代数多态性
Algebraic Polymorphisms
摘要：
本文考虑一类特殊的具有不变测度的多态性，-（cf. [1])-紧群的代数多态性。根据定义，一般的多态性是具有不变测度的多值映射，并且多态性的共轭算子是马尔可夫算子（即，保持常数的范数为1的$L^2$上的正算子）。在代数的情况下，多态性是一个对应关系的意义上的代数几何，但在这里，我们调查它从动力学的角度来看。最重要的例子是代数多态的环面，在那里我们引入了参数化的半群的合理矩阵和描述相应的马尔可夫算子的频谱的toral多态性。
英文摘要：
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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