Bernoulli shifts with bases of equal entropy are isomorphic

Abstract

<p style='text-indent:20px;'>We prove that if <inline-formula><tex-math id="M1">\begin{document}$ G $\end{document}</tex-math></inline-formula> is a countably infinite group and <inline-formula><tex-math id="M2">\begin{document}$ (L, \lambda) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ (K, \kappa) $\end{document}</tex-math></inline-formula> are probability spaces having equal Shannon entropy, then the Bernoulli shifts <inline-formula><tex-math id="M4">\begin{document}$ G \curvearrowright (L^G, \lambda^G) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ G \curvearrowright (K^G, \kappa^G) $\end{document}</tex-math></inline-formula> are isomorphic. This extends Ornstein's famous isomorphism theorem to all countably infinite groups. Our proof builds on a slightly weaker theorem by Lewis Bowen in 2011 that required both <inline-formula><tex-math id="M6">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> have at least <inline-formula><tex-math id="M8">\begin{document}$ 3 $\end{document}</tex-math></inline-formula> points in their support. We furthermore produce finitary isomorphisms in the case where both <inline-formula><tex-math id="M9">\begin{document}$ L $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ K $\end{document}</tex-math></inline-formula> are finite.</p>