hw0-ERG - DynSys MTG6401 Ergodic HW Efes Prof JLF King...

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Unformatted text preview: DynSys MTG6401 Ergodic HW Efes Prof. JLF King 4Dec2009 W1: Let K ⊂ C be the unit-circle in the complex plane, equipped with normalized arclength measure. Let S : K → K be the squaring-map z 7→ z 2 , which is a measure-preserving 2-to-1 map on K . Prove that S is measure-theoretically isomorphic to the 1-sided shift on head-tails coin-flip space, where Prob( Heads ) = 1 / 2 and Prob( Tails ) = 1 / 2 . Of course, S is not topologically isomorphic to the shift, since S lives on a circle, and the shift lives on a Cantor-set. W2: For B a complex number, let N B be the New- ton’s method map to find a zero of polynomial g B ( x ) := x 2- B 2 . Thus N B ( x ) = [ x 2 + B 2 ] [2 x ] . Use N () as a shorthand for N 1 (). Since N ( z ) equals 1 2 [ z + 1 z ], it averages a number and its reciprocal. a Argue that N B is well-defined as a map of the Riemann-sphere, ˙ C , to itself. b For B 6 = 0, show that N B is topologically conju- gate to N 1 . That is, find a homeomorphism f : ˙ C →...
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