Attractors, Foliations and Limit Cycles
International conference
dedicated to Yulij Ilyashenko’s 70th birthday
http://aflc.dyn-sys.org
Independent University of Moscow, Russia
January 13–17, 2014
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The conference is organized by
∙ Independent University of Moscow;
– Laboratoire J.-V.Poncelet;
∙ Higher School of Economics;
– Laboratory of algebraic geometry and its applications;
∙ Steklov Institute of Russian Academy of Sciences.
The scientific committee
∙ Dmitry Anosov (co-chair);
∙ Yakov Sinai (co-chair);
∙ Aleksander Bufetov;
∙ Sergey Yakovenko.
The organizing committee
∙ Alexey Glutsyuk;
∙ Anton Gorodetski;
∙ Alexey Klimenko;
∙ Yury Kudryashov;
∙ Olga Romaskevich;
∙ Maria Saprykina;
∙ Ilya Schurov (coordinator);
∙ Michael Tsfasman.
Schedule
Monday, January 13
10:00–10:45: Registration
10:50–11:00: Opening ceremony
11:00–11:45: Valerii Kozlov : Euler–Jacobi–Li Theorem
11:45–12:00: Coffee break
12:00–12:45: Robert Roussarie: Transitory canard cycles . . . . . . . . . . . . . . . 20
12:50–13:35: Vladlen Timorin: Smart criticality for cubic laminations . 21
13:35–15:30: Lunch
15:30–16:15: Christiane Rousseau: Modulus of analytic classification of unfoldings of non resonant irregular singularities of linear differential systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
16:20–17:05: Dmitry Turaev : Arnold Diffusion in a priori chaotic case . .22
17:05–17:30: Coffee break
17:30–18:15: Pierre Dehornoy: Which geodesic flows are left-handed? . . . 8
18:20–19:20: Welcome party
19:30–21:30: Concert by Ekaterina Antokolskaya: Johann Sebastian Bach
Suites for solo cello Nos. 1, 2, 5 and 6
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Tuesday, January 14
10:00–10:45: Lorenzo J. Díaz : Constructing ergodic non-hyperbolic measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
10:50–11:35: Vyacheslav Grines: On dynamics of diffeomorphisms with codimension one hyperbolic attractors and repellers . . . . . . . . . . . . . . . . . . . . 10
11:35–12:00: Coffee break
12:00–12:45: Vadim Kaloshin: Arnold diffusion via invariant cylinders and
Mather variational method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12
12:50–13:35: Konstantin Khanin: On Rigidity for Cyclic Nonlinear Interval
Exchange Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
13:35–15:30: Lunch
15:30–16:15: Grigori Olshanski : Continuous Young tableaux and Markov
dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
16:20–17:05: Frank Loray: Transversely projective foliations on projective
manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
17:05–17:30: Coffee break
17:30–19:00: Poster session . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Wednesday, January 15
09:00–09:45: Julio Rebelo: Quasi-invariant measures for non-discrete groups
on the circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
09:50–10:35: Sergey Voronin, Polina Fomina: Functional invariants of germs
of hyperbolic maps with resonances of Siegel’s type . . . . . . . . . . . . . . . . 22
10:35–11:00: Coffee break
11:00–11:45: Dmitry Treschev : On a class of locally integrable billiards 22
11:50–12:35: Misha Lyubich: Dynamics of dissipative polynomial automorphisms of C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
12:35–14:20: Lunch
14:20–17:00: Excursion to Kremlin (meeting at IUM).
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Thursday, January 16
10:00–10:45: Sergey Zelik : Inertial manifolds for 1D reaction-diffusion-advection problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24
10:50–11:35: Stefano Luzzatto: SRB measures for partially hyperbolic systems whose central direction is weakly expanding . . . . . . . . . . . . . . . . . . 14
11:35–12:00: Coffee break
12:00–12:45: Yanqi Qiu: Scaling limits for Christoffel-Darboux kernels associated with some modified Jacobi’s orthogonal ensembles . . . . . . . . . 18
12:50–13:35: Sergei Kuksin: Averaging for Hamiltonian PDE with resonances and the weak turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13
13:35–15:45: Lunch
15:45–17:30: Yulij Ilyashenko: Works of Yu. Ilyashenko in Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
17:30–18:20: Coffee break
18:20–19:20: Grigory Polotovskiy: Nizhni Novgorod mathematician Artemiy
Grigorievich Mayer and his course of the history of mathematics . . . 18
19:30–21:30: Banquet
Friday, January 17
10:00–10:45: Dmitry Novikov : Multiplicities of non-isolated intersections of
Noetherian functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
10:50–11:35: Alexander Prikhodko: Several dynamical system constructions
around Salem-Schaeffer measures and effects of spectral self-similarity
11:35–12:00: Coffee break
12:00–12:45: Olga Pochinka: On a Palis-Pugh’s problem and separation of
global attractor and repeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
12:50–13:35: Kristian Bjerklöv : Dynamics of a class of quasi-periodic Schrödinger cocycles
Talks
Which geodesic flows are left-handed?
Pierre Dehornoy (pierre.dehornoy@math.unibe.ch)
Institut Fourier, UJF Grenoble, France, UMR 5582
Monday, January 13, 17:30–18:15
Left-handedness is a topological property of 3-dimensional flows that implies
the existence of numerous sections for the flow, hence of many ways of seing the
flow as a suspension. A construction of Birkhoff suggests that geodesic flows
on unit tangent bundles to surfaces have good chances of being left-handed.
