Conjugating trivial automorphisms of 𝒫(ℕ)/`ch

A trivial automorphism of the Boolean algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P left-parenthesis double-struck upper N right-parenthesis slash normal upper F normal i normal n"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">F</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal P(\mathbb N) / \mathrm {Fin}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an automorphism induced by the action of some function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper N right-arrow double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo stretchy="false"> → </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb N \rightarrow \mathbb N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The forcing axiom <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper O sans-serif upper C sans-serif upper A Subscript normal upper T"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">O</mml:mi> <mml:mi mathvariant="sans-serif">C</mml:mi> <mml:mi mathvariant="sans-serif">A</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">T</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathsf {OCA}_{\mathrm {T}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> implies all automorphisms are trivial, and therefore two trivial automorphisms are conjugate if and only if they have the same (modulo finite) orbit structure. We show that the Continuum Hypothesis implies that two trivial automorphisms are conjugate if and only if there are no first-order obstructions to their conjugacy and their indices have the same parity, if and only if the given trivial automorphisms are conjugate in some forcing extension of the universe. To each automorphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P left-parenthesis double-struck upper N right-parenthesis slash normal upper F normal i normal n"> <mml:semantics> <mml:mrow> <mml:mpadded height="+1.5pt" depth="-1.5pt" voffset="+1.5pt"> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mstyle displaystyle="false" scriptlevel="1"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mstyle> </mml:mrow> </mml:mstyle> </mml:mpadded> <mml:mspace width="-0.056em"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mspace width="-0.056em"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mstyle displaystyle="false" scriptlevel="1"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">F</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> </mml:mstyle> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\raise 1.5pt\hbox {\(\scriptstyle \mathcal {P}(\mathbb {N})\)}\mkern -1mu/\mkern -1mu{\scriptstyle \mathrm {Fin}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we associate the first-order structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper A Subscript alpha Baseline equals left-parenthesis script upper P left-parenthesis double-struck upper N right-parenthesis slash normal upper F normal i normal n comma alpha right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">A</mml:mi> </mml:mrow> <mml:mi> α </mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mpadded height="+1.5pt" depth="-1.5pt" voffset="+1.5pt"> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mstyle displaystyle="false" scriptlevel="1"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mstyle> </mml:mrow> </mml:mstyle> </mml:mpadded> <mml:mspace width="-0.056em"/> <mml:mrow class="MJX-TeXAtom-ORD">