Differentiable dynamical systems

This is a survey article on the area of global analysis defined by differentiable dynamical systems or equivalently the action (differentiable) of a Lie group G on a manifold M. An action is a homomorphism G→Diff(M) such that the induced map G×M→M is differentiable. Here Diff(M) is the group of all diffeomorphisms of M and a diffeo- morphism is a differentiable map with a differentiable inverse. Everything will be discussed here from the C ∞ or C r point of view. All manifolds maps, etc. will be differentiable (C r , 1 ≦ r ≦ ∞) unless stated otherwise.

[1]  H. Poincaré,et al.  Sur les courbes définies par les équations différentielles(III) , 1885 .

[2]  Henri Poincaré,et al.  méthodes nouvelles de la mécanique céleste , 1892 .

[3]  J. Hadamard,et al.  Les surfaces a courbures opposees et leurs lignes geodesique , 1898 .

[4]  George D. Birkhoff,et al.  Dynamical systems with two degrees of freedom , 1917 .

[5]  G. Julia Mémoire sur l'itération des fonctions rationnelles , 1918 .

[6]  Harold Marston Morse A One-to-One Representation of Geodesics on a Surface of Negative Curvature , 1921 .

[7]  George D. Birkhoff,et al.  Surface transformations and their dynamical applications , 1922 .

[8]  George D. Birkhoff,et al.  On the periodic motions of dynamical systems , 1927, Hamiltonian Dynamical Systems.

[9]  O. Perron,et al.  Die Stabilitätsfrage bei Differentialgleichungen , 1930 .

[10]  M. Morse The Calculus of Variations in the Large , 1934 .

[11]  G. A. Hedlund,et al.  The dynamics of geodesic flows , 1939 .

[12]  S. M. Ulam,et al.  Measure-Preserving Homeomorphisms and Metrical Transitivity , 1941 .

[13]  N. Levinson,et al.  A Second Order Differential Equation with Singular Solutions , 1949 .

[14]  G. Mostow The Extensibility of Local Lie Groups of Transformations and Groups on Surfaces , 1950 .

[15]  H. Seifert Closed integral curves in 3-space and isotopic two-dimensional deformations , 1950 .

[16]  Y. Matsushima On the Discrete Subgroups and Homogeneous Spaces of Nilpotent Lie Groups , 1951, Nagoya Mathematical Journal.

[17]  John,et al.  Real Algebraic Manifolds , 1952 .

[18]  F. B. Fuller The Existence of Periodic Points , 1953 .

[19]  A. Kolmogorov On conservation of conditionally periodic motions for a small change in Hamilton's function , 1954 .

[20]  K. Nomizu On the Cohomology of Compact Homogeneous Spaces of Nilpotent Lie Groups , 1954 .

[21]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[22]  A. Selberg Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series , 1956 .

[23]  F. I. Mautner GEODESIC FLOWS ON SYMMETRIC RIEMANN SPACES , 1957 .

[24]  Solomon Lefschetz,et al.  Differential Equations: Geometric Theory , 1958 .

[25]  Raoul Bott,et al.  Homogeneous Vector Bundles , 1957 .

[26]  Shlomo Sternberg,et al.  Local Contractions and a Theorem of Poincare , 1957 .

[27]  W. H. Gottschalk,et al.  Minimal sets: An introduction to topological dynamics , 1958 .

[28]  Robert Ellis,et al.  Distal transformation groups. , 1958 .

[29]  Shlomo Sternberg,et al.  On the Structure of Local Homeomorphisms of Euclidean n-Space, II , 1958 .

[30]  C. Pugh,et al.  Structural stability , 2008, Scholarpedia.

[31]  Shlomo Sternberg,et al.  THE STRUCTURE OF LOCAL HOMEOMORPHISMS, III*1 , 1959 .

[32]  A. Weil On Discrete Subgroups of Lie Groups , 1960 .

[33]  E. B. Dynkin,et al.  The maximal subgroups of the classical groups , 1960 .

[34]  R. Palais,et al.  DEFORMATIONS OF COMPACT DIFFERENTIABLE TRANSFORMATION GROUPS. , 1960 .

[35]  S. Smale Morse inequalities for a dynamical system , 1960 .

[36]  B. Kostant Lie Algebra Cohomology and the Generalized Borel-Weil Theorem , 1961 .

[37]  S. Smale On Gradient Dynamical Systems , 1961 .

[38]  Armand Borel,et al.  Seminar on Transformation Groups. , 1961 .

[39]  Lawrence Markus STRUCTURALLY STABLE DIFFERENTIAL SYSTEMS , 1961 .