We will see that it is indeed the case for hyperbolic triangular orbifolds, but
not in general.
Constructing ergodic non-hyperbolic measures
Lorenzo J. Díaz (lodiaz@mat.puc-rio.br)
PUC-Rio, Brazil
Tuesday, January 14, 10:00–10:45
In the paper [4] was proposed a method for the construction of non-hyperbolic
ergodic measures for some skew-products. This method was used and adapted
to new settings opening the doors of a new line of research. Applications
include open sets of diffeomorphisms [5], generic dynamics, [2, 3], and skew
products with multiple zero exponents, [1]. Motivated and inspired by the
method by Gorodetski-Ilyashenko-Kleptsyn-Nalsky, we introduce an object
called a blender with flip-flops that allows us to construct a new class of
non-hyperbolic ergodic measures for an open and dense subsets of the class
of 𝐶 1 -robustly transitive non-hyperbolic diffeomorphisms. We present this
method and discuss its pros and cons.
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Håkan Eliasson
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Bibliography
[1]
J. Bochi, C. Bonatti, and L. Díaz. “Robust vanishing of all Lyapunov
exponents for iterated function systems”. to appear in Math. Zeit.
[2]
C. Bonatti, L. J. Díaz, and A. Gorodetski. “Non-hyperbolic ergodic measures with large support”. In: Nonlinearity 23.3 (2010), pp. 687–705. url:
http://stacks.iop.org/0951-7715/23/i=3/a=015.
[3]
L. J. Díaz and A. Gorodetski. “Non-hyperbolic ergodic measures for nonhyperbolic homoclinic classes”. In: Ergodic Theory Dynam. Systems 29.5
(Oct. 2009), pp. 1479–1513.
[4]
A. S. Gorodetski et al. “Nonremovability of zero Lyapunov exponents”.
In: Funct. Anal. Appl. 39.1 (2005), pp. 21–30.
[5]
V. A. Kleptsyn and M. B. Nalsky. “Stability of the existence of nonhyperbolic measures for 𝐶 1 -diffeomorphisms”. In: Funct. Anal. Appl. 4 (41
2007), pp. 271–283.
Analytic Diophantine KAM-tori are never isolated
Håkan Eliasson (hakan.eliasson@math.jussieu.fr)
IMJ-PRG, Université Paris Diderot, France
Talk is cancelled
A Diophantine KAM-torus for a nearly integrable Hamiltonian system is an
invariant Lagrangian torus whose induced dynamics is conjugated to a translational flow with a Diophantine frequency (=translation) vector. In a joint
work with B. Fayad and R. Krikorian we have proven that, in an analytic
Hamiltonian system, any analytic Diophantine KAM-torus is always accumulated by other analytic Diophantine KAM-tori with, in general, different
frequency vectors. A similar result holds for an analytic Hamiltonian system
near an irrational elliptic equilibrium: the equilibrium is always accumulated
by other analytic Diophantine KAM-tori. We don’t know if the corresponding
result holds in the 𝒞 ∞ category.
Often the set of KAM-tori not only accumulates the given KAM-torus (or
elliptic equilibrium) but is also of positive Lebesgue measure, but we are not
able to prove that this is always the case (a question raised by M. Herman in
his ICM lecture in Berlin 1996). We know however that such a stronger result
cannot hold in the 𝒞 ∞ category: there are examples of systems whose set of
KAM-tori has zero Lebesgue measure.
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On dynamics of diffeomorphisms with codimension one hyperbolic attractors and repellers
Vyacheslav Grines (vgrines@yandex.ru)
Nizhnii Novgorod State University
Tuesday, January 14, 10:50–11:35
We present an overview of results and problems on dynamics of diffeomorphisms given on closed orientable manifolds 𝑀 𝑛 , (𝑛 > 2) whose nonwandering set contains hyperbolic attractors and repellers with topological dimension
𝑛 − 1. We take attention on topological classification of important classes of
such diffeomorphisms and on interrelation between dynamics and topological
structure of ambient manifolds. For an introduction with the topic see for
example [1, 2, 3, 4].
The research is supported by RFBR grants No 12-01-00672-a and 13-0112452-ofi-m.
Bibliography
[1]
V. Grines. “On topological classification of 𝐴-diffeomorphisms of surfaces”. In: J. of Dynam. and Control Systems 6.1 (2000), pp. 97–126.
[2]
V. Z. Grines and Yu. A. Levchenko. “Topological classification of diffeomorphisms on 3-manifolds with surface attractors and repellers”. In:
Doklady Mathematics 86.3 (2012), pp. 747–749.
[3]
V. Z. Grines and O. V. Pochinka. Introduction to the topological classification of cascades on manifolds of dimension two and three. M.; Izhevsk:
Institute of Computer Science: Regular and Chaotic Dynamics, 2011.
[4]
V. Z. Grines and E. V Zhuzhoma. “Dynamical Systems with Nontrivially
Recurrent Invariant Manifolds”. In: Springer Proceedings in Mathematics.
Vol. 1: Dynamics, Games and Science I. 2011, pp. 421–470.
Pascal Hubert
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Some remarks on the Lyapunov exponents of the
Kontsevich–Zorich cocycle
Pascal Hubert (pascal.hubert@univ-amu.fr)
Laboratoire d’Analyse, Topologie et Probabilités, Université d’Aix-Marseille
Talk is cancelled
In this talk, we will recall the definition of the Kontsevich-Zorich cocycle.
