Orbit Representations from Linear mod 1 Transformations

Abstract

Abstract. We show that every point x 0 ∈ [0, 1] carries a representation of a C * -algebra that encodes the orbit structure of the linear mod 1 interval map f β,α (x) = βx + α. Such C * -algebra is generated by partial isometries arising from the subintervals of monotonicity of the underlying map f β,α . 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 α ∈ [0, 1[ and β ≥ 1.