Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 029, 9 pages      arXiv:1205.3553      https://doi.org/10.3842/SIGMA.2012.029

Orbit Representations from Linear mod 1 Transformations

Carlos Correia Ramos a, Nuno Martins b and Paulo R. Pinto b
a) Centro de Investigação em Matemática e Aplicações, R. Romão Ramalho, 59, 7000-671 Évora, Portugal
b) Department of Mathematics, CAMGSD, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Received March 14, 2012, in final form May 09, 2012; Published online May 16, 2012

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 $\beta\geq 1$.

Key words: interval maps; symbolic dynamics; $C^*$-algebras; representations of algebras.

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