We will also discuss why it is an important tool for the dynamics of interval
exchange transformations, polygonal billiards and translation surfaces. We
will explain how to find subspaces of moduli spaces of quadratic differentials
with zero Lyapunov exponents (this part is a joint work with Julien Grivaux).
Works of Yu. Ilyashenko in Dynamical Systems
Yulij Ilyashenko (yulijs@gmail.com)
IUM, HSE, MSU, Steklov Inst. and Cornell University
Thursday, January 16, 15:45–17:30
The talk will consists of two part: real and complex ones.
Part 1:
∙ Normal forms of local families.
∙ Nonlocal bifurcations.
∙ Skew products.
∙ Attractors
Part 2:
∙ Petrovski-Landis strategy.
∙ Zeros of Abelian integrals.
∙ Polynomial foliations in the complex plane.
∙ Nonlinear Stokes phenomena.
∙ Limit cycles: finiteness theorems and Hilbert-Arnold problem.
∙ Persistence of geometric objects related to polynomial dynamics.
Not only the results, but also the relations between them will be discussed.
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Arnold diffusion via invariant cylinders and
Mather variational method
Vadim Kaloshin (vadim.kaloshin@gmail.com)
University of Maryland
Tuesday, January 14, 12:00–12:45
The famous ergodic hypothesis claims that a typical Hamiltonian dynamics
on a typical energy surface is ergodic. However, KAM theory disproves this.
It establishes a persistent set of positive measure of invariant KAM tori. The
(weaker) quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says
that a typical Hamiltonian dynamics on a typical energy surface has a dense
orbit. This question is wide open. In early 60th Arnold constructed an example of instabilities for a nearly integrable Hamiltonian of dimension 𝑛 > 2
and conjectured that this is a generic phenomenon, nowadays, called Arnold
diffusion. In the last two decades a variety of powerful techniques to attack
this problem were developed. In particular, Mather discovered a large class of
invariant sets and a delicate variational technique to shadow them. In a series
of preprints: one joint with P. Bernard, K. Zhang and two with K. Zhang we
prove Arnold’s conjecture in dimension 𝑛 = 3.
On Rigidity for Cyclic Nonlinear
Interval Exchange Transformations
Konstantin Khanin (khanin@math.toronto.edu)
University of Toronto
Tuesday, January 14, 12:50–13:35
We’ll report on recent progress in rigidity theory for nonlinear interval exchange transformations corresponding to cyclic permutations. Such maps can
be viewed as circle homeomorphisms with multiple break points. We shall
discuss both recent results on renormalizations of such maps in case of one
break point (joint with S. Kocic and A. Teplinsky), and extension to the
multiple-break setting (based on work in progress with A. Teplinsky).
Sergei Kuksin
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Averaging for Hamiltonian PDE with resonances
and the weak turbulence
Sergei Kuksin (kuksin@math.jussieu.fr)
Université Paris-Diderot (Paris 7)
Thursday, January 16, 12:50–13:35
My talk will be dedicated to the study of long-time behaviour of the system,
given by a hamiltonian PDE, perturbed by vanishing random force and dissipation, which are scaled in such a way that the limiting solutions are small,
but not too small. The scaling agrees with that used in the theory of weak
turbulence. The limiting behaviour turns out to be non-trivial and non-linear.
It is described in terms of resonances in the spectrum of small oscillations in
the hamiltonian PDE, and of its non-linear part.
Transversely projective foliations on projective
manifolds
Frank Loray (frank.loray@univ-rennes1.fr)
CNRS / Université de Rennes 1
Tuesday, January 14, 16:20–17:05
This is a joint work with J.-V. Pereira and F. Touzet. We study singular holomorphic foliations of codimension one on compact complex manifolds. Among
them, transversely projective foliations play a special role: they are integrable
in the sense of Malgrange. Examples are foliations defined by a merormorphic
closed 1-form, but also Riccati foliations. Other rigid examples come from
arithmetic, such as Hilbert modular foliations. Assuming the ambient manifold 𝑀 is projective, we prove that these are essentially the only examples.
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SRB measures for partially hyperbolic systems
whose central direction is weakly expanding
Stefano Luzzatto (luzzatto@ictp.it)
Abdus Salam International Center for Theoretical Physics (ICTP), Trieste,
Italy
Thursday, January 16, 10:50–11:35
1
diffeomorphisms of compact Riemannian
We consider partially hyperbolic 𝐶+
manifolds of arbitrary dimension which admit a partially hyperbolic tangent
bundle decomposition 𝐸 𝑠 ⊕ 𝐸 𝑐𝑢 . Assuming the existence of a set of positive
Lebesgue measure on which 𝑓 satisfies a weak nonuniform expansivity assumption in the centre unstable direction, we prove that there exists at most a finite
number of transitive attractors each of which supports an SRB measure. As
part of our argument, we prove that each attractor admits a Gibbs-MarkovYoung geometric structure with integrable return times. We also characterize
in this setting SRB measures which are liftable to Gibbs-Markov-Young structures.
This is joint work with J. Alves, C. Dias and V. Pinheiro.
