Math 2.0

Dynamical systems and ergodic theory Definition

Hyperbolic set

Builds on Riemannian manifoldDiffeomorphismInvariant setDifferential of a smooth mapCompact space

Definition

Let MM be a Riemannian manifold with norm ∥⋅∥\norm{\cdot} on each tangent space, and let f ⁣:M→Mf \colon M \to M be a C1C^1 diffeomorphism with differential DfDf. A set Λ⊆M\Lambda \subseteq M is a hyperbolic set of ff if the following hold.

  1. Λ\Lambda is compact and f(Λ)=Λf(\Lambda) = \Lambda.
  2. For each x∈Λx \in \Lambda there is a splitting TxM=Exs⊕ExuT_x M = E^s_x \oplus E^u_x with Dfx(Exs)=Ef(x)sDf_x(E^s_x) = E^s_{f(x)} and Dfx(Exu)=Ef(x)uDf_x(E^u_x) = E^u_{f(x)}.
  3. There are constants C>0C > 0 and 0<λ<10 < \lambda < 1 such that, for all x∈Λx \in \Lambda and n≥0n \ge 0, ∥Dfxnv∥≤Cλn∥v∥\norm{Df^n_x v} \le C \lambda^n \norm{v} for v∈Exsv \in E^s_x and ∥Dfx−nv∥≤Cλn∥v∥\norm{Df^{-n}_x v} \le C \lambda^n \norm{v} for v∈Exuv \in E^u_x.

Remarks

The subspaces ExsE^s_x and ExuE^u_x are then unique and depend continuously on xx. Changing the Riemannian metric changes CC but not the property, because Λ\Lambda is compact; an adapted metric allows C=1C = 1. Example: a fixed point pp of ff is a hyperbolic set exactly when no eigenvalue of DfpDf_p has modulus 11. For f(x,y)=(x/2,2y)f(x, y) = (x/2, 2y) on ℝ2\R^2 and Λ={0}\Lambda = \set{0}, take EsE^s the xx-axis, EuE^u the yy-axis, C=1C = 1 and λ=1/2\lambda = 1/2. Non-example: for the rotation f(z)=eiθzf(z) = e^{i\theta} z of ℝ2=ℂ\R^2 = \C, the set {0}\set{0} is compact and invariant, but Df0Df_0 preserves lengths, so no splitting satisfies clause 3. A flow version replaces clause 2 by TxM=Exs⊕Ex0⊕ExuT_x M = E^s_x \oplus E^0_x \oplus E^u_x, where Ex0E^0_x is spanned by the vector field, as in the definition of an Anosov flow.

Reading path

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

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Hyperbolic_set
About this page
TagNY9MCZXY6N
URIm2://NY9MCZXY6N
Kinddefinition
WritingW0 stub
MSC37D05, 37D20
DefinesHyperbolic set
UsesRiemannian manifold, Diffeomorphism, Invariant set, Differential of a smooth map, Compact space

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