Matrices with cyclically monotone rows and Cantor numeration systems

Abstract

We study a class of square matrices with non-negative elements which have cyclically monotone rows in the sense that each row of a matrix from the class consists of a cyclically non-increasing sequence of numbers starting from a maximal element on the diagonal. We prove that if every diagonal element is strictly larger than all other elements in the respective row, then the matrix is regular. This property enables us to solve an open problem that comes from the theory of non-standard numeration systems, also called Cantor numeration systems. The problem concerns a one-to-one relationship between Cantor real bases, which are supposed to be alternate, that is, periodic with a period p, and lists of p sequences of non-negative integers satisfying the so-called Parry condition.