×

Polymorphisms, Markov processes, and quasi-similarity. (English) Zbl 1115.37002

Recall that a polymorphism \(\Pi\) with invariant measure is defined as a diagram \[ (X,\mu)@<{\pi_1}<<(X\times X,\nu)@>{\pi_2}>>(X, \mu), \] where \((X,\mu)\) is a Lebesgue space, \(\pi_i\) the projection on the \(i\)th coordinate \((i= 1,2)\) and \(\nu\) a measure on \(X\times X\) with \(\pi_i(\nu)= \mu\). The image \(\Pi(x)\) of \(x\in X\) is then defined as the conditional measure \(\nu^x\) on the fiber \(\{x\}\times X\). The author’s emphasis in this paper is on ergodic theory and dynamical aspects for these objects, and might be seen as commencement of the author’s studies in [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 72, 26–61 (1977; Zbl 0408.28014)] and [Vestn. Leningr. Univ., Math. 20, No. 3, 22–29 (1987; Zbl 0655.60010)]. In particular he is interested in the question wether polymorphisms and automorphisms are quasi-images of each other (\(\Pi_1\) is a quasi-image of \(\Pi_2\) if there exists a dense polymorphism \(\Gamma\) such that \(\Gamma\cdot\Pi_1= \Pi_1= \Pi_2\cdot\Gamma\)).
The focus in this paper is on the class of prime and nonmixing polymorphisms, where a polymorphism \(\Pi\) is prime (or exact) if there exist no nontrivial factor endomorphisms, and nonmixing if \(\Pi^n\) does not converge to the zero polymorphism (that is \(\nu\) is the product measure). The main result is the following. For every prime and nonmixing polymorphism \(\Pi\), there exists a \(K\)-automorphism \(T\) which is a quasiimage of \(\Pi\), and for every \(K\)-automorphism \(T\) there exist polymorphisms \(\Pi_1\), \(\Pi_2\) such that \(\Pi_1\) is a quasi-image of \(T\), and \(T\) is a quasi-image of \(\Pi_2\). Furthermore, in certain cases, \(\Pi_1= \Pi_2\).
The first part of this result is achieved by considering the Markov process associated with \(\Pi\), which then gives rise to the quasi-image \(T\) constructed from the residual automorphism of the process. The second part follows from a direct construction, and the related polymorphisms \(\Pi_1\), \(\Pi_2\) might be characterised as random perturbations of the given \(K\)-automorphism.

MSC:

37A05 Dynamical aspects of measure-preserving transformations
PDFBibTeX XMLCite
Full Text: DOI