The dynamics of vector fields with singularities

DOI10.1007/s10114-022-0471-5zbMath1494.37016OpenAlexW4210339993MaRDI QIDQ2075438

Da-Wei Yang

Publication date: 14 February 2022

Published in: Acta Mathematica Sinica. English Series (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s10114-022-0471-5

zbMATH Keywords

horseshoevector fieldsingularityMorse-Smale systemsingular hyperbolicty

Mathematics Subject Classification ID

Dynamics induced by flows and semiflows (37C10) Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) (37D20) Invariant manifold theory for dynamical systems (37D10) Dynamical systems with hyperbolic orbits and sets (37D05) Partially hyperbolic systems and dominated splittings (37D30) Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) (37D25) Morse-Smale systems (37D15) Dynamical systems with singularities (billiards, etc.) (37C83)

Cites Work

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• Homoclinic classes for sectional-hyperbolic sets
• Exponential decay of correlations for nonuniformly hyperbolic flows with a \(C^{1+\alpha}\) stable foliation, including the classical Lorenz attractor
• On the singular-hyperbolicity of star flows
• Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms
• Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems
• Nonuniform hyperbolicity for \(C^{1}\)-generic diffeomorphisms
• Connecting invariant manifolds and the solution of the \(C^ 1\) stability and \(\Omega\)-stability conjectures for flows
• Geometric Lorenz flows with historic behavior
• Structural stability on two-dimensional manifolds
• The Lorenz attractor is mixing
• Indices of singularities of robustly transitive sets
• The metric entropy of diffeomorphisms. I: Characterization of measures satisfying Pesin's entropy formula
• A proof of the \(C^ 1\) stability conjecture
• Structural stability of Lorenz attractors
• Lyapunov exponents, entropy and periodic orbits for diffeomorphisms
• The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms
• The ergodic theory of axiom A flows
• Positivity of entropy production in nonequilibrium statistical mechanics
• Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows.
• Homoclinic tangencies and hyperbolicity for surface diffeomorphisms
• \(C^1\) weak Palis conjecture for nonsingular flows
• Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers
• An ergodic closing lemma
• Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective
• On the \(C^ 1\) stability conjecture for flows
• Empirical measures of partially hyperbolic attractors
• Entropy theory for sectional hyperbolic flows
• Robust transitivity of singular hyperbolic attractors
• Star flows and multisingular hyperbolicity
• On the notions of singular domination and (multi-)singular hyperbolicity
• On the finiteness of uniform sinks
• On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis
• Lyapunov stability and sectional-hyperbolicity for higher-dimensional flows
• Diffeomorphisms with infinitely many sinks
• Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation
• A global perspective for non-conservative dynamics
• Nonsingular star flows satisfy Axiom A and the no-cycle condition
• A structural stability theorem
• Robustly transitive singular sets via approach of an extended linear Poincaré flow
• Dominated chain recurrent class with singularities
• Recent progress on sectional-hyperbolic systems
• Three-Dimensional Flows
• Sectional-hyperbolic systems
• Singular-hyperbolic attractors are chaotic
• Diffeomorphisms in ℱ¹(M) satisfy Axiom A
• A Measure Associated with Axiom-A Attractors
• Attracteurs de Lorenz de variété instable de dimension arbitraire
• Robust transitivity and topological mixing for 𝐶¹-flows
• A dichotomy for three-dimensional vector fields
• SRB measures for higher dimensional singular partially hyperbolic attractors
• Cherry flow: physical measures and perturbation theory
• Deterministic Nonperiodic Flow
• Morse-Smale systems and horseshoes for three dimensional singular flows
• The Lorenz Attractor, a Paradigm for Chaos
• The set of axiom A diffeomorphisms with no cycles
• On the existence of non-trivial homoclinic classes
• Open questions leading to a global perspective in dynamics
• The Closing Lemma
• Differentiable dynamical systems
• GIBBS MEASURES IN ERGODIC THEORY
• Shorter Notes: A 3-Dimensional Abraham-Smale Example
• Recurrence and genericity
• Equilibrium states and the ergodic theory of Anosov diffeomorphisms
• Invariant manifolds
• A second order differential equation with singular solutions
• What's new on Lorenz strange attractors?

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