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### slidesBowen

Course: WORKSHOP 0309, Fall 2009
School: UCLA
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Word Count: 6340

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in Entropy Measurable Dynamics Lewis Bowen UCLA, March 2009 Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 1 / 34 Notation Let (X , ) be a measure space with (X ) = 1. Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 2 / 34 Notation Let (X , ) be a measure space with (X ) = 1. Let G be a countable discrete group acting by measure-preserving transformations on (X , )....

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in Entropy Measurable Dynamics Lewis Bowen UCLA, March 2009 Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 1 / 34 Notation Let (X , ) be a measure space with (X ) = 1. Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 2 / 34 Notation Let (X , ) be a measure space with (X ) = 1. Let G be a countable discrete group acting by measure-preserving transformations on (X , ). Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 2 / 34 Notation Let (X , ) be a measure space with (X ) = 1. Let G be a countable discrete group acting by measure-preserving transformations on (X , ). The triple (G, X , ) is a dynamical system . Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 2 / 34 Notation Let (X , ) be a measure space with (X ) = 1. Let G be a countable discrete group acting by measure-preserving transformations on (X , ). The triple (G, X , ) is a dynamical system . Two systems (G, X1 , 1 ) and (G, X2 , 2 ) are isomorphic if there exists a measure-space isomorphism : X1 X2 with (gx) = g(x) for a.e. x X1 and for all g G. Main Problem: Classify systems up to isomorphism. Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 2 / 34 Bernoulli shifts Let K be a nite set, a probability measure on K . Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 3 / 34 Bernoulli shifts Let K be a nite set, a probability measure on K . K G = {x : G K }. Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 3 / 34 Bernoulli shifts Let K be a nite set, a probability measure on K . K G = {x : G K }. G is the product measure on K G . Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 3 / 34 Bernoulli shifts Let K be a nite set, a probability measure on K . K G = {x : G K }. G is the product measure on K G . G acts on K G by shifting. (gx)(f ) = x(g 1 f ) for all x K G , g, f G. Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 3 / 34 Bernoulli shifts Let K be a nite set, a probability measure on K . K G = {x : G K }. G is the product measure on K G . G acts on K G by shifting. (gx)(f ) = x(g 1 f ) for all x K G , g, f G. (G, K G , G ) is the Bernoulli shift over G with base measure . Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 3 / 34 von Neumanns question Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 4 / 34 von Neumanns question If |K | = n and is the uniform probability measure on K then (G, K G , G ) is the full n-shift over G. Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 4 / 34 von Neumanns question If |K | = n and is the uniform probability measure on K then (G, K G , G ) is the full n-shift over G. von Neumanns question: Is the full 2-shift over Z isomorphic to the full 3-shift over Z? Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 4 / 34 Ideas from Information Theory Let x X be a point unknown to us. Let E X . Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 5 / 34 Ideas from Information Theory Let x X be a point unknown to us. Let E X . Goal: quantify the amount of information we gain by being told that x E. Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 5 / 34 Ideas from Information Theory Let x X be a point unknown to us. Let E X . Goal: quantify the amount of information we gain by being told that x E. This amount, denoted I(E), should depend only on (E). So write I(E) = I((E)). Lewis Bowen (University of Hawaii) Entropy in Measurable Dynamics 5 / 34 Ideas from Information Theory Let x X be a point unknown to us. Let E X . Goal: quantify the amount of information we gain by being told that x E. This...
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