Math 2.0

Dynamical systems and ergodic theory Smooth and hyperbolic dynamics Problem

Smale's problems on dynamics

Builds on Closing lemmaNon-wandering setPeriodic point

Problem

In 1998 Smale published a list of eighteen problems for the twenty-first century. The following problems of the list concern dynamical systems.

  1. Problem 10, the CrC^r closing lemma. Let pp be a non-wandering point of a CrC^r diffeomorphism ff of a compact manifold, with r≥2r \ge 2. Can ff be CrC^r-approximated by diffeomorphisms gg for which pp is a periodic point?
  2. Problem 11: is one-dimensional dynamics generally hyperbolic? In particular, can a complex polynomial be approximated by polynomials of the same degree all of whose critical points tend to attracting periodic orbits?
  3. Problem 12, centralisers of diffeomorphisms: can every CrC^r diffeomorphism of a compact manifold be CrC^r-approximated by one that commutes only with its own powers?
  4. Problem 13: Hilbert's sixteenth problem, on the number and position of the limit cycles of polynomial vector fields in the plane.
  5. Problem 14: do the Lorenz equations have a strange attractor of the type of the geometric Lorenz model?

Remarks

Problem 10 is the higher-smoothness version of Pugh's closing lemma, which settles the case r=1r = 1; it remains open in general. For Problem 11, density of hyperbolicity among real polynomials was proved by Kozlovski, Shen and van Strien. The real quadratic case had been settled before by Lyubich and by Graczyk and Świątek. The complex case is the density of hyperbolicity conjecture and is open. Problem 12 was solved in the C1C^1 topology by Bonatti, Crovisier and Wilkinson. Problem 13 is open, even for quadratic vector fields. Problem 14 was solved positively by Tucker with a computer-assisted proof; see Smale's fourteenth problem.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

Show the path as a graph
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 lemmastatementSmale's problems on dynamics, problem, no evidence levelSmale's problems on d…problem
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
  23. Smale's problems on dynamics problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Smale%27s_problems
About this page
TagG14BVRT221
URIm2://G14BVRT221
Kindproblem
WritingW0 stub
MSC37C20, 37E05, 37D45
StatusThis copy of the site has no build data. Run m2 status G14BVRT221.
AboutSmooth dynamical system
UsesClosing lemma, Non-wandering set, Periodic point

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