Algebraic Polymorphisms

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 L 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. A toral polymorphism is an automorphism of T if and only if the associated rational matrix lies in GL(m,Z). We characterize toral polymorphisms which are factors of toral automorphisms. 1. Algebraic polymorphisms Definition 1.1. Let G be a compact group with Borel field BG, normalized Haar measure λG and identity element 1 = 1G. A closed subgroup P ⊂ G × G is an (algebraic) correspondence of G if π1(P) = π2(P) = G, where πi : G×G −→ G, i = 1, 2, are the coordinate projections (which are obviously group homomorphisms). Every correspondence P ⊂ G×G defines a map ΠP from G to the set of all nonempty closed subsets of G by ΠP(x) = {y : (x, y) ∈ P} (1.1) for every x ∈ G. Clearly, πi sends the Haar measure on P to Haar measure on G; in the terminology of [1], the correspondence P defines an (algebraic) polymorphism of G (more exactly, P determines a polymorphism of the measure space (G,BG, λG) to itself ). 1 The correspondence P ⊂ G × G and the polymorphism ΠP obviously determine each other. Algebraic polymorphisms from one compact group to another are defined similarly. A correspondence P ⊂ G × G is a factor of a correspondence P ⊂ G × G (and the polymorphism ΠP′ is a factor of ΠP) are isomorphic if there exists a surjective group homomorphism φ : G −→ G with (φ × φ)(P) = P. If φ can be chosen to be a group isomorphism then P and P (resp. ΠP and ΠP′) are isomorphic. This notion of factors is consistent with the terminology in [1]: if Π is a measurepreserving polymorphism of a probability space (X, S, μ) determined by a self-coupling ν 2000 Mathematics Subject Classification. 37A05,37A45.