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Extra resources for Cardinal functions in topology, ten years later

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46 3 Topological Dynamics 3. For the third inequality, we note that no dn -open ball of radius ε/2 contains two points at a dn -distance ε. Thus N (n, ε) ≤ M(n, ε/2). 4. Finally, let U1 , . . , Um be a cover of X by sets of dn -diameter less than ε/2, where m = C(n, ε/2). Now take a point pi ∈ Ui for each i. Clearly, Bn (pi , ε/2) ⊃ Ui and these dn -balls form a cover of X. Hence, M(n, ε/2) ≤ C(n, ε/2). This completes the proof of the proposition. Now we obtain several alternative formulas for the topological entropy of a dynamical system.

8) that C ∩ ω(x) = ∅. This contradiction shows that the set ω(x) is connected. In the case of flows, we have analogous results for the α-limit set. 3 Topological Recurrence 37 1. y ∈ α(x) if and only if there exists a sequence tk −∞ in R− such that ϕtk (x) → y when k → ∞; 2. if Φ is a topological flow, then α(x) is backward Φ-invariant. 8 Given a topological flow Φ = (ϕt )t∈R of X, if the negative semiorbit γ − (x) of a point x ∈ X has compact closure, then: 1. α(x) is compact, connected and nonempty; 2.

6 Given a topological semiflow Φ = (ϕt )t≥0 of X, if the positive semiorbit γ + (x) of a point x ∈ X has compact closure, then: 1. ω(x) is compact, connected and nonempty; 2. inf{d(ϕt (x), y) : y ∈ ω(x)} → 0 when t → +∞. 2. In order to show that ω(x) is connected, we proceed by contradiction. If ω(x) was not connected, then we could write it in the form ω(x) = A ∪ B for some nonempty sets A and B such that A ∩ B = A ∩ B = ∅. 36 3 Topological Dynamics Fig. 3 The set C ∩ {ϕs (x) : s > t} Since ω(x) is closed, we have A = A ∩ ω(x) = A ∩ (A ∪ B) = (A ∩ A) ∪ (A ∩ B) = A and analogously B = B.

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