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By Goffman C.

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FI / be a subshift of order 2 and suppose that every letter occurs in some word in F. Consider the next assertions. FI / is irreducible. FI / is forward transitive. Then (i) implies (ii). If F does not have isolated points or if the shift is two-sided, then (ii) implies (i). Proof. 35. It suffices to consider open sets U and V intersecting F that are of the form « ˚ « ˚ and V D x W x0 D v0 ; : : : ; xm D vm U D x W x0 D u 0 ; : : : ; xn D u n for some n; m 2 N0 , u0 ; : : : ; un ; v0 ; : : : ; vm 2 f0; : : : ; k 1g.

KI '/ be an invertible topological system, with K metrizable. , there is a point x 2 K with dense orbit. U/ \ V ¤ ;. Let us turn to our central examples. 36 (Rotation Systems). GI a/ be a left rotation system. GI a/ is topologically forward transitive. (ii) Every point of G has dense forward orbit. GI a/ is topologically transitive. (iv) Every point of G has dense total orbit. 3 Topological Transitivity 25 Proof. g// h W g 7! gh/ for every g; h 2 G. Taking closures we obtain the equivalences (i) , (ii) and (iii) , (iv).

LI / coincides with the projective limit system associated with this projective system (cf. Exercise 18). MI / ! KI '/ is a factor map. MI / ! LI / with D ı Q . ) Chapter 3 Minimality and Recurrence Point set topology is a disease from which the human race will soon recover. Henri Poincaré1 In this chapter, we study the existence of nontrivial subsystems of topological systems and the intrinsically connected phenomenon of regularly recurrent points. It was Birkhoff who discovered this connection and wrote in (1912): THÉORÈME III.

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