نظریهٔ مدرن سیستم‌های دینامیکی: ادای احترام به دیمیتری ویکتورویچ آنوسوف (ریاضیات معاصر)

This volume is a tribute to one of the founders of modern theory of dynamical systems, the late Dmitry Victorovich Anosov. It contains both original papers and surveys, written by some distinguished experts in dynamics, which are related to important themes of Anosov's work, as well as broadly interpreted further crucial developments in the theory of dynamical systems that followed Anosov's original work. Also included is an article by A. Katok that presents Anosov's scientific biography and a picture of the early development of hyperbolicity theory in its various incarnations, complete and partial, uniform and nonuniform. Cover Title page Contents Preface Dmitry Viktorovich Anosov: His life and mathematics 1. Personal recollections 2. Mathematical legacy References D. V. Anosov and our road to partial hyperbolicity References Escape from large holes in Anosov systems 1. Introduction 2. Set up 3. Induced open dynamical system 4. Algebraic automorphism of the torus 5. The survival probabilities and escape rates 6. Multidimensional automorphisms 7. Nonlinear systems Acknowledgement References A dynamical decomposition of the torus into pseudo-circles 1. Introduction 2. A criterion for the existence of a partition into pseudo-circles 2.a–Crooking 2.b–Elements of the construction 2.c–Inductive properties 2.d–Proof of the theorem 3. Inductive construction 3.1–Preliminary constructions 3.b–Construction of H_{n+1} 3.c–Choice of b_{n+1}, ε_{n+1} and N_{n+1} 4. Uniqueness of the semi-conjugacy, non-existence of wandering curves and further remarks Added in Proof References On irreducibility and disjointness of Koopman and quasi-regular representations of weakly branch groups 1. Introduction 2. Preliminaries 3. Stabilizers and centralizers of weakly branch group actions 4. Irreducibility of Koopman representations of weakly branch groups References Isolated elliptic fixed points for smooth Hamiltonians Introduction 1. Notations 2. Orbits accumulating the axis and diffusive orbits 3. Proof for the case d=2 4. Proofs for the case d=4 5. Proofs for the case d=3. References Nonlocally maximal and premaximal hyperbolic sets 1. Introduction 2. Background 3. Nonlocally maximal sets for flows 4. Premaximality References Rotation numbers for S2 diffeomorphisms 1. The function \cR_{f} for fixed points. 2. The complete symmetries of \cR_{f}. 3. Proof of Theorem (2.3) References Path connectedness and entropy density of the space of hyperbolic ergodic measures 1. Introduction 2. Examples 3. Proofs References Around Anosov-Weil Theory Introduction 1. Mathematical background 2. Historical background 3. Nonlocal behavior of curves on universal coverings 4. Asymptotic properties of special curves 5. Applications to foliations and dynamical systems References Attractors and skew products 1. Attractors 2. Gorodetski-Ilyashenko strategy 3. New robust properties of \diffs of closed manifolds and manifolds with boundary Acknowledgments References Thermodynamic formalism for some systems with countable Markov structures 1. Motivation: Folklore Theorem in dimension 1 2. Model under consideration. Geometric and hyperbolicity conditions 3. Distortion conditions 4. Additional hyperbolicity and distortion conditions and statement of the main theorem 5. Hölder properties of log(D^{u}F(z)) 6. Some sufficient conditions for exponential decay of correlations in countable shifts 7. Proof of the exponential decay of correlations References Non-uniform measure rigidity for Z^{k} actions of symplectic type 1. Introduction 2. Preliminary results on behavior of semi-conjugacies 3. Formulation of results 4. Absolute continuity of μ 5. Existence of proper periodic orbits 6. Differentiable rigidity of real analytic actions. References On a differentiable linearization theorem of Philip Hartman 1. Introduction 2. The Associated Algebraic Problem 3. The α-Contracting Case 4. Preliminary Constructions 5. Proof of Theorem 1.5 6. Continuous Dependence on Parameters 7. Vector Fields and Motivation References Time change invariants for measure preserving flows References Spectral boundary value problems for Laplace–Beltrami operator: Moduli of continuity of eigenvalues under domain deformation 1. Introduction. 2. Basic notions. 3. Necessary background, constructions and estimates. 4. Proof of the basic results. References Measure-theoretical properties of center foliations 1. Introduction 2. Dimension theory for the center foliaton 3. Semiconjugacy to the linear model 4. Upper absolute continuity Appendix A. Atomic disintegration References Back Cover Dmitry Viktorovich Anosov: His Life And Mathematics / Anatole Katok -- D.v. Anosov And Our Road To Partial Hyperbolicity / Michael Brin And Yakov Pesin -- Escape From Large Holes In Anosov Systems / Valentin Afraimovich And Leonid Bunimovich -- A Dynamical Decomposition Of The Torus Into Pseudo-circles / François Béguin, Sylvain Crovisier, And Tobias Jäger -- On Irreducibility And Disjointedness Of Koopman And Quasi-regular Representations Of Weakly Branch Groups / Artem Dudko And Rostislav Grigorchuk -- Isolated Elliptic Fixed Points For Smooth Hamiltonians / Bassam Fayad And Maria Saprikina -- Nonlocally Maximal And Premaximal Hyperbolic Sets / T. Fisher, T. Petty, And S. Tikhomirov -- Rotation Numbers For S2 Diffeomorphisms / John Franks -- Path Connectedness And Entropy Density Of The Space Of Hyperbolic Ergodic Measures / Anton Gorodetski And Yakov Pesin -- Around Anosov-weil / V. Grines And E. Zhuzhoma -- Attractors And Skew Products / Yu. Ilyashenko And I. Shilin -- Thermodynamic Formalism For Some Systems With Countable Markov Structures / Michael Jakobson -- Non-uniform Measure Rigidity For Z[superscript K] Actions Of Symplectic Type / Anatole Katok And Federico Rodriguez Hertz -- On A Differentiable Linearization Theorem Of Philip Hartman / Sheldon E. Newhouse -- Time Change Invariants For Measure Preserving Flows / Marina Ratner -- Spectral Boundary Value Problems For Laplace-beltrami Operator: Moduli Of Continuity Of Eigenvalues Under Domain Deformation / A. Stepin And I. Zilin -- Measure-theoretical Properties Center Foliations / Marcelo Viana And Jiagang Yang. Anatole Katok, Yakov Pesin, Federico Rodriguez Hertz, Editors. Includes Bibliographical References.