A Franks’ lemma that preserves invariant manifolds

DOI10.1017/etds.2014.101zbMath1369.37025arXiv0912.1121OpenAlexW2963623516WikidataQ124845449 ScholiaQ124845449MaRDI QIDQ2813945

Nicolas Gourmelon

Publication date: 17 June 2016

Published in: Ergodic Theory and Dynamical Systems (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0912.1121

zbMATH Keywords

dominated splittinghomoclinic tangencyperturbation, stable/unstable manifold

Mathematics Subject Classification ID

Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics (37C25) Generic properties, structural stability of dynamical systems (37C20) Invariant manifold theory for dynamical systems (37D10) Homoclinic and heteroclinic orbits for dynamical systems (37C29)

Related Items (12)

Dirac physical measures on saddle-type fixed points ⋮ Hyperbolicity versus weak periodic orbits inside homoclinic classes ⋮ The entropy of 𝐶¹-diffeomorphisms without a dominated splitting ⋮ Ergodic measures with multi-zero Lyapunov exponents inside homoclinic classes ⋮ Thermodynamical u-formalism I: measures of maximal u-entropy for maps that factor over Anosov ⋮ Local perturbations of conservativeC¹diffeomorphisms ⋮ Stably inverse shadowable transitive sets and dominated splitting ⋮ Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere ⋮ Lyapunov stable homoclinic classes for smooth vector fields ⋮ Codimension one structurally stable chain classes ⋮ Bi-Lyapunov stable homoclinic classes for \(C^1\) generic flows ⋮ Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes

Cites Work

• Connecting invariant manifolds and the solution of the \(C^ 1\) stability and \(\Omega\)-stability conjectures for flows
• The \(C^{1+\alpha}\) hypothesis in Pesin theory
• Periodic orbits and chain-transitive sets of \(C^1\)-diffeomorphisms
• On maximal transitive sets of generic diffeomorphisms
• A \(C^1\)-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks of sources
• Homoclinic tangencies and hyperbolicity for surface diffeomorphisms
• An ergodic closing lemma
• Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective
• Generation of homoclinic tangencies by \(C^1\)-perturbations
• Création de connexions en topologie C¹
• Perturbation of the Lyapunov spectra of periodic orbits
• Perturbations of the derivative along periodic orbits
• Generic bi-Lyapunov stable homoclinic classes
• Differential Topology
• 𝐶¹ Connecting Lemmas
• Homoclinic tangencies and dominated splittings
• Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication
• The Closing Lemma
• Recurrence and genericity

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