# A treatise on the analytical geometry of the point, line, by John Casey

By John Casey

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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.