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arXiv · 0705.3706

Algebraic Polymorphisms

Klaus Schmidt·Anatoly Vershik
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Abstract

In this paper we consider a special class of polymorphisms with invariant measure, - (cf.[1])- 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 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.

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Klaus Schmidt, Anatoly Vershik. 2007-05-25. Algebraic Polymorphisms. https://arxiv.org/abs/0705.3706

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