Math 2.0

Dynamical systems and ergodic theory Topological dynamics Theorem

Shadowing lemma

Builds on Riemannian manifoldDiffeomorphismHyperbolic setExpansive homeomorphism

Statement

Let MM be a Riemannian manifold with distance dd, let f ⁣:M→Mf \colon M \to M be a C1C^1 diffeomorphism, and let Λ⊆M\Lambda \subseteq M be a compact hyperbolic set for ff. For ε>0\varepsilon > 0, an ε\varepsilon-pseudo-orbit is a sequence (xn)n∈ℤ(x_n)_{n \in \Z} in MM with d(f(xn),xn+1)<εd(f(x_n), x_{n+1}) < \varepsilon for all n∈ℤn \in \Z. A point x∈Mx \in M δ\delta-shadows it if d(fn(x),xn)<δd(f^n(x), x_n) < \delta for all n∈ℤn \in \Z. Then there is an open neighbourhood UU of Λ\Lambda with the property that follows. For every δ>0\delta > 0 there is ε>0\varepsilon > 0 such that every ε\varepsilon-pseudo-orbit contained in UU is δ\delta-shadowed by some x∈Mx \in M. Moreover, there is δ0>0\delta_0 > 0 such that for δ<δ0\delta < \delta_0 the shadowing point xx is unique.

Remarks

The idea of the proof: a pseudo-orbit is a near-solution of the equation xn+1=f(xn)x_{n+1} = f(x_n) in a space of sequences. Hyperbolicity makes the linearised equation invertible with a bounded inverse, contracting forward along stable directions and backward along unstable ones, and a fixed-point argument gives a true solution nearby. Uniqueness comes from expansiveness near Λ\Lambda: two orbits that stay δ\delta-close for all times coincide. For an Anosov diffeomorphism of a compact manifold, Λ=M\Lambda = M, so every pseudo-orbit with small enough errors is shadowed. Hyperbolicity is needed: consider the rotation of the circle by an irrational angle α\alpha. The pseudo-orbit that rotates by α+ε/2\alpha + \varepsilon/2 at each step drifts away from every true orbit at rate ε/2\varepsilon/2 per step.

Going deeper

For a student

Shadowing is the main tool for building orbits with prescribed behaviour in hyperbolic dynamics. One glues pieces of orbits into a pseudo-orbit and shadows it. This gives the Anosov closing lemma, which finds a periodic orbit near any orbit segment that nearly closes up. It also gives the Markov partitions used to code the dynamics by subshifts of finite type. It also gives structural stability of Anosov diffeomorphisms, since a perturbed orbit is a pseudo-orbit of the original map.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectExpansive homeomorphism, concept, no evidence levelExpansive homeomorphi…conceptFunction, concept, no evidence levelFunctionconceptHyperbolic set, concept, no evidence levelHyperbolic setconceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptDiffeomorphism, concept, no evidence levelDiffeomorphismconceptShadowing lemma, statement, no evidence levelShadowing lemmastatement
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. Expansive homeomorphism concept
  3. Function concept
  4. Hyperbolic set concept
  5. Riemannian manifold concept
  6. Total derivative concept
  7. Function of several variables concept
  8. Differentiable function of several variables concept
  9. Diffeomorphism concept
  10. Shadowing lemma statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Shadowing_lemma
About this page
TagN7C06T3KCZ
URIm2://N7C06T3KCZ
Kindstatement, theorem
WritingW0 stub
MSC37C50, 37D05
StatusThis copy of the site has no build data. Run m2 status N7C06T3KCZ.
AboutHyperbolic set
UsesRiemannian manifold, Diffeomorphism, Hyperbolic set, Expansive homeomorphism
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