Download Dynamical systems 4. exercise 30.9.2016 1. Prove Theorem 2.2.5. 2

Dynamical systems
4. exercise 30.9.2016
1. Prove Theorem 2.2.5.
2. Prove that (Σ+
m , σ) is topologically mixing.
3. Construct an example of a topological dynamical system (X, T ) such
that limn→∞ T n (x) = x0 for all x ∈ X but {x0 } is not an attractor.
4. Show that if π : X → Y is a topological semiconjugacy and (X, T )
is topologically mixing, then (Y, S) is topologically mixing.
5. Prove that the map π : S → Φ in Example 2.3.7 is continuous and
π ◦ h = idΦ .
6. Let Λ = {x ∈ [0, 1] | Tµn (x) ∈ [0, 1] for all n ∈ N}, where Tµ is the
logistic map.
√ Show that Tµ : Λ → Λ is topologically mixing provided
µ > 2 + 5.
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