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Additional info for A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples

Example text

L. Barreira, C. 1) 27 28 3 Topological Dynamics Now we consider the case of continuous time. 2 Any flow (respectively, any semiflow) ϕt : X → X such that the map (t, x) → ϕt (x) is continuous in R × X (respectively, in R+ 0 × X) is said to be a topological flow (respectively, a topological semiflow). Any topological flow or semiflow is also said to be a topological dynamical system with continuous time or simply a topological dynamical system. In particular, the continuity assumptions imply that each map ϕt : X → X is continuous (in the case of flows it is even a homeomorphism).

If f and g are topologically conjugate, then h(f ) = h(g). Proof Let H : X → Y be a homeomorphism such that H ◦ f = g ◦ H. 20) where dX and dY are, respectively, the distances on X and Y . We note that δ → 0 when ε → 0. 19) that H f m (x) = g m (H (x)) for m ∈ N and x ∈ X. 20), if p1 , . . , pm ∈ Y are such that max dY g m (qi ), g m (qj ) : m = 0, . . , n − 1 ≥ ε for i = j, where qi = H (pi ), then max dX f m (pi ), f m (pj ) : m = 0, . . , n − 1 ≥ δ for i = j. 8. 4 Topological Entropy 45 for each ε > 0.

6) that d f mk (x), y ≥ δ for k ∈ N and y ∈ ω(x) and thus, d(p, y) ≥ δ for y ∈ ω(x). But this is impossible since p ∈ ω(x). This contradiction yields the last property in the proposition. For invertible maps, we have the following results for the α-limit set. 3 Given an invertible map f : X → X, for each x ∈ X the following properties hold: ∞ in N such that 1. y ∈ α(x) if and only if there exists a sequence nk f −nk (x) → y when k → ∞; 2. if f has a continuous inverse, then α(x) is backward f -invariant.

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