Dynamics of dissipative polynomial
automorphisms of C2
Misha Lyubich (mlyubich@math.sunysb.edu)
Stony Brook University
Wednesday, January 15, 11:50–12:35
Two-dimensional complex dynamics displays a number of phenomena that are
not observable in dimension one. However, if 𝑓 is moderately dissipative then
there are more similarities between the two fields. For instance, dynamics on
an invariant Fatou component admits a nearly complete description:
Theorem 1. Any invariant component 𝐷 of the Fatou set is either an attracting basin or parabolic basin, or the basin of a rotation domain (Siegel disk
or Herman ring).
Theorem 2. In the first two cases, 𝐷 contains a “critical point”.
In complex and real one-dimensional world, structurally stable maps are
dense. In dimension two this fails because of the Newhouse phenomenon
Dmitry Novikov
15
caused by homoclinic tangencies. Palis conjectured that in the real twodimensional case this is the only reason for failure. We prove a complex
version of this conjecture:
Theorem 3. Any moderately dissipative polynomial automorphism of C2 is
either “weakly stable” or it can be approximated by a map with homoclinic
tangency.
Theorem 1 is joint with Han Peters, the rest is joint with Romain Dujardin.
Multiplicities of non-isolated intersections of Noetherian functions
Dmitry Novikov (dmitry.novikov@weizmann.ac.il)
Weizmann Institute of Science
Friday, January 17, 10:00–10:45
A subring 𝐾 ⊂ 𝒪(𝑈 ), 𝑈 ⊂ C𝑛 is called a ring of Noetherian functions if it
contains the polynomial ring C[𝑥1 , . . . , 𝑥𝑛 ], is finitely generated as a module
𝜕
, 𝑖 = 1, . . . , 𝑛.
over it, and it is closed under the partial derivatives 𝜕𝑥
𝑖
Noetherian functions have natural complexity parameters as 𝑛 and the
degrees of polynomials used in their definition. A natural goal is to give an
upper bound for a multiplicity of a common zero of 𝑛 Noeterian functions in
terms of these parameters. For an isolated zero, this was done by Gabrielov
and Khovanskii [1]. We extend this result by giving an upper bound for a
suitably defined deformation multiplicity of a common non-isolated zero.
This is a joint work with Gal Binyamini, University of Toronto.
Bibliography
[1]
A. Gabrielov and A. Khovanskii. “Multiplicity of a Noetherian intersection”. In: Geometry of differential equations. Vol. 186. Amer. Math. Soc.
Transl. 2. Amer. Math. Soc., Providence, RI, 1998, pp. 119–130.
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Continuous Young tableaux and Markov dynamics
Grigori Olshanski (olsh2007@gmail.com)
Institute for Information Transmission Problems, Independent University of
Moscow, National Research University — Higher School of Economics
Tuesday, January 14, 15:30–16:15
Young tableaux are fundamental combinatorial objects, widely used in algebraic combinatorics and representation theory. They were discovered by
Alfred Young more than a century ago. Quite recently, Alexei Borodin invented a continuous analog of Young tableaux. I will explain what are Gibbs
measures on the space of continuous Young tableaux and how to construct a
related model of Markov dynamics in 1 + 1 dimensions with infinitely many
interacting particles.
The talk is based on recent papers [1, 2, 3].
Bibliography
[1]
A. Borodin and G. Olshanski. An interacting particle process related to
Young tableaux. Oct. 2013. arXiv: 1303.2795.
[2]
A. Borodin and G. Olshanski. “Markov dynamics on the Thoma cone:
a model of time-dependent determinantal processes with infinitely many
particles”. In: Electronic Journal of Probability 18.75 (2013), pp. 1–43.
arXiv: 1303.2794.
[3]
A. Borodin and G. Olshanski. “The Young bouquet and its boundary”.
In: Moscow Mathematical Journal 13.2 (2013), pp. 193–232. arXiv: 1110.
4458.
On a Palis-Pugh’s problem and separation
of global attractor and repeller
Olga Pochinka (olga-pochinka@yandex.ru)
Nizhny Novgorod State University
Friday, January 17, 12:00–12:45
These results were obtained in collaboration with V. Grines [2] and related to
a problem of Palis-Pugh [5] on the existence of an arc with a finite or countable
Olga Pochinka
17
set of bifurcations, connecting two Morse-Smale systems on a smooth closed
manifold 𝑀 𝑛 . S. Newhouse and M. Peixoto [4] showed that for flows such
arc exists for any 𝑛 and moreover it is simple. However, there are isotopic
diffeomorphisms that can not be connected by a simple arc. For 𝑛 = 1 it is
connected with Poincare rotation number, and for 𝑛 = 2 is connected with the
possibility of the existence of periodic points with different periods and heteroclinic orbits [1, 3]. We show that there is a new obstruction for the existence
of a simple arc in dimension 𝑛 = 3 which is associated with the wild embedding of the separatrices of the saddle points. Moreover we prove that 2-sphere
separation of global attractor and repeller for a Morse-Smale diffeomorphism
without heteroclinic intersections gives necessary and sufficient conditions to
connect it with the diffeomorphism “source-sink” by a simple arc.
The research is supported by RFBR grants No 12-01-00672-a and 13-0112452-ofi-m, and by The Ministry of education and science of Russia 2012-2014
grant No 1.1907.2011.
Bibliography
[1]
P. R. Blanchard. “Invariants of the NPT isotopy classes of Morse-Smale
diffeomorphisms of surfaces”. In: Duke Math. J. 47.1 (1980), pp. 33–46.
