cles
containing
more
than
one
singular
point.
Let
us
denote
by
Q
H
and
Q
R
the
Hamiltonian
class
and
reversible
class
of
quadratic
integrable
systems.
There
are
several
topological
types
for
systems
belonging
to
Q
H
\Q
R
.
One
of
them
is
the
case
where
the
corresponding
system
has
two
heteroclinic
loops
with
2
saddles
(2-saddle-loops),
sharing
one
saddle{connection,
which
is
a
line
segment
(the
other
saddle-connections
form
an
ellipse).
In
a
recent
work
in
collaboration
with
Chengzhi
Li,
we
prove
that
the
maximal
number
of
limit
cycles
which
bifurcate
from
the
union
of
the
2-
saddle-loops
with
respect
to
quadratic
perturbations
is
two,
in
the
reversible
direction.
This
means
that
every
bifurcating
limit
cycle
is
related
to
a
zero
of
Abelian
integral.
We
also
describe
the
corresponding
bifurcation
diagram.
This
diagram
can
not
be
deduced
from
the
bifurcation
diagram
for
Abelian
integral
zeros.
It
must
be
noticed
that
the
bifurcation
diagram
is
not
completely
established,
up
to
now,
outside
a
conic
sector
around
the
reversible
direction.
36
On
systems
with
homoclinic
loops
of
saddle-focus L.
P.
Shilnikov 37

Rectiable
pencils
of
conics
Vladlen
Timorin
A
set
of
curves
with
a
common
point
is
called
rectiable
if
there
is
a
local
dieomorphism
(dened
in
a
neighborhood
of
the
common
point)
that
sends
all
curves
from
this
set
to
straight
lines.
Rectiable
pencils
of
curves
appear,
for
example,
in
Riemannian
geometry|
geodesics
passing
through
a
given
point
are
rectiable
by
the
inverse
of
the
exponential
map.
The
following
is
one
type
of
questions
I
am
particularly
interested
in.
For
some
nice
class
of
curves,
determine
what
pencils
of
curves
from
this
class
are
rectiable.
Almost
all
rectiable
analytic
pencils
of
conics
in
the
plane
look
as
fol-
lows.
All
conics
from
the
pencil
(except
for
3
degenerate
conics,
which
are
straight
lines)
have
3
points
of
tangency
with
the
same
curve,
which
is
the
dual
to
a
cubic.
38
Homoclinic
bifurcations
and
dimension
of
attractors
of
evolutionary
PDE's
Dmitry
Turaev
A
new
method
of
estimating
the
attractor's
dimension
is
suggested
which
involves
analysis
of
homoclinic
bifurcations,
and
which
allows
for
expressing
both
upper
and
lower
bounds
for
the
attractor's
deimension
in
terms
of
Lyapunov
dimension.
The
method
is
applied
for
obtaining
sharp
estimates
of
the
attractor's
dimension
for
a
class
of
damped
wave
equations
which
are
beyond
the
reach
of
the
classical
methods.
This
is
a
joint
work
with
S.Zelik.
39

Monodromy
groups
of
degenerate
singularities
of
holomorphic
foliations
S.
M.
Voronin
We
shall
study
a
generalization
of
the
usual
monodromy
group
of
holo-
morphic
foliations.
This
group
is
obtained
by
glueing
the
monodromy
groups
of
pasted
spheres
under
a
good
blowing-up.
40
Center
problem
for
Abel
equation,
Moments
and
Compositions
Y.
Yomdin
The
Weizmann
Institute
of
Science
Consider
the
system
of
dierential
equations
(
_ x
=
y
+F(x;y);
_ y
=
x+G(x;y)
(1)
with
F(x;
y)
and
G(x;
y)
homogeneous
polynomials
of
a
given
degree
d

