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Algebraic polymorphisms. (English) Zbl 1154.37304

Ergodic Theory Dyn. Syst. 28, No. 2, 633-642 (2008); erratum ibid. 29, No. 4, 1369-1370 (2009).
Summary: We consider a special class of polymorphisms with invariant measure, the algebraic polymorphisms of compact groups. A general polymorphism is-by definition-a many-valued map with invariant measure, and the conjugate operator of a polymorphism is a Markov operator (i.e. a positive operator on \(L^{2}\) of norm 1 which preserves the constants). In the algebraic case a polymorphism is a correspondence in the sense of algebraic geometry, but here we investigate it from a dynamical point of view. The most important examples are the algebraic polymorphisms of a torus, where we introduce a parametrization of the semigroup of toral polymorphisms in terms of rational matrices and describe the spectra of the corresponding Markov operators. A toral polymorphism is an automorphism of \(\mathbb T^m\) if and only if the associated rational matrix lies in GL\((m,\mathbb Z)\). We characterize toral polymorphisms which are factors of toral automorphisms.

MSC:

37A15 General groups of measure-preserving transformations and dynamical systems
37A30 Ergodic theorems, spectral theory, Markov operators
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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References:

[1] Vershik, Discrete Contin. Dyn. Syst. 13 pp 1305– (2005)
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