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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 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.

Key words: interval maps; symbolic dynamics; C∗-algebras; representations of algebras.

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