×

zbMATH — the first resource for mathematics

Gaussian automorphisms whose ergodic self-joinings are Gaussian. (English) Zbl 0977.37003
Let \((X_i,\mathcal B_i,\mu_i, T_i)\) \((i=1,2)\) be dynamical systems, and denote by \(U_{T_i}\) the corresponding unitary operators. For a probability measure \(\lambda\) whose marginal distributions are \(\mu_i\), define \(\Phi_\lambda\) by \((\Phi_\lambda f|g)=\int f(x_1)\overline{g(x_2)} dx_1 dx_2\). Let \(H_i=H_i^r+iH_i^r\) (Gaussian space), where each component of \(H_i^r\) is a real function with mean 0 and has Gaussian distribution, and moreover the smallest \(\sigma\)-algebra which makes \(H^r\) measurable is \(\mathcal B_i\). \(T_i\) is called a Gaussian automorphism when \(H_i\) is invariant under \(U_{T_i}\).
The mapping \(\lambda\mapsto\Phi_\lambda|_{H_1}\) establishes a 1-1 correspondence between Gaussian joinings of \(T_1\) and \(T_2\) and real operators from \(H_1\) to \(H_2\) of norm at most 1 intertwining \(U_{T_1}|_{H_1}\) and \(U_{T_2}|_{H_2}\). A joining \(\lambda\) is called Gaussian joining when for \(f_i\in H_i^r\) \(f_1+f_2\) has a Gaussian distribution whenever it is not identically 0. A Gaussian automorphism is called GAG if its set of ergodic self-joinings coincides with the set of Gaussian self joinings.
\(T_\sigma\) is a GAG if and only if it has commuting self-joinings. In particular, if it has simple spectrum then it is a GAG. \(C^g(T)\) is a Gaussian centralizer which is the subset of centralizer preserving Gaussian space. \(\widehat{\mathcal A}\supset{\mathcal A}\) is the smallest factor such that \(\mathcal B\to\widehat{\mathcal A}\) is relatively weakly mixing and \(\widehat{\mathcal A}\to{\mathcal A}\) is distal. If \(\mathcal B(H_1)\) is a Gaussian factor and \(\mathcal K_1\) is a compact subgroup of \(C^g(T|_{\mathcal B(H_1)})\), then \({\mathcal A}=\mathcal B(H_1)/\mathcal K_1\) is called classical.
Let \(T\) be a GAG. For every factor \({\mathcal A}\) of \(T\) there exists a compact subgroup \(\mathcal K\subset C^g(T_{\widehat{\mathcal A}})\) such that \({\mathcal A}=\widehat{\mathcal A}/\mathcal K\). In other words, each factor of a GAG is classical. If there exists only one joining \(\mu_1\otimes\mu_2\), \(T_i\) is called disjoint and expressed by \(T_1\bot T_2\).
Let \(S\) and \(T\) be automorphisms from \((X,\mathcal B,\mu)\) and \((Y,\mathcal C,\nu)\) into themselves, respectively. If \(S\) and \(T\) are not disjoint then \(T\) has a common factor with an infinite self-joining of \(S\). If \(T\) and \(S\) are additionally ergodic then \(T\) has a common factor with an ergodic self-joining of \(S\). \(T\) is Gaussian automorphism of type \(\sigma\) if the maximal spectral type of \(T\) is \(\exp\sigma\).
Let \(T\) be a GAG of type \(\sigma\) and \(S\) be an ergodic automorphism. Assume that \(T\bot S\). Then \(S\) and \(T_\sigma^\infty\) have a common factor. In particular, if \(T_\sigma^\infty\bot S\) then \(S\) and \(T_\sigma^\infty\) itself have a common factor.
Let \(T\) be a GAG of type \(\sigma\) and let \(T_\tau\) be a standard Gaussian automorphism. Suppose that \(T\) and \(T_\tau\) are not disjoint. Then there exists \(z_0\in\mathbb T\) such that \(\sigma\bot\tau*\delta_{z_0}\).

MSC:
37A05 Dynamical aspects of measure-preserving transformations
37A30 Ergodic theorems, spectral theory, Markov operators
37A50 Dynamical systems and their relations with probability theory and stochastic processes
60G15 Gaussian processes
PDF BibTeX XML Cite
Full Text: EuDML