2.
This
system
has
a
center
at
the
origin
if
all
its
trajectories
around
the
origin
are
closed.
A
(version
of)
the
classical
Poincar e's
Center-Focus
problem
is
to
nd
conditions
on
F
and
G
necessary
and
suôcient
for
the
system
(1)
to
have
a
center.
Closely
related
to
this
question
is
the
second
part
of
Hilbert's
16-th
problem
which
asks
for
the
maximal
possible
number
of
isolated
closed
trajectories
(limit
cycles)
in
the
above
system.
These
classical
problems
can
be
naturally
reformulated
for
the
Abel
dierential
equation
y
0
=
p(x)y
2
+q(x)y
3
(2)
This
equation
is
said
to
have
a
center
at
a
pair
of
complex
numbers
(a;
b)
if
y(a)
=
y(b)
for
every
solution
y(x)
of
(2)
with
the
initial
value
y(a)
small
enough.
The
Center-Focus
problem
is
to
give
necessary
and
suôcient
conditions
on
p;
q
for
(2)
to
have
a
center.
The
analogue
of
the
Hilbert
16-th
problem
for
Abel
equation
(called
sometimes
\Pugh
problem")
is
to
bound
the
number
of
isolated
solutions
y(x)
of
(2)
satisfying
y(a)
=
y(b)
(the
\periodic"
ones).
It
is
a
general
belief
that
the
Abel
equation
version
of
the
Hilbert
and
the
Center-Focus
problems
re ects
the
main
diôculties
of
the
original
ones
while
suggesting
serious
technical
simplications.
Moreover,
it
opens
new
relations
with
classical
Analysis
and
Algebra.
Indeed,
recently
center
con-
ditions
for
Abel
equation
(2)
have
been
related
to
the
composition
factor-
ization
of
P
=
R
p
and
Q
=
R
q
on
one
side,
and
to
the
vanishing
conditions
41

for
the
moments
mk
=
Z
b
a
P
k
(x)q(x)dx
(3)
on
the
other.
(Finding
these
vanishing
conditions
for
mk;
k
=
0;
1;
:
:
:
we
call
a
\Moment
problem".
It
turns
out
to
be
a
tangential
version
of
the
Center-Focus
problem).
Let
us
assume
that
the
coeôcients
p
=
P
0
and
q
=
Q
0
of
Abel
equation
(2)
are
complex
polynomials.
P
and
Q
are
said
to
satisfy
a
Polynomial
Composition
condition
if
there
exist
polynomials
~ P,
~
Q,W
such
thatW(a)
=
W(b)
and
P(x)
=
~
P(W(x));
Q(x)
=
~
Q(W(x)):
(4)
A
change
of
variables
in
(2)
and
(3)
according
to
(4)
shows
that
the
Com-
position
condition
implies
both
the
center
for
(2)
and
the
vanishing
of
all
the
moments
(3).
We
do
not
know
any
counterexample
to
the
following
Composition
Conjecture.
The
Polynomial
Composition
condition
is
nec-
essary
and
suôcient
for
Abel
equation
(2)
to
have
a
center.
This
conjecture
has
been
veried
for
small
degrees
of
p;
q
and
in
some
special
cases
(M.A.M.
Alwash,
N.G.
Lloyd,
M.
Briskin,
J.-P.
Francoise,
Y.Y.,
C.
Christopher,
M.
Blinov,
N.
Roytvarf,
Yang
Lijun
and
Tang
Yun).
Below
we
give
some
very
recent
results
further
supporting
this
conjecture.
For
some
time
a
similar
conjecture
for
the
Moment
problem
has
been
discussed.
It
turns
out
to
be
true
in
many
special
cases
but
in
general
the
vanishing
of
all
the
moments
(3)
does
not
imply
(4)
(F.
Pakovich).
Let
us
call
\denite"
on
a;
b
the
polynomials
P
for
which
the
vanishing
of
mk;
k
=
0;
1;
:
:
:
with
any
polynomial
q
=
Q
0
does
imply
Composition
condition
(4).
All
polynomials
P
up
to
degree
5
are
denite.
In
the
space
Vl
of
polyno-
mials
P
of
a
xed
degree
l

