Math 2.0

Dynamical systems and ergodic theory Low-dimensional dynamics Example

Tent map

Builds on Discrete-time dynamical systemInvariant measureLebesgue measureTopological entropyLyapunov exponent

Example

For a parameter μ∈[0,2]\mu \in [0, 2], the tent map is Tμ ⁣:[0,1]→[0,1],Tμ(x)=μmin⁡(x,1−x).T_\mu \colon [0, 1] \to [0, 1], \qquad T_\mu(x) = \mu \min(x, 1 - x). Its graph is a tent: it rises with slope μ\mu on [0,1/2][0, 1/2] and falls with slope −μ-\mu on [1/2,1][1/2, 1]. Its top value is μ/2≤1\mu/2 \le 1, so it maps [0,1][0, 1] into itself and is a discrete-time dynamical system.

Check that Lebesgue measure λ\lambda is invariant for μ=2\mu = 2. For 0≤a<b≤10 \le a < b \le 1, the preimage T2−1([a,b])T_2^{-1}([a, b]) is [a/2,b/2]∪[1−b/2,1−a/2][a/2, b/2] \cup [1 - b/2, 1 - a/2], two intervals of length (b−a)/2(b - a)/2 each. So λ(T2−1[a,b])=b−a=λ([a,b])\lambda(T_2^{-1}[a, b]) = b - a = \lambda([a, b]), and this extends from intervals to all Borel sets.

Remarks

For μ=2\mu = 2 the map is ergodic for Lebesgue measure, has topological entropy log⁡2\log 2, and has Lyapunov exponent log⁡2\log 2 at every point whose orbit avoids 1/21/2, since ∣T2′∣=2\abs{T_2'} = 2. For 0≤μ<10 \le \mu < 1 every orbit tends to the fixed point 00. The map T2T_2 is conjugate to the logistic map x↦4x(1−x)x \mapsto 4x(1 - x).

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptDiscrete-time dynamical system, concept, no evidence levelDiscrete-time dynamic…conceptEuclidean space, object, no evidence levelEuclidean spaceobjectInvariant measure, concept, no evidence levelInvariant measureconceptLyapunov exponent, concept, no evidence levelLyapunov exponentconceptOuter measure, concept, no evidence levelOuter measureconceptTopological entropy, concept, no evidence levelTopological entropyconceptLebesgue measure, object, no evidence levelLebesgue measureobjectTent map, example, no evidence levelTent mapexample
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  1. Borel sigma-algebra concept
  2. Discrete-time dynamical system concept
  3. Euclidean space object
  4. Invariant measure concept
  5. Lyapunov exponent concept
  6. Outer measure concept
  7. Topological entropy concept
  8. Lebesgue measure object
  9. Tent map example

Where this leads

9 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Tent_map
About this page
TagD0A22MEB1F
URIm2://D0A22MEB1F
Kindexample
WritingW0 stub
MSC37E05, 37A05
Instance ofDiscrete-time dynamical system
UsesDiscrete-time dynamical system, Invariant measure, Lebesgue measure, Topological entropy, Lyapunov exponent
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