Recent Research (arranges by topics):

 

 

                  Instabilities in Hamiltonian systems and Arnold diffusion

 

o    ``Geometric proofs of Mather's accelerating and connecting theorems’’, Topics in Dynamics and Ergodic Theory (eds. S. Bezuglyi and S. Kolyada), London Mathematical Society, Lecture Notes Series, Cambridge University Press, 2003, 81—106, [PDF]; 

 

o    ``Mather theory, weak KAM, and viscosity solutions of Hamilton—Jacobi  PDE's’’,  EQUADIFF 2003, 3948, World Sci. Publ., Hackensack, NJ, 2005, [PDF];

 

o    ``Instability of totally elliptic points of symplectic maps in dimension 4'', Instability of resonant totally elliptic points of symplectic maps in dimension 4, with J. Mather and E. Valdinoci,  Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes. II. Asterisque, no. 297, (2004), 79—116,  [PDF];

 

o    ``Diffusion for Hamiltonian perturbations of integrable systems in high dimensions'', with J. Bourgain, Journ. of Func. Analysis, 229,  (2005), no 1, 1—61, [PDF];

 

o    ``An example of Arnold diffusion for nearly integrable Hamiltonian systems’’, with Mark Levi, Bulletin of AMS, Vol. 45, no. 3, July 2008, 409427, [PDF];

 

o    ``Geometry of Arnold diffusion’’, with Mark Levi,  SIAM Review, Vol. 50, no. 4, 2008, 702720, [PDF];

 

o    ``Arnold diffusion in a pendulum lattice’’, with Mark Levi and Maria Saprykina,  preprint,  2010,    22pp, submitted, [PDF];

 

o    ``An example of a nearly integrableHamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension, with Maria Saprykina, preprint, 2010, 51pp, to appear in CMP, [PDF];

 

o    ``Almost dense orbit on energy surface’’, with Ke Zhang and Yong Zheng, Proceedings of XVITH International Congress on Mathematical Physics.  Held 3-8 August 2009 in Prague, Czech Republic. Edited by Pavel Exner (Doppler Institute, Prague, Czech Republic). Published by World Scientific Publishing Co, 314—322, [PDF];

 

o    `` Arnold diffusion in arbitrary degrees of freedom and 3-dimensional normally hyperbolic invariant cylinders’’, with P. Bernard and K. Zhang, preprint, 2011, 70pp, [PDF];

 

o    ``Diffusion along mean motion resonance for the restricted planar three body problem’’, with J. Fejoz, M. Guardia, P. Roldan, preprint, 2011, 68pp, submitted, [PDF];

 

o    `` Normally normally hyperbolic invariant manifolds near strong double resonance’’, with K. Zhang, preprint, 2011, 44pp, [PDF];

 

                  Growth of Sobolev norms for Hamiltonian PDEs

o      `` Growth of Sobolev norms in the cubic defocusing nonlinear Schodinger equation’’, with M. Guardia, preprint,  73pp, 2012, [PDF];

 

       Marked length spectrum and rigidity for convex billiards   

o      `` On conjugacy of convex billiards’’, with A. Sorrentino, preprint,  23pp, 2012, [PDF];

 

 

                       Growth of the number of periodic orbits

 

o    ``An extension of Artin-Mazur theorem'', Annals of Mathematics, vol. 150, no. 2 (1999), 429441, [PDF];

 

o    ``Generic diffeomorphisms with superexponential growth of number of periodic points’’, Comm. in Math. Physics, Vol. 211, (2000), 1, 253271, [PDF];

 

o    ``Growth of the number of periodic points’’, in Normal forms, Bifurcations and Finiteness Problems in Differential Equations (eds. Y. Ilyashenko and C. Rousseau), Kluwer 2004, 355385, [PDF];

 

o    ``A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms'', I & II. Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1727 [PDF]; & 2836, [PDF];

 

o    ``A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I’’, with B. Hunt, Annals of Mathematics, 165 (2007), 89—170, [PDF];

 

o    ``A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II’’, preprint, 2002, 85pp, [PDF];

 

o    ``How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency’’, with A. Gorodetski, Advances in Mathematics, 208 (2007), 710—797, [PDF];

 

o    ``Newton interpolation polynomials, discretization method,  and  certain prevalent properties in dynamical system’’,  with B. Hunt and V. Kaloshin,  Vol. 2,   Proceedings of ICM 2006, Madrid, Spain, European Math Society (2006), 27—55, [PDF];

 

o    ``A C^r unimodal map with an arbitrary fast growth of the number of periodic points'',  with O. Kozlovsky, preprint, (2002), 7pp, Ergodic Th. & Dynam. Systems, 2011, [PDF];

 

             Celestial mechanics  and instabilities for 3 body problem

 

o    `` Remote Periodic and Quasiperiodic Motions in the Planar Circular Restricted 3-Body Problem of KAM and Aubry-Mather Type’’, with T. Nguyen, preprint, 2005, 35pp, [PDF];

 

o    ``Destruction of invariant curves in the restricted planar circular three body problem using action comparison’’, with J. Galante, Duke Math J, Vol. 159, no. 2 (2011), 275327, [PDF];