[2]
V. Grines and O. Pochinka. “On the simple isotopy class of a source-sink
diffeomorphism on the 3-sphere”. In: Mathematical Notes 94.5-6 (2013),
pp. 862–875.
[3]
S. Matsumoto. “There are two isotopic Morse-Smale diffeomorphism which
can not be joined by simple arcs”. In: Invent. Math. 51.1 (1979), pp. 1–7.
[4]
S. Newhouse and M. M. Peixoto. “There is a simple arc joining any two
Morse-Smale flows”. In: Asterisque (31 1976), pp. 15–41.
[5]
J. Palis and C. Pugh. “Fifty problems in dynamical systems/”. In: Dynamical Systems–Warwick 1974. Ed. by A. Manning. Vol. 468. Lecture
Notes in Math. 1975, pp. 345–353.
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Nizhni Novgorod mathematician Artemiy Grigorievich Mayer and his course of the history of
mathematics
Grigory Polotovskiy (polotovsky@gmail.com)
Lobachevsky State University of Nizhny Novgorod
Thursday, January 16, 18:20–19:20
Professor Artemiy Grigoryevich Mayer was one of the closest employees of
academician A. A. Andronov. He is the author of a number of classical results
(about dynamic systems on surfaces, about the central trajectories in Birkgof’s
problem, etc.). The short review of scientific results of Mayer will be given and
it will be told about the little-known facts of his biography, including history
of persecution of A. G. Mayer in 1950 by reason of his “ideological mistakes
in a course of history of mathematics”.
Scaling limits for Christoffel-Darboux kernels associated with some modified Jacobi’s orthogonal
ensembles
Yanqi Qiu (yqi.qiu@gmail.com)
I2M — Université Aix-Marseille
Thursday, January 16, 12:00–12:45
We investigate the scaling limits for the modified Jacobi orthogonal polynomial ensembles. Explicit scaling limits for the Christoffel-Darboux kernels are
obtained, the limit kernels are related to the ergodic decomposition of infinite
Pickrell measures.
This is a joint work with Alexander Bufetov.
The research is supported by A*MIDEX grant.
Jean-Pierre Ramis
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Linear and non-linear Galoisian obstructions to
the integrability of dynamical systems
Jean-Pierre Ramis (jean-pierre.ramis@math.univ-toulouse.fr)
Mathématiques Fondamentales, Institut de Mathématiques de Toulouse
Talk is cancelled
Nearly 15 years ago Jean-Pierre Ramis and Juan Morales-Ruiz discovered
some Galoisian obstructions to the integrability of Hamiltonian systems, generalizing the monodromy obstructions of S.L. Ziglin. At the beginning only
linear Galois differential theory and only the first variational equations were
used and the notion of integrability was the classical one (Liouville integrability). Later the results were extended to higher variational equations, nonlinear Galois theory (Malgrange version) was used and efficient algorithms
were discovered and implemented on computer algebra systems. The case of
non Hamiltonian systems and discrete dynamical systems were investigated
and various notions of integrability were used. Many authors developped a
lot of applications (more than 120 papers).
In the lecture I will give a short history of the problem, explain basics on
the necessary tools and I will describe the state of the art.
Quasi-invariant measures for non-discrete groups
on the circle
Julio Rebelo (rebelo@math.univ-toulouse.fr)
Université Paul Sabatier, Toulouse
Wednesday, January 15, 09:00–09:45
We shall discuss the structure of quasi-invariant measures for finitely generated
non-discrete groups of analytic diffeomorphisms of the circle. Though we
focus on analytic diffeomorphisms, suitable versions of most statements hold
for smooth diffeomorphisms as well and, besides, some higher dimensional
extensions are also possible. The purpose will be to provide some criteria
to detect absolutely continuous quasi-invariant measures as well as to obtain
additional information concerning the fractal nature of singular quasi-invariant
measures. Some applications to stationary measures will also be given.
20
Transitory canard cycles
Robert Roussarie (roussari@u-bourgogne.fr)
Institut de Mathématiques de Bourgogne, UMR 5584 du CNRS
Monday, January 13, 12:00–12:45
Transitory canard cycles appear at the boundary of open layers of generic
canard cycles. Peter De Maesschalk, Freddy Dumortier and myself have begun
recently the study of these canard cycles. We consider two simple possible
cases: slow-fast canard cycle and fast-fast canard cycle. In the first case
the cycle passes from the slow curve to a fast orbit and in the second case
the cycle passes from a fast orbit to another fast orbit, through a transitory
contact point.
The slow divergence integral, which is a function of the limit layer value
corresponding to the canard cycle, controls bifurcations of generic canard cycles in limit cycles. But, as this function is non differentiable at the limit
layer value corresponding to a transitory canard cycle, it is not possible to
deduce even its cyclicity (i.e. the upper bound of the number of bifurcating
limit cycles) by a direct simple argument, as it was possible for generic canard
cycles.
The idea is to blow-up the system at the transitory contact point. The
transitory canard cycle is then replaced by a 1-parameter family of secondary
canard cycles and we study the difference map (between the left and the right
transition) on a section transverse to the critical locus of the blowing-up.
In the two cases, the difference map is studied on different charts and the
difficulties are more or less serious, depending on the case and also on the
chart.
In the slow-fast case, the map along the fast side dominates the map along
the slow side. In this case, the codimension of the slow divergence integral is
equal to one or two and the cyclicity of the canard cycle is equal to two or
three.
