Generic mixing transformations are rank 1
Let \((X,\mathcal{B},\mu)\) be a non-atomic Lebesgue space. By the classical Rhoklin theorem, \((X,\mathcal{B},\mu)\) is isomorphic to \(([0,1],\mathcal{B}([0,1]),\lambda)\) where \(\mathcal{B}([0,1])\) is the Borel algebra and \(\lambda\) is the Lebesgue measure. Let us denote by \(\mathfrak{T}\) the space of all invertible measure-preserving transformations acting on \((X,\mathcal{B},\mu)\). A map \(T\) is in \(\mathfrak{T}\) if and only if \(\mu(T^{-1}A)=\mu(A)\), for every \(A \in \mathcal{B}\). We denote by \(\mathfrak{M}\) the sublass of mixing transformations. A map \(T\) is in \(\mathfrak{M}\) if and only if \(\displaystyle \lim_{n \longrightarrow +\infty}\mu(T^nA \cap B)=\mu(A)\mu(B).\) The weak topology on \(\mathfrak{T}\) is generated by each of the following metrics \[ d(T,S)=\sum_{i \in \mathbb{N}} \frac1{2^i}\Big(\mu(TA_i \Delta SA_i)+\mu(T^{-1}A_i \Delta S^{-1}A_i)\Big), \] \[ a(T,S)=\sum_{i,j \in \mathbb{N}}\frac1{2^{i+j}}\Big|\mu(TA_i \Delta A_j)-\mu(SA_i \Delta A_j)\Big|, \] where \((A_i)\) is a countable collection of Borel sets generating the \(\sigma\)-algebra \(\Sigma\). The space \(\mathfrak{M}\) equipped with those metrics is not complete. To make it a complete metric space, \textit{S. V. Tikhonov} [Math. Notes 95, No. 2, 253--266 (2014); translation from Mat. Zametki 95, No. 2, 282--299 (2014; Zbl 1370.37004)] introduced the following metric \[ \tau(T,S)=d(T,S)+\sup_{n \in \mathbb{N}}a(T^n,S^n). \] The corresponding topology is called the leash topology. In this paper the author proves that the conjugacy class of any mixing map \(S\) is dense in \(\mathfrak{M}\). We remind that the conjugacy class of \(S\) is given by \[ c(S)=\Big\{T,\quad T=R^{-1}SR,\, \text{for some } R \in \mathfrak{T}\Big\}. \] As a consequence, the author obtains that the mixing rank-one maps are dense in \(\mathfrak{M}\).
scientific article; zbMATH DE number 6158177
scientific article