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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
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