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SIGMA, 2012, Volume 8, 029, 9 pages (Mi sigma706)  

Orbit representations from linear mod 1 transformations

Carlos Correia Ramosa, Nuno Martinsb, Paulo R. Pintob

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

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 $\beta\geq 1$.

Keywords: interval maps, symbolic dynamics, $C^*$-algebras, representations of algebras.

DOI: https://doi.org/10.3842/SIGMA.2012.029

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Full text: http://emis.mi.ras.ru/.../029
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Bibliographic databases:

ArXiv: 1205.3553
MSC: 46L55, 37B10, 46L05
Received: March 14, 2012; in final form May 9, 2012; Published online May 16, 2012
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Citation: Carlos Correia Ramos, Nuno Martins, Paulo R. Pinto, “Orbit representations from linear mod 1 transformations”, SIGMA, 8 (2012), 029, 9 pp.

Citation in format AMSBIB
\Bibitem{RamMarPin12}
\by Carlos Correia Ramos, Nuno Martins, Paulo R. Pinto
\paper Orbit representations from linear mod 1 transformations
\jour SIGMA
\yr 2012
\vol 8
\papernumber 029
\totalpages 9
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\crossref{https://doi.org/10.3842/SIGMA.2012.029}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2942810}
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