in Entropy Measurable Dynamics
Lewis Bowen
UCLA, March 2009
Lewis Bowen (University of Hawaii)
Entropy in Measurable Dynamics
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Notation
Let (X , ) be a measure space with (X ) = 1.
Lewis Bowen (University of Hawaii)
Entropy in Measurable Dynamics
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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
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Bernoulli shifts
Let K be a nite set, a probability measure on K .
Lewis Bowen (University of Hawaii)
Entropy in Measurable Dynamics
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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
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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
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von Neumanns question
Lewis Bowen (University of Hawaii)
Entropy in Measurable Dynamics
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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
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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
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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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