Math 2.0

Dynamical systems and ergodic theory Smooth and hyperbolic dynamics Theorem

Closing lemma

Builds on Non-wandering setPeriodic pointDiffeomorphism

Statement

Let MM be a compact smooth manifold and let f ⁣:M→Mf \colon M \to M be a C1C^1 diffeomorphism. Let pp be a non-wandering point of ff. Give the space of C1C^1 diffeomorphisms of MM the C1C^1 topology, and let 𝒰\mathcal{U} be a neighbourhood of ff in it. Then there is g∈𝒰g \in \mathcal{U} such that pp is a periodic point of gg.

Remarks

The result is due to Pugh, and it is often called Pugh's closing lemma. Since pp is non-wandering, some orbit starting near pp returns near pp. The difficulty is to close this orbit by a perturbation that is small in C1C^1. A naive push changes the derivative by about the size of the gap divided by the size of the region where one pushes. Pugh's argument spreads the push over many returns of the orbit, so that each step is small in C1C^1. A consequence is the general density theorem: for a C1C^1-generic diffeomorphism, the periodic points are dense in the non-wandering set. The analogous statement with the CrC^r topology for r≥2r \ge 2 is open in general; it is the tenth of Smale's problems.

Going deeper

For a student

The closing lemma is the basic perturbation tool of C1C^1 generic dynamics. Together with the Kupka–Smale theorem it shows that a C1C^1-generic diffeomorphism has dense hyperbolic periodic points in its non-wandering set. It is a key ingredient in Mañé's proof of the stability conjecture. Related tools are the connecting lemma of Hayashi and Franks' lemma, which also rely on C1C^1-small perturbations.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction composition, concept, no evidence levelFunction compositionconceptNon-wandering set, concept, no evidence levelNon-wandering setconceptPeriodic point, concept, no evidence levelPeriodic pointconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionChain rule in several variables, statement, no evidence levelChain rule in several…statementHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionJacobian determinant, definition, no evidence levelJacobian determinantdefinitionDerivative of the inverse function, statement, no evidence levelDerivative of the inv…statementC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementInverse function theorem, statement, no evidence levelInverse function theo…statementDiffeomorphism, definition, no evidence levelDiffeomorphismdefinitionClosing lemma, statement, no evidence levelClosing lemmastatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Function composition concept
  4. Non-wandering set concept
  5. Periodic point concept
  6. Fréchet derivative definition
  7. Function of several variables concept
  8. Limit of a function of several variables concept
  9. Partial derivative concept
  10. Differentiable function of several variables definition
  11. Partial derivative definition
  12. Differentiable on an open set definition
  13. Jacobian matrix definition
  14. Chain rule in several variables statement
  15. Higher partial derivatives definition
  16. Iterated derivative definition
  17. Jacobian determinant definition
  18. Derivative of the inverse function statement
  19. C^k function characterised by continuous derivatives statement
  20. Inverse function theorem statement
  21. Diffeomorphism definition
  22. Closing lemma statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Pugh%27s_closing_lemma
About this page
TagE1Z444F6TV
URIm2://E1Z444F6TV
Kindstatement, theorem
WritingW0 stub
MSC37C20, 37C25
StatusThis copy of the site has no build data. Run m2 status E1Z444F6TV.
AboutNon-wandering set, Periodic point
UsesNon-wandering set, Periodic point, Diffeomorphism
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