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 ,ν ), where Y is a nv-cpt metric space with Borel field Y and non-atomic prob.meas ν , 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 α ∈ X so that z is the...
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