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hw3-ERG - Ergodic Theory MTG6401 Ergodic HW Trois Prof JLF...

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Unformatted text preview: Ergodic Theory MTG6401 Ergodic HW Trois Prof. JLF King 2Nov2009 ( Due Wednesday, 30Sep2009. Please staple this sheet as the first page of your write-up. ) Notation. Our prob. space X := ( X, X ,μ ) en- genders a complex Hilbert space H := L 2 ( μ, C ). H6: With T is a bi-mpt on X , define U = U T by U f := f ◦ T , a unitary op on H . a For each convergence notion Reg/AbsCes/Ces` aro of a sequence of numbers, prove that if h f, U n g i -→ n →∞ h f, 1 i · h 1 ,g i * : for all fncs f,g in some dense subset of H , then it holds for every pair f,g ∈ H . b Prove that T is mixing/weak-mixing/ergodic as ∀ f,g ∈ H : h f, U n g i -→ n →∞ h f, 1 i · h 1 ,g i ** : in the Regular/AbsCes` aro/Ces` aro sense. Our defn/conclusion-of-thm, for these three properties, is that ( ** ) holds when f and g are indicator-fncs of sets. H7: Here, Z 1 ,Z 2 ,Z 3 ,... ⊂ N are zero-density sets....
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