In the fast-fast case the two sides of the difference map are completely
similar in a central chart. We obtain that the cyclicity in this chart is two as a
consequence of the concavity of the transition time at infinity for the blown-up
vector field: 𝑥˙ = 𝑦 − 𝑥2 , 𝑦˙ = 1, deduced from properties of Airy functions.
Moreover in this fast-fast case the codimension of the slow divergence integral
can take any value 𝑘 ∈ N. If 𝑘 > 2, the cyclicity of the transitory canard cycle
is less than 𝑘 + 4.
Christiane Rousseau
21
Modulus of analytic classification of unfoldings
of non resonant irregular singularities of linear
differential systems
Christiane Rousseau (rousseac@dms.umontreal.ca)
University of Montreal
Monday, January 13, 15:30–16:15
In this lecture we will present a complete modulus of analytic classification
under analytic equivalence for germs of generic unfoldings of linear differential
systems 𝑥𝑘+1 𝑦 ′ = 𝐴(𝑥) with a non resonant irregular singularity of Poincaré
rank k. The formal part of the modulus consists essentially in the eigenvalues
of the singular points, and the analytic part is given by the unfolding of the
Stokes matrices. To define the modulus, one constructs sectors in 𝑥-space
and on each of them a change of coordinates to the normal form, which is
unique up to composition with diagonal transformations. The Stokes matrices
are obtained from comparing these normalizing changes of coordinates. The
construction is performed on 𝐶(𝑘) sectoral domains in parameter space, which
cover the open set of parameter values for which the singular points are distinct
(𝐶(𝑘) is the 𝑘-th Catalan number). This study sheds a new light on the
meaning of the Stokes matrices in the limit situation.
Smart criticality for cubic laminations
Vladlen Timorin (vtimorin@hse.ru)
Mathematical department of the NRU “Higher School of Economics”
Monday, January 13, 12:50–13:35
Invariant laminations (collections of chords in the disk satisfying certain properties) were introduced by W. Thurston as models for the topological dynamics of polynomials on their Julia sets (provided that the latter are locally
connected). Chords of a lamination are called leaves. A quadratic invariant
lamination has one or two longest leaves, called major(s). The image of a
longest leaf is called a minor. A crucial property is that distinct minors do
not cross inside the unit disk; this gives rise to a combinatorial model of the
Mandelbrot set. We will discuss some partial generalizations of this result to
invariant laminations of higher degree.
The talk is based on a joint project with A. Blokh, L. Oversteegen and R.
Ptacek.
22
On a class of locally integrable billiards
Dmitry Treschev (treschev@mi.ras.ru)
Steklov Mathematical Institute and Moscow State University
Wednesday, January 15, 11:00–11:45
Can a billiard map be locally conjugated to a rigid rotation? We prove that
the answer to this question is positive in the category of formal series. We also
present numerical evidence that for “good” rotation angles the answer is also
positive. We present other numerical results. Some of them look surprising.
Arnold Diffusion in a priori chaotic case
Dmitry Turaev (dturaev@imperial.ac.uk)
Imperial College, London
Monday, January 13, 16:20–17:05
Let a real-analytic symplectic map have a two-dimensional normally-hyperbolic cylinder 𝐴 such that the restriction of the map on the cylinder has a twist
property. Let the stable and unstable manifolds of the cylinder intersect transversely at some homoclinic cylinder 𝐵, and let the cylinder 𝐵 be transverse
to the strong stable and strong unstable foliations. The homoclinic channel
is a small neighbourhood of the union of the cylinder 𝐴 and the orbit of the
cylinder 𝐵. We show that generically (in the real-analytic category) there
always exist orbits in the channel which deviate from the initial condition
unboundedly.
Functional invariants of germs of hyperbolic maps
with resonances of Siegel’s type
Sergey Voronin (voron@csu.ru), Polina Fomina (fominapa@csu.ru)
Chelyabinsk State University, Russia
Wednesday, January 15, 09:50–10:35
A germ of holomorphic map 𝐹 : (C2 , 0) → (C2 , 0) is hyperbolic with Siegel’s
type resonances if its multiplicators are hyperbolic and its product is 1. For
formally linearizable germs of such type formal and analytic classifications
are coincide and for nonlinearizable are not [1]. We will show that analytic
Sergey Voronin, Polina Fomina
23
classification of germs with nonlinear formal normal form has functional invariants. For simplicity we describe its for one partial case (a namely, for
the formal equivalence class ℱ of the 1-time map 𝐹0 = 𝑔𝑣1 of the vector field
𝜕
𝜕
+ 𝑦(−1 + 𝑥𝑦) 𝜕𝑦
).
𝑣 = 𝑥 𝜕𝑥
It is not difficult to verify that a germ of ℱ by a holomorphic change of
variables can be reduced to the form
𝐹 : (𝑥, 𝑦) ↦→ 𝐹0 (𝑥, 𝑦) + (𝑜(𝑥3 ), 𝑜(𝑦 3 ))
Let ℱ0 be a class of germs of this form. Two germs of ℱ0 are strongly equivalent, if one of them can be transformed to another by a local change of
variables of type (𝑥, 𝑦) ↦→ (𝑥 + 𝑜(𝑥2 ), 𝑦 + 𝑜(𝑦 2 )), 𝑥, 𝑦 → 0.
Let us consider the class ℳ consisting of collections (Φ𝑖± , Ψ± , 𝜙± , 𝜓± ),
where Φ± and Ψ± (𝜙± and 𝜓± ) are germs of holomorphic functions in (C2 , 0)
(correspondently, in (C, 0)).