6
non-denite
polynomials
belong
to
a
certain
proper
algebraic
subset.
More
specically,
all
indecomposable
P
are
denite
(for
every
a
6=
b),
as
well
as
all
P
with
P
0
(a)
6=
0,
P
0
(b)
6=
0.
Chebyshev
polynomial
T6
is
not
denite
with
respect
to
a
=
p
3=2,
b
=
p
3=2.
Many
additional
classes
of
denite
polynomials
are
known
(M.
Briskin,
J.-P.
Fran-
coise,
Y.Y.,
C.
Christopher,
M.
Blinov,
N.
Roytvarf,
F.
Pakovich).
The
following
results
(Blinov,
Briskin,
Roytvarf,
Y)
support
the
Compo-
sition
Conjecture
above
and
illustrate
the
role
of
denite
polynomials
(and
of
the
moments
in
general)
in
the
Center
problem
for
the
Abel
equation.
In
all
these
results
the
polynomial
p
=
P
0
is
xed
while
the
polynomial
q
of
a
given
degree
d
varies
inside
the
space
Vd
.
42
Theorem
1.
Let
a
polynomial
p
=
P
0
be
xed,
with
P
denite.
Let
the
maximal
degree
d
of
the
polynomials
q
=
Q
0
be
xed.
There
exists

=
(p;
d;
a;
b)
>
0
depending
only
on
p,
a;
b
and
d
such
that
for
any
q
of
degree
d
with
k
q
k

Abel
equation
(2)
has
a
center
at
a;
b
if
and
only
if
P
and
Q
satisfy
(4).
A
special
case
of
this
result
for
p(x)
=
x
has
been
obtained
by
Yang
Lijun
and
Tang
Yun.
For
P
denite
the
Moment
Bautin
index
N
=
N(P;
d;
a;
b)
is
the
minimal
number
of
the
moments
mk
whose
vanishing
implies
(4)
for
any
q
of
degree
at
most
d.
This
number
can
be
explicitly
computed
(or
bounded).
Theorem
2.
Let
p
=
P
0
be
xed,
with
P
denite.
The
local
Bautin
ideal
I
of
Abel
equation
(2)
on
a;
b
(generated
on
the
ball
B

Vd
by
all
the
Taylor
coeôcients
v
k
(q)
of
the
Poincar e
rst
return
mapping
of
(2))
coincides
with
the
ideal
J
generated
by
the
moments
mi(q);
i
=
0;
1;
:
:
:
;N.
The
Bautin
index
of
(2)(i.e.
the
minimal
number
of
v
k
generating
I)
is
equal
to
N.
This
implies
a
bound
on
the
number
of
\small"
periodic
solutions
of
(2):
Corollary
1.
There
is
ô
=
ô(P;
d;
a;
b)
>
0
such
that
for
any
q
with
k
q
k

the
number
of
solutions
y
of
the
Abel
equation
(2.1)
satisfying
y(a)
=
y(b)
and
jy(a)j

ô
does
not
exceed
N(P;
d;
a;
b).
The
momentsmk(q)
form
the
linear
parts
of
the
Center
equations
v
k
(q)
=
0,
k
=
2;
:
:
:
.
As
Theorems
1
and
2
show,
their
structure
essentially
deter-
mines
the
structure
of
the
Center
equations
near
the
origin
in
Vd
.
It
turns
out
that
the
same
is
true
\at
innity"
(i.e.
in
a
neighborhood
of
the
in-
nite
hyperplane
in
the
projectivization
PVd
of
the
space
Vd
).
Combining
this
fact
with
the
local
information
above
we
can
show
in
many
cases
that
globally
the
set
of
q
=
Q
0
for
which
Abel
equation
(2)
has
a
center
consists
of
those
q
for
which
P;Q
satisfy
Composition
condition
(4)
and
possibly
of
a
nite
number
of
additional
points
q
j
.
43

The
Bogdanov-Takens
singularity
Henryk
_
Zo
l
adek
I
will
present
some
results
about
clasication
of
germs
of
vector
elds
in
the
complex
plane
with
singular
point
having
nilpotent
linear
part.
Firstly,
the
classical
Takens
normal
form
can
be
obtained
in
an
analytic
way.
Recall
that
the
Takens
form
is
not
the
nal
one,
when
considering
it
from
the
formal
orbital
equivalence
point
of
view.
I
will
present
a
complete
formal
orbital
classication
of
such
singualarities.
These
results
were
obtained
together
with
Ewa
Str o_
zyna.
44