 

o    `` Finiteness of central configurations of five bodies in the plane’’, with A. Albouy, 54pp, to appear in  Annals of Mathematics, 2012, [PDF];

 

o    ``Diffusion along mean motion resonance for the restricted planar three body problem’’, with J. Fejoz, M. Guardia, P. Roldan, preprint, 2011, 67pp, [PDF];

 

o    `` The method of spreading cumulative twist and application to the restricted planar circular three body problem’’, with J. Galante, preprint, 2010, 48pp, [PDF];

 

o    ``Destruction of invariant curves in the restricted planar circular three body problem using ordering condition, with Joseph Galante, preprint, 2010, 28pp, [PDF];

 

o    ``Conservative homoclinic bifurcations and some applications’’, with Anton Gorodetski,  Steklov Institute Proceedings, Vol. 267 (2009), dedicated to the 70th anniversary of Vladimir Arnold, [PDF];

 

o    ``Hausdorff dimension of oscillatory motions for restricted three body problems’’, with A. Gorodetski, preprint, 2011, 70pp, [PDF];

 

 

 

              The Hilbert 16-th problem and bifurcation theory

 

o    ``Hilbert-Arnold Problem and an estimate for cyclicity of polycycles on the plane and in the space’’,  Func Anal and its Appl, 35, no. 2, (2001), 146—147, [PDF];

 

o    ``Bifurcations of planar and spatial polycycles: Arnold's program and its development'', with Yu.S.Ilyashenko, Fields Institute Communications, 24, (1999), 241—271, [PDF];

 

o    ``The Existential Hilbert Problem and an estimate on cyclicity of elementary polycycles'', Inventiones Mathematica, 151 (2003), no. 3, 451—512, [PDF];

 

o    `` Around Hilbert-Arnold Problem'',  in ``On finiteness in differential equations and Diophantine geometry'', 111—162, CRM Monogr. Ser., 24, Amer. Math. Soc., Providence, RI, 2005,  [PDF];

 

         Random flows & random walks along orbits of dynamical systems

 

o    ``Hausdorff Dimension of Linear Escape points for periodic Stochastic Dispersions’’, with D. Dolgopyat and L. Koralov, J. Stat. Physics, Vol 108, (2002), 943—971, [PDF];

 

o    `` Sample Path Properties of Stochastic Flows’’, with D. Dolgopyat and L. Koralov, Annals of Probability, Vol. 32, no. 1A, (2004), 1—27, [PDF];

 

o    ``A Limit Shape Theorem for Stochastic Dispersions’’, with D. Dolgopyat and L. Koralov, Comm. in Pure and Appl. Math., Vol. 57, no. 9, (2004), 1127—1158, [PDF];

 

o    ``Long time behaviour of periodic stochastic disperions’’, with D. Dolgopyat and L. Koralov, XIVth International Congress on Mathematical Physics,  290—295, World Sci. Publ., Hackensack, NJ, 2005, [PDF];

 

o    ``Nonsymmetric random walks along orbits of ergodic automorphisms'', with Ya.G.Sinai, Amer. Math. Soc. Transl.(2), 198, (2000), 109—115, [PDF];

 

o    ``Simple random walks along orbits of Anosov diffeomorphisms'', with Ya.G.Sinai, Proc. of Steklov Math. Inst., 228, (2000), 224—233, [PDF];  

 

                Properties of Fractal Sets under Projections

 

o    `` How projections affect the dimension spectrum of fractal measures'', with B. Hunt, Nonlinearity, 10, (1997), no. 5, 1031—1046,  [PDF];

 

o    ``Regularity of of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces'', with B. Hunt, Nonlinearity, Vol. 12, (1999), no. 5, 1263—1275, [PDF];

 

o    ``The effect of projections on fractal sets and measures in Banach spaces'', with B. Hunt and W. Ott, Ergodic Theory and Dynamical Systems, 26, no. 3, (2006), 869—891, [PDF];

 

Notion of Prevalence or Probability one in nonlinear infinite-dimensional spaces

 

o    ``Some prevalent properties of smooth dynamical systems'', Proceedings of the Steklov Institute of Mathematics, 213, (1997), 123—151;

 

o    ``Prevalence in spaces of smooth mappings and a prevalent Kupka-Smale theorem'' (in Russian),  Dokl. Akad. Nauk, 355, (1997), no. 3, 306—307;

 

o    ``Prevalence in spaces of finitely smooth mappings'', Functional Analysis and its Application, 31, (1997), no. 2, 95—99;

 

o    ``Prevalence’’, with B. Hunt , chapter 2, Handbook in dynamical systems, edited by H. Broer, F. Takens, B. Hasselblatt, Vol. 3, 2010, pg. 43—87, [PDF]; see also http://www.sciencedirect.com/science/article/pii/S1874575X10003103

 

                               Diophantine properties of SO(3).

 

o    ``Diophantine Properties of Elements of SO(3)’’, with I.Rodnianski, Geom and Func Anal, 11, (2001), 953—970, [PDF];

 

                                        Whitney stratification

o    ``A Geometric Proof of Existence of Whitney Stratifications’’, Moscow Math. Journ., 5 (2005), no.1, 125—133, [PDF].