Theorem 1. The space ℳ is a moduli space of the strong analytic classification of germs of ℱ0 .
Note that a germ 𝐹 ∈ ℱ0 is embeddable in a flow (i.e. 𝐹 is 1-time shift
for a holomorphic vector field 𝑣) iff the components Φ± , Ψ± of its moduli
vanish. Moreover then the pair (𝜙+ , 𝜙− ) is a Martinet-Ramis moduli (of
orbital classification) [2] of the corresponding vector field 𝑣, and the pair
(𝜓+ , 𝜓− ) is an additional moduli (of non-orbital classification) [3] of 𝑣.
The research is supported by RFBR grant No 13-01-00512a.
Bibliography
[1]
A. D. Bruno. “Analytic form of differential equations”. In: Tr. Mosk. Mat.
Obs. 26 (1972), pp. 199–239.
[2]
J. Martinet and J. Ramis. “Classification analytique des équations différentielles non linéaires resonnantes du premier ordre”. In: Ann. Sci.
École norm. supér. 16.4 (1983), pp. 571–621. issn: 0012-9593. url: http:
//www.numdam.org/item?id=ASENS_1983_4_16_4_571_0.
[3]
S. Voronin and A. Grinchii. “An analytic classification of saddle resonant
singular points of holomorfic vector fields in the complex plane”. In: J.
Dynam. Control Systems 2.1 (1996), pp. 21–53.
24
Inertial manifolds for 1D reaction-diffusion-advection problems
Sergey Zelik (s.zelik@surrey.ac.uk)
University of Surrey, UK
Thursday, January 16, 10:00–10:45
The results concerning the existence of inertial manifolds for one dimensional
reaction-diffusion-advection systems on the interval endowed by the Dirichlet
boundary conditions will be presented.
Noted that the spectral gap condition is not satisfied for such equations, so
the inertial manifold cannot be constructed in a straightforward way. However,
with the help of a non-trivial (non-local in space) change of variables, these
equations can be reduced to a class of abstract semilinear parabolic equations
where the spectral gap condition is satisfied. The related open problems will
be also discussed.
Posters
Necessary and sufficient conditions for the topological conjugacy of structurally stable diffeomorphisms of 𝑀 3 with two-dimensional surface basic
sets
Yulia Levchenko (ulev4enko@gmail.com)
Nizhnii Novgorod State University
The report is devoted to consideration of diffeomorphisms on 3-manifolds
which satisfy S. Smale’s axiom 𝐴 under assuming that non-wandering set
consists of two-dimensional surface basic sets. It was investigated the interrelation between the dynamics of such a diffeomorphism and the topology of
the ambient manifold. Moreover under certain restrictions on the asymptotic
behavior of two-dimensional invariant manifolds of points from basic sets necessary and sufficient conditions of topological conjugacy of structurally stable
diffeomorphisms from considered class were obtained (see [1]). The author appreciates V.Z. Grines for the formulation of the problem and O.V. Pochinka
for useful discussions.
The research is supported by RFBR grants No 12-01-00672-a and 13-0112452-ofi-m.
Bibliography
[1]
V. Z. Grines and Yu. A. Levchenko. “Topological classification of diffeomorphisms on 3-manifolds with surface attractors and repellers”. In:
Doklady Mathematics 86.3 (2012), pp. 747–749.
25
26
Renormalization and Universality of Rotation
Sets for Lorenz Type Systems
Mikhail Malkin (malkin@unn.ru)
Lobachevsky State University of Nizhni Novgorod
We consider rotation sets in one-parameter families of multidimensional perturbations of one-dimensional Lorenz-type maps. More precisely, let
Φ𝜆 (𝑦𝑛 , 𝑦𝑛+1 , . . . , 𝑦𝑛+𝑚 ) = 0,
𝑛∈Z
be a difference equation of order 𝑚 with parameter 𝜆. It is assumed that the
non-perturbed operator Φ𝜆0 depends on two variables only, i.e.,
Φ𝜆0 (𝑦0 , . . . , 𝑦𝑚 ) = 𝜓(𝑦𝑁 , 𝑦𝑀 ),
where 0 6 𝑁 , 𝑀 6 𝑚 and 𝜓 is a piecewise monotone piecewise 𝐶 2 -function.
It is also assumed that for the equation 𝜓(𝑥, 𝑦) = 0, there is a branch 𝑦 = 𝜙(𝑥)
which represent a one-dimensional Lorenz-type map. We prove results on continuous dependence of rotation sets with respect to both the initial Lorenztype maps and its renormalizations under multidimensional perturbations.
Numerical results show universality phenomena in bifurcations responsible for
birth of nontrivial rotation intervals of renormalized maps. Our technique is
based on approximations of chaotic orbits for perturbations of singular difference equations [1].
The research is supported by RFBR grants No 13-01-00589 and 12-0100672.
Bibliography
[1]
J. Juang, M.-C. Li, and M. Malkin. “Chaotic difference equations in two
variables and their multidimensional perturbations”. In: Nonlinearity 21.5
(2008), pp. 1019–1040.
