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• W1961328222 abstract "Furstenberg showed that if two topological systems <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper X comma upper T right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(X,T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper Y comma upper S right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(Y,S)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are disjoint, then one of them, say <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper Y comma upper S right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(Y,S)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is minimal. When <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper Y comma upper S right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(Y,S)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is nontrivial, we prove that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper X comma upper T right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(X,T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> must have dense recurrent points, and there are countably many maximal transitive subsystems of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper X comma upper T right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(X,T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that their union is dense and each of them is disjoint from <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper Y comma upper S right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(Y,S)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Showing that a weakly mixing system with dense periodic points is in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M Superscript up-tack> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>⊥<!-- ⊥ --></mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>{mathcal {M}}^{perp }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the collection of all systems disjoint from any minimal system, Furstenberg asked the question to characterize the systems in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M Superscript up-tack> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>⊥<!-- ⊥ --></mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>{mathcal {M}}^{perp }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that a weakly mixing system with dense regular minimal points is in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M Superscript up-tack> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>⊥<!-- ⊥ --></mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>{mathcal {M}}^{perp }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and each system in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M Superscript up-tack> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>⊥<!-- ⊥ --></mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>{mathcal {M}}^{perp }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has dense minimal points and it is weakly mixing if it is transitive. Transitive systems in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M Superscript up-tack> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>⊥<!-- ⊥ --></mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>{mathcal {M}}^{perp }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and having no periodic points are constructed. Moreover, we show that there is a distal system in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M Superscript up-tack> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>⊥<!-- ⊥ --></mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>{mathcal {M}}^{perp }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Recently, Weiss showed that a system is weakly disjoint from all weakly mixing systems iff it is topologically ergodic. We construct an example which is weakly disjoint from all topologically ergodic systems and is not weakly mixing." @default.
• W1961328222 created "2016-06-24" @default.
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• W1961328222 date "2004-04-16" @default.
• W1961328222 modified "2023-10-16" @default.
• W1961328222 title "Dynamical systems disjoint from any minimal system" @default.
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