Orbit Representations from Linear mod 1 Transformations

Abstract

We show that every point $x_0\in [0,1]$ carries a representation of a $C^{\ast }$-algebra that encodes the orbit structure of the linear mod 1 interval map $f_{\beta ,\alpha }(x)=\beta x+\alpha$. Such $C^{\ast }$-algebra is generated by partial isometries arising from the subintervals of monotonicity of the underlying map $f_{\beta ,\alpha }$. 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 $\alpha \in [0,1[$ and $\beta \ge 1$.