Type: Article
Publication Date: 2020-10-30
Citations: 9
DOI: https://doi.org/10.1090/btran/53
To every one-sided shift space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we associate a cover <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper X overTilde"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> <mml:mo>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\widetilde {\mathsf {X}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a groupoid <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper G Subscript sans-serif upper X"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {G}_\mathsf {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi mathvariant="normal">C</mml:mi> <mml:mo>β<!-- β --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {C^*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O Subscript sans-serif upper X"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {O}_\mathsf {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We characterize one-sided conjugacy, eventual conjugacy and (stabilizer-preserving) continuous orbit equivalence between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {X}</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="sans-serif upper Y"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {Y}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of isomorphism of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper G Subscript sans-serif upper X"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {G}_\mathsf {X}</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="script upper G Subscript sans-serif upper Y"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">Y</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {G}_\mathsf {Y}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and diagonal-preserving <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi /> <mml:mo>β<!-- β --></mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-isomorphism of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O Subscript sans-serif upper X"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {O}_\mathsf {X}</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="script upper O Subscript sans-serif upper Y"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">Y</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {O}_\mathsf {Y}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also characterize two-sided conjugacy and flow equivalence of the associated two-sided shift spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript sans-serif upper X"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Ξ<!-- Ξ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\Lambda _\mathsf {X}</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="normal upper Lamda Subscript sans-serif upper Y"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Ξ<!-- Ξ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">Y</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\Lambda _\mathsf {Y}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of isomorphism of the stabilized groupoids <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper G Subscript sans-serif upper X Baseline times script upper R"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> </mml:msub> <mml:mo>Γ<!-- Γ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {G}_\mathsf {X}\times \mathcal {R}</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="script upper G Subscript sans-serif upper Y Baseline times script upper R"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">Y</mml:mi> </mml:mrow> </mml:msub> <mml:mo>Γ<!-- Γ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {G}_\mathsf {Y}\times \mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and diagonal-preserving <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi /> <mml:mo>β<!-- β --></mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-isomorphism of the stabilized <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi mathvariant="normal">C</mml:mi> <mml:mo>β<!-- β --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {C^*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebras <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O Subscript sans-serif upper X Baseline circled-times double-struck upper K"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> </mml:msub> <mml:mo>β<!-- β --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {O}_\mathsf {X}\otimes \mathbb {K}</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="script upper O Subscript sans-serif upper Y Baseline circled-times double-struck upper K"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">Y</mml:mi> </mml:mrow> </mml:msub> <mml:mo>β<!-- β --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {O}_\mathsf {Y}\otimes \mathbb {K}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our strategy is to lift relations on the shift spaces to similar relations on the covers. Restricting to the class of sofic shifts whose groupoids are effective, we show that it is possible to recover the continuous orbit equivalence class of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from the pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis script upper O Subscript sans-serif upper X Baseline comma upper C left-parenthesis sans-serif upper X right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathcal {O}_\mathsf {X}, C(\mathsf {X}))</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the flow equivalence class of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript sans-serif upper X"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Ξ<!-- Ξ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\Lambda _\mathsf {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from the pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis script upper O Subscript sans-serif upper X Baseline circled-times double-struck upper K comma upper C left-parenthesis sans-serif upper X right-parenthesis circled-times c 0 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> </mml:msub> <mml:mo>β<!-- β --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>β<!-- β --></mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathcal {O}_\mathsf {X}\otimes \mathbb {K}, C(\mathsf {X})\otimes c_0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, continuous orbit equivalence implies flow equivalence for this class of shift spaces.