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By David R. Morrison, Janos Kolla Summer Research Institute on Algebraic Geometry

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Additional info for Algebraic Geometry Santa Cruz 1995: Summer Research Institute on Algebraic Geometry, July 9-29, 1995, University of California, Santa Cruz (Proceedings of Symposia in Pure Mathematics) (Pt. 2)

Example text

13 is topologically conjugate to the expanding map E2 . 18)) that h(f ) = h(E2 ) = log 2. 3 Alternative Characterizations In this section we describe several alternative characterizations of topological entropy. These are particularly useful in the computation of the entropy. 10 Given n ∈ N and ε > 0, we denote by M(n, ε) the least number of points p1 , . . , pm ∈ X such that each x ∈ X satisfies dn (x, pi ) < ε for some i. 11 Given n ∈ N and ε > 0, we denote by C(n, ε) the least number of elements of a cover of X by sets U1 , .

This shows that the rotation Rα is not topologically mixing. 9 Now we consider the expanding map E2 : S 1 → S 1 . 4, there exists a point x ∈ S 1 whose positive semiorbit γ + (x) is dense in S 1 . 1 that the map E2 is topologically transitive. Now we show that E2 is also topologically mixing. x1 x2 · · · xn 11 · · · , 40 3 Topological Dynamics with the endpoints written in base-2. x1 x2 · · · xn y1 y2 · · · ∈ I is in E2−n U since E2n (x) = y. Therefore, E2−n U ∩ V ⊃ E2−n U ∩ I = ∅. This shows that the map E2 is topologically mixing.

5 Let ϕt be a flow determined by a differential equation x = f (x) for some C 1 function f : R2 → R2 . Show that if L ⊂ R2 is a transversal to f (that is, a line segment such that for each x ∈ L the directions of L and f (x) generate R2 ), then for each x ∈ R2 the set ω(x) ∩ L contains at most one point. 6 Show that no increasing homeomorphism f : I → I , where I ⊂ R is an interval, is topologically transitive. 7 Let f : I → I be a continuous onto map, where I ⊂ R is an interval. Show that the following properties are equivalent: 1.

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