By Tanja Eisner, Bálint Farkas, Markus Haase, Rainer Nagel

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**Example text**

Hint: Consider the iterates of the points . ) 19. Ki I 'i //i ; . 18. KI '/. K/ and the Koopman Operator Explain this to me on a simple example; the difficult example I will be able to do on my own. Israel M. Gelfand In the previous two chapters we introduced the concept of a topological dynamical system and discussed certain basic notions such as minimality, recurrence, and transitivity. However, a deeper study requires a change of perspective: Instead of the state space transformation ' W K !

X0 /. x0 ; 1/. x0 ; 1/ for all n 2 N. GI h/, so if V is a neighborhood of 1, then hn 2 V for some n 2 N. x0 ; hn / 2 U V for any neighborhood U of x0 . HI /. An analogous result is true for uniformly recurrent points. 16. HI / the group extension along ˚ W K ! G. x0 ; g/ 2 H is uniformly recurrent in H for all g 2 G. Proof. x0 ; h/ is uniformly recurrent for one h 2 G. KI '/ is minimal. 5). The projection W H ! H 0 / is a '-invariant subset in K, and therefore must be equal to K. x0 ; h/ 2 H 0 .

16. 8. GI a/ be a group rotation and let be a closed subgroup of G. GI a/, cf. 19. 9. 25. 10 (Dyadic Adding Machine). Let K WD f0; 1gN0 be endowed with the product topology, and define « W K ! x0 C Z2 ; x0 C 2x1 C Z4 ; x0 C 2x1 C 4x2 C Z8 ; : : : : : : /: Show that « is a homeomorphism. KI '/ ! A2 I 1/ into an isomorphism. Describe the action of ' on a 0-1-sequence x 2 K. 11. KI '/ be a topological system. a) Show that if A Â K is invariant/stable, then A is invariant/stable, too. b) Show that the intersection of arbitrarily many (bi-)invariant subsets of K is again (bi-)invariant.