Weakly mixing transformation that is Shannon orbit equivalent to a given ergodic transformation

Abstract We prove that every ergodic transformation is Shannon orbit equivalent to a weakly mixing transformation. The proof is based on the techniques introduced by Fieldsteel and Friedman to show that there is a mixing transformation for a given ergodic transformation T that is, for all <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>a</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:math> $a\geq 1$ a greater than or equals 1 , weak- a -equivalent to T and, for all <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> ${b\in (0,1)}$ b element of left parenthesis 0 comma 1 right parenthesis , strong- b -equivalent to T . In particular, we adapt the construction of Fieldsteel and Friedman by which they permute the columns of each Rokhlin tower in a sequence of rapidly growing Rokhlin towers so that the corresponding cocycles converge to an orbit equivalence cocycle of T such that the resulting transformation and orbit equivalence have the desired properties. In addition to this, we demonstrate a flexible method for obtaining actions of <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> $\mathbb {Z}^{2} $ double struck upper Z squared that are Shannon orbit equivalent to a given ergodic transformation.