266 Sloan
Mathematics 253-37
California Institute of Technology
Pasadena, CA 91125
ph.  (626) 395 4342
fax  (626) 585 1728
E-mail: kaloshin@its.caltech.edu
 
 
 

Research Summary (.pdf)

Spring 2004: Mathematical Chaos (Undegraduate)
Fall 2003: Weak KAM Theory (Graduate)
Spring 2003: Aubry-Mather and KAM theory (Graduate)
 

My research interests include smooth and random dynamical systems and Hamiltonian systems,  in particular, Aubry-Mather theory and Arnold diffusion. 

Research papers sorted by topics:

1. ``An extension of Artin-Mazur theorem'', Annals of Mathematics, Vol 150, No. 2, (1999), 429-441, online ;
2. ``Generic diffeomorphisms with superexponential growth of number of periodic points'', Comm. in Math. Physics,Vol 211, (2000), 1, 253--271, online;
3. ``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, 355--385, online;
4. (with B.Hunt) ''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), 17--27 online; & 28--36, online;
5. (with B.Hunt) ``A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms'', part I, to appear in Annals of Mathematics, 94pp, online;
6. ``A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms'', part II, preprint, (2002), 85pp, online;
7. (with A. Gorodetski) ``How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency'', to appear in Advances in Mathematics, 99p, online;
8. (with O. Kozlovsky) ``A C^r unimodal map with an arbitrary fast growth of the number of periodic points'', preprint, (2002), 5pp, online;
9. (with M. Saprykina) ``Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits'', Discrete Contin. Dyn. Syst. 15 (2006), no. 2, 611--640, online;
10. (with A. Gorodetski, B.Hunt) ``Newton interpolation polynomials, discretization method, and certain prevalent properties in dynamical systems'', notes for the 2006 ICM talk, 29pp, online;

II. Instabilities in Hamiltonian systems and Arnold diffusion

1. ``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, online;
2. ``Mather theory, weak KAM, and viscosity solutions of Hamilton--Jacobi PDE's'',  EQUADIFF 2003, 39--48, World Sci. Publ., Hackensack, NJ, 2005, online;
3. (with J. Bourgain) ``Diffusion for Hamiltonian perturbations of integrable systems in high dimensions'', J. of Func Analysis, 229,  (2005), no 1, 1--61, online;
4. (with J. Mather and E. Valdinoci) ``Instability of totally elliptic points of symplectic maps in dimension 4'', Instability of resonant totally elliptic points of symplectic maps in dimension 4. Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes. II. Astérisque No. 297 (2004), 79--116, online;
5. (with T. Nguyen) ``Remote Periodic and Quasiperiodic Motions in the Planar Circular Restricted 3-Body Problem of KAM and Aubry-Mather Type'', preprint, 35pp, online;

III. The Hilbert 16-th problem and bifurcation theory

1. ``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, online;
2. (with Yu.S.Ilyashenko) ``Bifurcations of planar and spatial polycycles: Arnold's program and its development'', Fields Institute Communications, 24, (1999), 241-271,  online;
3. ``The Existential Hilbert Problem and an estimate on cyclicity of elementary polycycles'', Inventiones Mathematica, 151 (2003), no. 3, 451--512, online;
4. ``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, online;

IV. Random flows and random walks along orbits of dynamical systems

1. (with D. Dolgopyat and L. Koralov) ``Hausdorff Dimension of Linear Escape points for periodic Stochastic Dispersions'', J. Stat. Physics, Vol 108, (2002), 943--971, online;
2. (with D. Dolgopyat and L. Koralov) ``Sample Path Properties of Stochastic Flows'', Annals of Probability, Vol. 32, no. 1A, (2004), 1--27, online;
3. (with D. Dolgopyat and L. Koralov) ``A Limit Shape Theorem for Stochastic Dispersions'', Comm. in Pure and Appl. Math., vol. 57, no. 9, (2004), 1127-1158, online;
4. (with D. Dolgopyat and L. Koralov) ``Long time behaviour of periodic stochastic disperions'', to appear in Proceedings of ICMP 2002, 8pp, online;
5. (with Ya.G.Sinai) ``Nonsymmetric random walks along orbits of ergodic automorphisms'', Amer. Math. Soc. Transl.(2), 198, (2000), 109--115, online;
6. (with Ya.G.Sinai) ``Simple random walks along orbits of Anosov diffeomorphisms'', Proc. of Steklov Math. Inst., 228, (2000), 224--233, online;

1. (with B. Hunt) ``How projections affect the dimension spectrum of fractal measures'', Nonlinearity, 10, (1997), no. 5, 1031--1046, online;
2. (with B. Hunt) ``Regularity of of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces'', Nonlinearity,Vol 12, (1999), no. 5, 1263--1275 online;
3. (with B.Hunt and W.Ott) ``The effect of projections on fractal sets and measures in Banach spaces'', to appear in Ergodic Theory and Dynamical Systems, (2003), 23pp, online;

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

1. ``Some prevalent properties of smooth dynamical systems'', Proceedings of the Steklov Institute of Mathematics, 213, (1997), 123--151;
2. ``Prevalence in spaces of smooth mappings and a prevalent Kupka-Smale theorem'' (in Russian), Dokl. Akad. Nauk, 355, (1997), no. 3, 306--307;
3. ``Prevalence in spaces of finitely smooth mappings'', Functional Analysis and its Application, 31, (1997), no. 2, 95--99;

VII. Diophantine properties of SO(3).

1. (with I.Rodnianski) ``Diophantine Properties of Elements of SO(3)'', Geom and Func Anal, 11, (2001), 953--970, online;

VIII. Whitney stratification

1. ``A Geometric Proof of Existence of Whitney Stratifications'', Mosc. Math. J. 5 (2005), no. 1, 125--133, online;

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