Orbit Representations from Linear mod 1 Transformations

Abstract

We show that every point x0 2 [0; 1] carries a representation of a C -algebra that encodes the orbit structure of the linear mod 1 interval map f ; (x) = x + . Such C -algebra is generated by partial isometries arising from the subintervals of monotonicity of the underlying map f ; . Then we prove that such representation is irreducible. Moreover two such of representations are unitarily equivalent if and only if the points belong to the same generalized orbit, for every 2 [0; 1[ and 1.