Math 2.0

Dynamical systems and ergodic theory Theorem

Spectral decomposition theorem

Builds on Axiom ANon-wandering setTopological transitivityTopological mixing

Statement

Let MM be a compact smooth manifold and let f ⁣:M→Mf \colon M \to M be a C1C^1 diffeomorphism that satisfies Axiom A, with non-wandering set Ω(f)\Omega(f). Then:

  1. There are pairwise disjoint closed sets Λ1,…,Λk\Lambda_1, \ldots, \Lambda_k with Ω(f)=Λ1∪⋯∪Λk\Omega(f) = \Lambda_1 \cup \cdots \cup \Lambda_k such that f(Λi)=Λif(\Lambda_i) = \Lambda_i and f∣Λif|_{\Lambda_i} is topologically transitive for each ii.
  2. Each Λi\Lambda_i is a union of pairwise disjoint closed sets Xi,1,…,Xi,niX_{i,1}, \ldots, X_{i,n_i} such that f(Xi,j)=Xi,j+1f(X_{i,j}) = X_{i,j+1} for j<nij < n_i, f(Xi,ni)=Xi,1f(X_{i,n_i}) = X_{i,1}, and fni∣Xi,jf^{n_i}|_{X_{i,j}} is topologically mixing for each jj.

Remarks

The sets Λi\Lambda_i are called basic sets; the decomposition into them is unique. The idea of the proof is to group periodic points whose stable and unstable manifolds meet transversally in both directions. The local product structure of the hyperbolic set makes the closures of these groups open and closed in Ω(f)\Omega(f). Compactness then makes them finite in number. Part 1 is due to Smale, and part 2 to Bowen. Example: for the north-south map of the circle, the basic sets are the two hyperbolic fixed points, with ni=1n_i = 1. For an Anosov diffeomorphism of the torus, Ω(f)\Omega(f) is the whole torus and there is one basic set. Without hyperbolicity the conclusion fails: for the identity of the circle every point is non-wandering, and Ω(f)\Omega(f) is connected but ff is not transitive on it.

Going deeper

For a student

The theorem is the starting point of the theory of Axiom A systems. Orbits outside Ω(f)\Omega(f) go from one basic set to another. When there are no cycles among the basic sets, this gives a partial order on them, as in Conley's theory of chain recurrence. Each basic set admits a Markov partition, so the dynamics on it is a factor of a subshift of finite type. This leads to equilibrium states, Sinai–Ruelle–Bowen measures and the counting of periodic orbits.

Reading path

To understand this page from the ground up, read these 81 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. Topological mixing concept
  16. Topological transitivity concept
  17. Trajectory concept
  18. Set definition
  19. Ordinary differential equation definition
  20. Fréchet derivative definition
  21. Total derivative concept
  22. Function of several variables concept
  23. Lipschitz condition definition
  24. Grönwall's inequality statement
  25. Conjunction definition
  26. Disjunction definition
  27. Implication definition
  28. Predicate definition
  29. Riemannian manifold definition
  30. Empty set object
  31. Family of sets definition
  32. Ordered pair definition
  33. Intersection definition
  34. Limit of a function of several variables concept
  35. Solution of an ordinary differential equation definition
  36. Union definition
  37. Partial derivative concept
  38. Existential quantifier definition
  39. Universal quantifier definition
  40. Cartesian product definition
  41. Autonomous system definition
  42. Differentiable function of several variables definition
  43. Initial value problem definition
  44. Partial derivative definition
  45. Intersection of a family of sets definition
  46. Subset definition
  47. Jacobian matrix concept
  48. Union of a family of sets definition
  49. Uniqueness quantifier definition
  50. Differentiable on an open set definition
  51. Chain rule in several variables statement
  52. Binary relation definition
  53. Universal set definition
  54. Higher partial derivatives definition
  55. Topological space definition
  56. Jacobian matrix definition
  57. Uniqueness of solutions for Lipschitz vector fields statement
  58. Differential of a smooth map definition
  59. Iterated derivative definition
  60. Complement definition
  61. Derivative of the inverse function statement
  62. Maximal solution definition
  63. Function definition
  64. Jacobian determinant definition
  65. C^k function characterised by continuous derivatives statement
  66. Flow of a vector field definition
  67. Metric space definition
  68. Open ball definition
  69. Inverse function theorem statement
  70. Invariant set definition
  71. Diffeomorphism definition
  72. Metric topology definition
  73. Open set definition
  74. Closed set definition
  75. Open cover definition
  76. Compact space definition
  77. Closure definition
  78. Dense set definition
  79. Hyperbolic set definition
  80. Axiom A definition
  81. Spectral decomposition theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Axiom_A
About this page
Tag6MTVY6NYY7
URIm2://6MTVY6NYY7
Kindstatement, theorem
WritingW0 stub
MSC37D20
StatusThis copy of the site has no build data. Run m2 status 6MTVY6NYY7.
AboutAxiom A
UsesAxiom A, Non-wandering set, Topological transitivity, Topological mixing

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