Orbit Representations from Linear mod 1 Transformations - Correia Ramos, C.; Martins, N.; Pinto, P.R. (Symmetry, Integrability and Geometry: Methods and Applications, 2012)
We show that every point x0∈[0,1] carries a representation of a C∗-algebra that encodes the orbit structure of the linear mod 1 interval map fβ,α(x)=βx+α. Such C∗-algebra is generated by partial isometries arising from the subintervals of monotonicity of the underlying map fβ,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.