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Standardness of sequences of σ-fields given by certain endomorphisms

Jacob FeldmanDaniel Rudolph — 1998

Fundamenta Mathematicae

 Let E be an ergodic endomorphism of the Lebesgue probability space X, ℱ, μ. It gives rise to a decreasing sequence of σ-fields , E - 1 , E - 2 , . . . A central example is the one-sided shift σ on X = 0 , 1 with 1 2 , 1 2 product measure. Now let T be an ergodic automorphism of zero entropy on (Y, ν). The [I|T] endomorphismis defined on (X× Y, μ× ν) by ( x , y ) ( σ ( x ) , T x ( 1 ) ( y ) ) . Here ℱ is the σ-field of μ× ν-measurable sets. Each field is a two-point extension of the one beneath it. Vershik has defined as “standard” any decreasing sequence of σ-fields isomorphic...

If the [T,Id] automorphism is Bernoulli then the [T,Id] endomorphism is standard

Christopher HoffmanDaniel Rudolph — 2003

Studia Mathematica

For any 1-1 measure preserving map T of a probability space we can form the [T,Id] and [ T , T - 1 ] automorphisms as well as the corresponding endomorphisms and decreasing sequence of σ-algebras. In this paper we show that if T has zero entropy and the [T,Id] automorphism is isomorphic to a Bernoulli shift then the decreasing sequence of σ-algebras generated by the [T,Id] endomorphism is standard. We also show that if T has zero entropy and the [T²,Id] automorphism is isomorphic to a Bernoulli shift then the...

The Morse minimal system is finitarily Kakutani equivalent to the binary odometer

Mrinal Kanti RoychowdhuryDaniel J. Rudolph — 2008

Fundamenta Mathematicae

Two invertible dynamical systems (X,,μ,T) and (Y,,ν,S), where X and Y are Polish spaces and Borel probability spaces and T, S are measure preserving homeomorphisms of X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping ϕ from a subset X₀ of X of measure one onto a subset Y₀ of Y of full measure such that (1) ϕ | X is continuous in the relative topology on X₀ and ϕ - 1 | Y is continuous in the relative topology on Y₀, (2) ϕ ( O r b T ( x ) ) = O r b S ( ϕ ( x ) ) for μ-a.e. x ∈ X. (X,,μ,T) and...

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