Math 2.0

Dynamical systems and ergodic theory Definition

Axiom A

Builds on Hyperbolic setNon-wandering setPeriodic pointDense setDiffeomorphism

Definition

Let MM be a compact smooth manifold and let f ⁣:M→Mf \colon M \to M be a C1C^1 diffeomorphism. Let Ω(f)\Omega(f) be the non-wandering set of ff. Then ff satisfies Axiom A if both of the following hold.

  1. Ω(f)\Omega(f) is a hyperbolic set of ff.
  2. The set of periodic points of ff is dense in Ω(f)\Omega(f).

Remarks

Every periodic point is non-wandering, so clause 2 says that Ω(f)\Omega(f) is the closure of the set of periodic points. Example: the north-south map of the circle, the time-one map of θ′=sin⁡θ\theta' = \sin\theta, has Ω(f)={0,π}\Omega(f) = \set{0, \pi}. Both fixed points are hyperbolic, with derivatives ee and e−1e^{-1}, so both clauses hold. Non-example: an irrational rotation of the circle has Ω(f)\Omega(f) equal to the whole circle, but it has no periodic points, and the derivative 11 is not hyperbolic; both clauses fail. Another non-example is the time-one map of θ′=1−cos⁡θ\theta' = 1 - \cos\theta: its only non-wandering point 00 is fixed, but the derivative there is 11, so clause 1 fails. For flows, Axiom A asks that the non-wandering set be hyperbolic in the flow sense and that closed orbits be dense in it.

Reading path

To understand this page from the ground up, read these 78 pages in order. The foundations come first.

  1. Differential equation concept
  2. Element definition
  3. Euclidean space object
  4. Function concept
  5. Function composition concept
  6. Interval concept
  7. Non-wandering set concept
  8. Periodic point concept
  9. Proposition concept
  10. Real numbers object
  11. Riemannian metric concept
  12. Smooth function concept
  13. Smooth manifold concept
  14. Tangent bundle concept
  15. Trajectory concept
  16. Set definition
  17. Ordinary differential equation definition
  18. Fréchet derivative definition
  19. Total derivative concept
  20. Function of several variables concept
  21. Lipschitz condition definition
  22. Grönwall's inequality statement
  23. Conjunction definition
  24. Disjunction definition
  25. Implication definition
  26. Predicate definition
  27. Riemannian manifold definition
  28. Empty set object
  29. Family of sets definition
  30. Ordered pair definition
  31. Intersection definition
  32. Limit of a function of several variables concept
  33. Solution of an ordinary differential equation definition
  34. Union definition
  35. Partial derivative concept
  36. Existential quantifier definition
  37. Universal quantifier definition
  38. Cartesian product definition
  39. Autonomous system definition
  40. Differentiable function of several variables definition
  41. Initial value problem definition
  42. Partial derivative definition
  43. Intersection of a family of sets definition
  44. Subset definition
  45. Jacobian matrix concept
  46. Union of a family of sets definition
  47. Uniqueness quantifier definition
  48. Differentiable on an open set definition
  49. Chain rule in several variables statement
  50. Binary relation definition
  51. Universal set definition
  52. Higher partial derivatives definition
  53. Topological space definition
  54. Jacobian matrix definition
  55. Uniqueness of solutions for Lipschitz vector fields statement
  56. Differential of a smooth map definition
  57. Iterated derivative definition
  58. Complement definition
  59. Derivative of the inverse function statement
  60. Maximal solution definition
  61. Function definition
  62. Jacobian determinant definition
  63. C^k function characterised by continuous derivatives statement
  64. Flow of a vector field definition
  65. Metric space definition
  66. Open ball definition
  67. Inverse function theorem statement
  68. Invariant set definition
  69. Diffeomorphism definition
  70. Metric topology definition
  71. Open set definition
  72. Closed set definition
  73. Open cover definition
  74. Compact space definition
  75. Closure definition
  76. Dense set definition
  77. Hyperbolic set definition
  78. Axiom A definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Axiom_A
About this page
TagS31RTTJPDC
URIm2://S31RTTJPDC
Kinddefinition
WritingW0 stub
MSC37D20
DefinesAxiom A
UsesHyperbolic set, Non-wandering set, Periodic point, Dense set, Diffeomorphism

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