Entropy of automorphisms of \(\text{II}_1\)-factors arising from the dynamical systems theory (Q5937640)

Suppose that a countable amenable group \(G\) acts freely and ergodically by automorphisms \((S_g)\) of a Lebesgue space \((X,\mu)\) and suppose that \(S_g\) preserves the measure \(\mu\). Let \(M\) be the crossed product of \(L^\infty(X,\mu)\) and \(G\), which is a \(\text{II}_1\)-factor. Suppose that \(T\) is a measure preserving automorphism of \((X,\mu)\) such that \(T S_g = S_{\beta(g)} T\) for some automorphism \(\beta\) of \(G\). NEWLINENEWLINENEWLINEBecause \(T\) extends to an automorphism \(\alpha_T\) of \(M\), it is a natural idea to compare the Connes-Størmer entropy \(H(\alpha_T)\) [\textit{A. Connes} and \textit{E. Størmer}, Acta Math. 134, 289-306 (1975; Zbl 0326.46032)] with the Kolmogorov-Sinai entropy \(h(T)\) of \(T\). If \(T\) actually commutes with \((S_g)\), it is proven that \(H(\alpha_T)=h(T)\). Next, examples are constructed showing that, when \(\beta\) is non-trivial, any values \(h(T)=s\) and \(H(\alpha_T)=t\) for \(0 \leq s \leq t \leq \infty\) are possible. NEWLINENEWLINENEWLINEUsing their general construction, the authors prove the existence of ergodic automorphisms of an amenable equivalence relation which are non-isomorphic, but have the same Kolmogorov-Sinai entropy. More specifically, the authors construct automorphisms of the hyperfinite \(\text{II}_1\)-factor which are not conjugate, but have the same Connes-Størmer entropy.