Orbit Representations from Linear mod 1 Transformations

TL;DR

This paper constructs and analyzes $C^*$-algebra representations derived from linear mod 1 interval maps, revealing their irreducibility and classifying unitary equivalence via generalized orbits.

Contribution

It introduces a novel $C^*$-algebra framework for linear mod 1 maps and characterizes the irreducibility and orbit-based equivalence of their representations.

Findings

Representations encode orbit structure of linear mod 1 maps.

All such representations are irreducible.

Unitary equivalence corresponds to points in the same generalized orbit.

Abstract

We show that every point $x_0\in [0,1]$ carries a representation of a $C^*$-algebra that encodes the orbit structure of the linear mod 1 interval map $f_{\beta,\alpha}(x)=\beta x +\alpha$. Such $C^*$-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 $\beta\geq 1$.