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Algebraic polymorphisms

Published online by Cambridge University Press:  01 April 2008

KLAUS SCHMIDT
Affiliation:
Mathematics Institute, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria (email: [email protected])
ANATOLY VERSHIK
Affiliation:
Steklov Institute of Mathematics at St. Petersburg, 27 Fontanka, St. Petersburg 191011, Russia (email: [email protected])

Abstract

In this paper 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 L2 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 if and only if the associated rational matrix lies in . We characterize toral polymorphisms which are factors of toral automorphisms.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Vershik, A. M.. Polymorphisms, Markov processes, and quasi-similarity. Discrete Contin. Dyn. Syst. 13 (2005), 13051324.CrossRefGoogle Scholar