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• W1569462149 abstract "While there is, up to homeomorphism, only one Cantor space, i.e. one zero-dimensional, perfect, compact, nonempty metric space, there are many measures on Cantor space which are not topologically equivalent. The <italic>clopen values set</italic> for a full, nonatomic measure <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu> <mml:semantics> <mml:mi>μ<!-- μ --></mml:mi> <mml:annotation encoding=application/x-tex>mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the countable dense subset <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace mu left-parenthesis upper U right-parenthesis colon upper U> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>U</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>U</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>{ mu (U) : U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is clopen<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=right-brace> <mml:semantics> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:annotation encoding=application/x-tex>}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the unit interval. It is a topological invariant for the measure. For the class of good measures it is a complete invariant. A full, nonatomic measure <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu> <mml:semantics> <mml:mi>μ<!-- μ --></mml:mi> <mml:annotation encoding=application/x-tex>mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>good</italic> if whenever <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper U comma upper V> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>,</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>U, V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are clopen sets with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu left-parenthesis upper U right-parenthesis greater-than mu left-parenthesis upper V right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>U</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>></mml:mo> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mu (U) > mu (V)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding=application/x-tex>W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a clopen subset of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding=application/x-tex>V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu left-parenthesis upper W right-parenthesis equals mu left-parenthesis upper U right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>W</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>U</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mu (W) = mu (U)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. These measures have interesting dynamical properties. They are exactly the measures which arise from uniquely ergodic minimal systems on Cantor space. For some of them there is a unique generic measure-preserving homeomorphism. That is, within the Polish group of such homeomorphisms there is a dense, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G Subscript delta> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>δ<!-- δ --></mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>G_{delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> conjugacy class." @default.
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• W1569462149 date "2004-04-16" @default.
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• W1569462149 title "Good measures on Cantor space" @default.
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