Boundaries of stability in the space of germs
Natal’ya Medvedeva (medv@csu.ru)
Chelyabinsk State University
It is known that the monodromy map of a monodromic singular point has
a linear principal term of asymptotics ∆(𝜌) = 𝑐𝜌(1 + 𝑜(𝜌)). The equation
Tat’yana Mitryakova
27
𝑐 = 0 gives a boundary of stability in the corresponding class of monodromic
germs, defining by means of Newton diagramms. But if all the edges of all
Newton diagramms involving to the definition of the class, are even, then 𝑐 is
identically equal to zero. In this case it is necessary to find the second term of
asymptotics of the monodromy map. A number of formulas for the coefficient
of the second term of the asymptotics of the monodromy map are obtained.
One of obtained formulas allows to construct the boundary of stability in some
class of monodromic germs which is not a semianalytic subset of the space of
germs. This implies that the problem of stability of the singular point of the
vector field in the plain is not analytically solvable [1].
Bibliography
[1]
N. B. Medvedeva. “On analytic insolubility of the stability problem on
the plane”. In: Russian Math. Surveys 68.5 (2013), pp. 923–949.
Energy function for structurally stable cascades
of surfaces with non-trivial one-dimensional basic
sets
Tat’yana Mitryakova (tatiana.mitryakova@yandex.ru)
Lobachevsky State University of Nizhny Novgorod
The results were obtained in collaboration with V.Z. Grines and O.V. Pochinka.
Let 𝑆(𝑀 ) be the set of structurally stable diffeomorphisms 𝑓 : 𝑀 → 𝑀 of a
closed surface 𝑀 such that each non-trivial basic set of 𝑓 is one-dimensional
(attractor or repeller). Recall that Lyapunov function for a structurally stable
diffeomorphism 𝑓 : 𝑀 → 𝑀 is a continuous function which strictly decreases
along wandering trajectories and is constant on the basic sets. For a diffeomorphism 𝑓 ∈ 𝑆(𝑀 ) we construct a Lyapunov function 𝜙 such that it is a Morse
function out of non-trivial basic sets and its set of critical points coincides
with trivial basic sets. The existence of such function follows from topological classification of structurally stable diffeomorphisms on surface with onedimensional attractors and repellers [1] and construction of an energy function
for Morse-Smale diffeomorphism on a surface [2].
The research is supported by RFBR grant No 12-01-00672-a and by The
Ministry of education and science of Russia 2012-2014 grant No 1.1907.2011.
28
Bibliography
[1]
V. Grines. “On the topological classification of structurally stable diffeomorphisms of surfaces with one-dimensional attractors and repellers”. In:
Sb. Math. 188.4 (1997), pp. 537–569.
[2]
D. Pixton. “Wild unstable manifolds”. In: Topology 16.2 (1977), pp. 167–
172.
Utmost topological rigidity for generic quadratic
foliations on C𝑃 2
Valente Ramirez (valente@math.cornell.edu)
Department of Mathematics, Cornell University
In this poster we consider holomorphic foliations of C𝑃 2 which in a fixed
affine chart are induced by a quadratic vector field. In the generic case these
foliations have isolated singularities and an invariant line at infinity.
The object of this poster is to present the following result: In the generic
case two such foliations may be topologically equivalent if and only if they
are conjugate by an affine map on C2 . In fact, the analyitic conjugacy class
of the monodromy group at infinity is the modulus of both topological and
affine classification.
Topological rigidity of polynomial foliations was, until now, understood to
be a heuristic idea rather than a formal statement. The idea of topological
rigidity is that topological equivalence implies analytic equivalence. The first
rigidity property for polynomial foliations was discovered by Ilyashenko in [1]
and claims that two generic and topologically equivalent polynomial foliations
are affine equivalent provided they are close enough in the space of foliations
and the linking homeomorphism is close enough to the identity map. Our
new result shows that for quadratic foliations all hypothesis can be dropped.
This shows for the first time that the paradigm of topological rigidity of polynomial foliations may be formalized: if two generic quadratic foliations are
topologically equivalent then they are affine equivalent.
The research is supported by PAPIIT grants No IN103010 and IN102413.
Bibliography
[1]
Yu. S. Ilyashenko. “Topology of phase portraits of analytic differential
equations on a complex projective plane”. In: Trudy Sem. Petrovsk 4
(1978), pp. 83–136.
Evgenii Zhuzhoma
29
[2]
A. S. Pyartli. “Quadratic vector fields on C𝑃 2 with solvable monodromy
group at infinity”. In: Proceedings of the Steklov Institute of Mathematics
254.1 (2006), pp. 121–151.
[3]
V. Ramírez. “Strong topological invariance of the monodromy group at
infinity for quadratic vector fields”. Publicaciones Preliminares del Instituto de Matemáticas UNAM Id. Num. 906. 2011.
[4]
V. Ramírez. “The utmost rigidity property for generic quadratic foliations
on C𝑃 2 ”.
Solenoidal basic sets of Smale-Vietoris axiom A
diffeomorphisms
Evgenii Zhuzhoma (zhuzhoma@mail.ru)
Nizhny Novgorod State University, Russia
We introduce Smale-Vietoris diffeomorphisms that include the classical DEmappings with Smale solenoids. The main result is a correspondence between
basic sets of axiom A Smale-Vietoris diffeomorphism and the corresponding
nonsingular axiom A endomorphism. We construct two bifurcations between
these diffeomorphisms with different types of dynamics.
The research is supported by RFBR grants No 13-01-12452-ofi-m and 1201-00672-a. We would like to thank K. Kirsenko (musician and businessman)
for financial support.
Supporting organizations