Geodesic
Orbit representations from linear mod 1 transformations
Symmetry, integrability and geometry: methods and applications, Tome 8 (2012) Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a28/}
}
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Carlos Correia Ramos; Nuno Martins; Paulo R. Pinto. Orbit representations from linear mod 1 transformations. Symmetry, integrability and geometry: methods and applications, Tome 8 (2012). http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a28/

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