[40]  Richard S. Palais,et al.  On the Existence of Slices for Actions of Non-Compact Lie Groups , 1961 .

[41]  R. Palais Equivalence of nearby differentiable actions of a compact group , 1961 .

[42]  A. Weil On Discrete Subgroups of Lie Groups (II) , 1962 .

[43]  D. Anosov Roughness of Geodesic Flows on Compact Riemannian Manifolds of Negative Curvature , 2020 .

[44]  M. Peixoto,et al.  Structural stability on two-dimensional manifolds☆ , 1962 .

[45]  V. I. Arnol'd,et al.  PROOF OF A THEOREM OF A.?N.?KOLMOGOROV ON THE INVARIANCE OF QUASI-PERIODIC MOTIONS UNDER SMALL PERTURBATIONS OF THE HAMILTONIAN , 1963 .

[46]  Louis Auslander,et al.  Flows on Homogeneous Spaces , 1963 .

[47]  H. Furstenberg,et al.  The Structure of Distal Flows , 1963 .

[48]  S. Smale Dynamical systems and the topological conjugacy problem for diffeomorphisms , 1963 .

[49]  Arthur J. Schwartz,et al.  Errata: A Generalization of a Poincare-Bendixson Theorem to Closed Two-Dimensional Manifolds , 1963 .

[50]  J. Moser Perturbation Theory for Almost Periodic Solutions for Undamped Nonlinear Differential Equations , 1963 .

[51]  Elon L. Lima Common singularities of commuting vector fields on 2-manifolds , 1964 .

[52]  P. Hartman Ordinary Differential Equations , 1965 .

[53]  S. Lang Introduction to Differentiable Manifolds , 1964 .

[54]  H. Rosenberg A generalization of Morse-Smale inequalities , 1964 .

[55]  SMOOTH FOLIATIONS ON THREE-DIMENSIONAL MANIFOLDS , 1964 .

[56]  THE TOPOLOGY SUMMER INSTITUTE (Seattle, U.S.A., 1963) , 1965 .

[57]  Common periodic points of commuting functions. , 1965 .

[58]  V. V. Nemytskii,et al.  SOME MODERN PROBLEMS IN THE QUALITATIVE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS , 1965 .

[59]  THE CONSTRUCTION OF MINIMAL DISCRETE FLOWS. , 1965 .

[60]  Richard S. Palais,et al.  Homeomorphic conjugacy of automorphisms on the torus , 1965 .

[61]  R. Sacksteder Foliations and Pseudogroups , 1965 .

[62]  Barry Mazur,et al.  On Periodic Points , 1965 .

[63]  A. Dold Fixed point index and fixed point theorem for Euclidean neighborhood retracts , 1965 .

[64]  COMMUTING VECTOR FIELDS ON SI , 1965 .

[65]  Degeneracy of orbits of actions ofRm on a manifold , 1966 .

[66]  S. Smale,et al.  Structurally Stable Systems are not Dense , 1966 .

[67]  H. Rosenberg Actions ofRn on manifolds , 1966 .

[68]  W. Hsiang,et al.  Differentiable actions of compact connected classical groups , 1967 .

[69]  The entropy of Chebyshev polynomials , 1966 .

[70]  P. Billingsley,et al.  Ergodic theory and information , 1966 .

[71]  E. Grosswald Topics from the theory of numbers , 1966 .

[72]  C. Pugh An Improved Closing Lemma and a General Density Theorem , 1967 .

[73]  Vladimir Igorevich Arnolʹd,et al.  Problèmes ergodiques de la mécanique classique , 1967 .

[74]  M. Peixoto,et al.  On an approximation theorem of Kupka and Smale , 1967 .

[75]  Robert Hermann,et al.  Lie Groups for Physicists , 1967 .

[76]  Ralph Abraham,et al.  Foundations Of Mechanics , 2019 .

[77]  F. B. Fuller,et al.  An Index of Fixed Point Type for Periodic Orbits , 1967 .

[78]  C. Pugh,et al.  The Closing Lemma , 1967 .

[79]  Remarks on Differentiable Actions of Non-Compact Semi-Simple Lie Groups on Euclidean Spaces , 1968 .

[80]  Fixed points for iterates. , 1968 .

[81]  C. Pugh,et al.  Structurally Stable Systems on Open Manifolds are Never Dense , 1968 .

[82]  K. Meyer ENERGY FUNCTIONS FOR MORSE SMALE SYSTEMS. , 1968 .

[83]  R. Hermann The formal linearization of a semisimple Lie algebra of vector fields , 1968 .

[84]  C. Moore DISTAL AFFINE TRANSFORMATION GROUPS.1 , 1968 .