b-hm-ERG - DynSys MTG6401 Home-B Prof. JLF King 27Dec2009...

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Unformatted text preview: DynSys MTG6401 Home-B Prof. JLF King 27Dec2009 Intro . Due, no later than 11AM,Monday, 14Dec , slid completely under my office door, LIT402. A family A P ( X ) is an algebra if A is sealed under complement, pairwise union and pairwise intersection. For a map f : X X , use f n for f n = f f n ... f . B1: Exhibit an ( S : Y, Y , ), where Y is a nv-cpt metric space with Borel field Y and non-atomic prob.meas , and S is a bi-mpt homeomorphism. Construct S so that S S is both topologically conjugate, and isomorphic, to S . Produce an example where S is ergodic; you may quote theorems from class. [ Hint: What are the ergodic-multipliers? What properties are sealed under projective limits? ] B2: Voila f : X , an isometry of a complete metric space ( X, d , e ). Suppose we have a point z X whose orbit { z n } n Z is dense, where z n := f n ( z ). a Construct ( with proof, natch ) an abelian group operation on X and element X so that z is the...
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