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.

pdf (361 kb)   tex (29 kb)

References

1. Abe M., Kawamura K., Recursive fermion system in Cuntz algebra. I. Embeddings of fermion algebra into Cuntz algebra, Comm. Math. Phys. 228 (2002), 85-101, math-ph/0110003.
2. Bratteli O., Jorgensen P.E.T., Iterated function systems and permutation representations of the Cuntz algebra, Mem. Amer. Math. Soc. 139 (1999), no. 663, 89 pages, funct-an/9612002.
3. Bratteli O., Jorgensen P.E.T., Ostrovs'ky V., Representation theory and numerical AF-invariants. The representations and centralizers of certain states on $\mathcal{O}_d$, Mem. Amer. Math. Soc. 168 (2004), no. 797, 178 pages, math.OA/9907036.
4. Carlsen T.M., Silvestrov S., $C^*$-crossed products and shift spaces, Expo. Math. 25 (2007), 275-307, math.OA/0512488.
5. Correia Ramos C., Martins N., Pinto P.R., On $C^*$-algebras from interval maps, Complex Anal. Oper. Theory, to appear.
6. Correia Ramos C., Martins N., Pinto P.R., Orbit representations and circle maps, in Operator Algebras, Operator Theory and Applications, Oper. Theory Adv. Appl., Vol. 181, Birkhäuser Verlag, Basel, 2008, 417-427.
7. Correia Ramos C., Martins N., Pinto P.R., Sousa Ramos J., Cuntz-Krieger algebras representations from orbits of interval maps, J. Math. Anal. Appl. 341 (2008), 825-833.
8. Cuntz J., Krieger W., A class of $C^*$-algebras and topological Markov chains, Invent. Math. 56 (1980), 251-268.
9. Daubechies I., Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
10. Dutkay D.E., Jorgensen P.E.T., Wavelet constructions in non-linear dynamics, Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 21-33, math.DS/0501145.
11. Exel R., A new look at the crossed-product of a $C^*$-algebra by an endomorphism, Ergodic Theory Dynam. Systems 23 (2003), 1733-1750, math.OA/0012084.
12. Jorgensen P.E.T., Certain representations of the Cuntz relations, and a question on wavelets decompositions, in Operator Theory, Operator Algebras, and Applications, Contemp. Math., Vol. 414, Amer. Math. Soc., Providence, RI, 2006, 165-188, math.CA/0405372.
13. Marcolli M., Paolucci A.M., Cuntz-Krieger algebras and wavelets on fractals, Complex Anal. Oper. Theory 5 (2011), 41-81, arXiv:0908.0596.
14. Martins N., Sousa Ramos J., Cuntz-Krieger algebras arising from linear mod one transformations, in Differential Equations and Dynamical Systems (Lisbon, 2000), Fields Inst. Commun., Vol. 31, Amer. Math. Soc., Providence, RI, 2002, 265-273.
15. Matsumoto K., On $C^*$-algebras associated with subshifts, Internat. J. Math. 8 (1997), 357-374.
16. Milnor J., Thurston W., On iterated maps of the interval, in Dynamical Systems (College Park, MD, 1986-1987), Lecture Notes in Math., Vol. 1342, Springer, Berlin, 1988, 465-563.
17. Pedersen G.K., $C^*$-algebras and their automorphism groups, London Mathematical Society Monographs, Vol. 14, Academic Press Inc., London, 1979.