Periodic orbits and chain-transitive sets of \(C^1\)-diffeomorphisms

DOI10.1007/s10240-006-0002-4zbMath1226.37007OpenAlexW2070667751MaRDI QIDQ859794

Sylvain Crovisier

Publication date: 19 January 2007

Published in: Publications Mathématiques (Search for Journal in Brave)

Full work available at URL: http://www.numdam.org/item?id=PMIHES_2006__104__87_0

zbMATH Keywords

periodic orbitsshadowing\(C^1\)-generic diffeomorphismshomoclinic classeschain-recurrence classes

Mathematics Subject Classification ID

Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics (37C25) Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics (37C50) Dynamical systems involving smooth mappings and diffeomorphisms (37C05)

Related Items (56)

Robustly chain transitive diffeomorphisms ⋮ Robust chain transitive vector fields ⋮ Eventual shadowing for chain transitive sets of $C^1$ generic dynamical systems ⋮ Orbital shadowing property for generic divergence-free vector fields ⋮ Hyperbolicity versus weak periodic orbits inside homoclinic classes ⋮ On the dominated splitting of Lyapunov stable aperiodic classes ⋮ Weakly shadowable vector fields on non-oriented surfaces ⋮ \(C^1\) weak Palis conjecture for nonsingular flows ⋮ \(C^r\)-closing lemma for partially hyperbolic diffeomorphisms with 1D-center bundle ⋮ Vector fields with the asymptotic orbital pseudo-orbit tracing property ⋮ Partial hyperbolicity far from homoclinic bifurcations ⋮ Dominated splitting of differentiable dynamics with \(\mathrm{C}^1\)-topological weak-star property ⋮ Generalized hyperbolic sets on Banach spaces ⋮ An entropy dichotomy for singular star flows ⋮ Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems ⋮ Theory of pseudo-orbit shadowing in dynamical systems ⋮ On the partial hyperbolicity of robustly transitive sets with singularities ⋮ Local topological stability for diffeomorphisms ⋮ Hyperbolicity and types of shadowing for \(C^1\) generic vector fields ⋮ CENTER MANIFOLDS FOR PARTIALLY HYPERBOLIC SETS WITHOUT STRONG UNSTABLE CONNECTIONS ⋮ Stably asymptotic average shadowing property and dominated splitting ⋮ Usual limit shadowable homoclinic classes of generic diffeomorphisms ⋮ Orbital shadowing property on chain transitive sets for generic diffeomorphisms ⋮ On the centralizer of vector fields: criteria of triviality and genericity results ⋮ Shadowing is generic -- a continuous map case ⋮ Stable weakly shadowable volume-preserving systems are volume-hyperbolic ⋮ On the singular-hyperbolicity of star flows ⋮ Nondensity of the orbital shadowing property in $C^{1}$-topology ⋮ A rescaled expansiveness for flows ⋮ Orbital shadowing and stability for vector fields ⋮ Orbital shadowing for 3-flows ⋮ SURVEY Towards a global view of dynamical systems, for the C¹-topology ⋮ Stably weakly shadowing transitive sets and dominated splittings ⋮ s-limit shadowing is \(\mathcal C^0\)-dense ⋮ On the existence of non-trivial homoclinic classes ⋮ Codimension one structurally stable chain classes ⋮ Conservative flows with various types of shadowing ⋮ Singular robustly chain transitive sets are singular volume partial hyperbolic ⋮ On the hyperbolicity of \(\operatorname{C}^1\)-generic homoclinic classes ⋮ Star flows and multisingular hyperbolicity ⋮ Symmetries of vector fields: the diffeomorphism centralizer ⋮ Bi-Lyapunov stable homoclinic classes for \(C^1\) generic flows ⋮ A Franks’ lemma that preserves invariant manifolds ⋮ Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes ⋮ A type of the shadowing properties for generic view points ⋮ Newhouse phenomenon and homoclinic classes ⋮ Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes ⋮ The closure of periodic orbits in the Gromov-Hausdorff space ⋮ Flows with the (asymptotic) average shadowing property on three-dimensional closed manifolds ⋮ Morse-Smale systems and horseshoes for three dimensional singular flows ⋮ Topological horseshoes for surface homeomorphisms ⋮ Uniform hyperbolicity along periodic orbits ⋮ Asymptotic orbital shadowing property for diffeomorphisms ⋮ Average Shadowing Property With Non Uniformly Hyperbolicity on Periodic Points ⋮ Shadowable chain transitive sets ⋮ Structurally stable homoclinic classes

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