Math 2.0

Dynamical systems and ergodic theory Definition

Smooth dynamical system

Builds on Smooth manifoldDiffeomorphismDynamical system

Definition

Let MM be a smooth manifold and let r≥1r \ge 1 be an integer or ∞\infty. A smooth dynamical system of class CrC^r on MM is one of the following.

  1. Discrete time: a CrC^r diffeomorphism f ⁣:M→Mf \colon M \to M, with the action n↦fnn \mapsto f^n of ℤ\Z on MM.
  2. Continuous time: a CrC^r map φ ⁣:ℝ×M→M\varphi \colon \R \times M \to M, written φt(x)=φ(t,x)\varphi_t(x) = \varphi(t, x), such that φ0=id⁡M\varphi_0 = \id_M and φs+t=φs∘φt\varphi_{s+t} = \varphi_s \circ \varphi_t for all s,t∈ℝs, t \in \R.

Remarks

In case 2 each φt\varphi_t is a CrC^r diffeomorphism with inverse φ−t\varphi_{-t}, and φ\varphi is called a CrC^r flow. It is the flow of the vector field X(x)=ddtφt(x)∣t=0X(x) = \frac{d}{dt}\varphi_t(x)\big|_{t=0}, which is of class Cr−1C^{r-1}. Conversely, on a compact manifold every CrC^r vector field generates a complete CrC^r flow. Some authors also allow non-invertible CrC^r maps f ⁣:M→Mf \colon M \to M in case 1. Example: on the circle M=ℝ/ℤM = \R/\Z, the rotation f(x)=x+α mod 1f(x) = x + \alpha \bmod 1 is a C∞C^\infty diffeomorphism, and φt(x)=x+t mod 1\varphi_t(x) = x + t \bmod 1 is a C∞C^\infty flow. Non-example: f(x)=2x mod 1f(x) = 2x \bmod 1 on the circle is smooth but two-to-one, so it is not a diffeomorphism and fails case 1 in this definition.

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Dynamical system, concept, no evidence levelDynamical systemconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction composition, concept, no evidence levelFunction compositionconceptHomeomorphism, concept, no evidence levelHomeomorphismconceptSmooth function, concept, no evidence levelSmooth functionconceptTopological manifold, concept, no evidence levelTopological manifoldconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptSmooth manifold, definition, no evidence levelSmooth manifolddefinitionLimit 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 levelDiffeomorphismdefinitionSmooth dynamical system, definition, no evidence levelSmooth dynamical syst…definition
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  1. Dynamical system concept
  2. Euclidean space object
  3. Function concept
  4. Function composition concept
  5. Homeomorphism concept
  6. Smooth function concept
  7. Topological manifold concept
  8. Fréchet derivative definition
  9. Function of several variables concept
  10. Smooth manifold definition
  11. Limit of a function of several variables concept
  12. Partial derivative concept
  13. Differentiable function of several variables definition
  14. Partial derivative definition
  15. Differentiable on an open set definition
  16. Jacobian matrix definition
  17. Chain rule in several variables statement
  18. Higher partial derivatives definition
  19. Iterated derivative definition
  20. Jacobian determinant definition
  21. Derivative of the inverse function statement
  22. C^k function characterised by continuous derivatives statement
  23. Inverse function theorem statement
  24. Diffeomorphism definition
  25. Smooth dynamical system definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Dynamical_system
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Kinddefinition
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MSC37C05, 37C10
DefinesSmooth dynamical system
UsesSmooth manifold, Diffeomorphism, Dynamical system

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