Algebraic Polymorphisms

TL;DR

This paper studies algebraic polymorphisms of compact groups, especially tori, focusing on their structure, parametrization, and spectral properties of associated Markov operators from a dynamical systems perspective.

Contribution

It introduces a parametrization of toral polymorphisms via rational matrices and analyzes the spectra of their Markov operators, linking algebraic and dynamical viewpoints.

Findings

Parametrization of toral polymorphisms using rational matrices

Spectral analysis of associated Markov operators

Connection between algebraic correspondences and dynamical systems

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.