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Finite dimensional representations of \(*\)-algebras arising from a quadratic map. (English) Zbl 1142.37341

Summary: Consider an operator \(X\) satisfying the algebraic relation \(XX^{*} = f(X^{*}X)\), where \(f\) is the one-parameter family of quadratic maps \(f_b(x) = 4bx(1 - x)\) with \(b\in [0, 1]\). There is a correspondence between the periodic orbits of the dynamical system \(([0, 1], f_b)\) and the unitary classes of matrices \(X\) satisfying \(XX^{*} = f(X^{*}X)\). Using the symbolic dynamics theory for interval maps, we describe in detail the combinatorial structure of the finite dimensional representations associated to this relation.

MSC:

37E15 Combinatorial dynamics (types of periodic orbits)
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
37B10 Symbolic dynamics
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[1] Guckenheimer, John, On the bifurcation of maps of the interval, Invent Math, 39, 2, 165-178 (1977) · Zbl 0354.58013
[2] Guckenheimer, John, Sensitive dependence to initial conditions for one-dimensional maps, Commun Math Phys, 70, 2, 133-160 (1979) · Zbl 0429.58012
[3] Hao, B.-L.; Zheng, W.-M., Applied symbolic dynamics and chaos. Applied symbolic dynamics and chaos, Directions in chaos, 7 (1998), World Scientific Publishing Co., Inc.: World Scientific Publishing Co., Inc. River Edge, NJ · Zbl 0914.58017
[4] Lampreia JP, Rica da Silva A, Sousa Ramos J. Subtrees of the unimodal maps tree. Boll Un Mat Ital C (6) 1986;5(1):159-67 (1987).; Lampreia JP, Rica da Silva A, Sousa Ramos J. Subtrees of the unimodal maps tree. Boll Un Mat Ital C (6) 1986;5(1):159-67 (1987). · Zbl 0653.05024
[5] Milnor John, Thurston William. On iterated maps of the interval. Dynamical systems (College Park, MD, 1986-87), p. 465-563. Lecture Notes Math, 1342, Springer, Berlin, 1988.; Milnor John, Thurston William. On iterated maps of the interval. Dynamical systems (College Park, MD, 1986-87), p. 465-563. Lecture Notes Math, 1342, Springer, Berlin, 1988. · Zbl 0664.58015
[6] Ostrovskyi, V.; Samoilenko, Yu, Structure theorems for a pair of unbounded self-adjoint operators satisfying a quadratic relation. Representation theory and dynamical systems, (Adv Soviet Math, vol. 9 (1992), Amer Math Soc: Amer Math Soc Providence, RI), 131-149
[7] Ostrovskyi, V.; Samoilenko, Yu., Introduction to the theory of representations of finitely presented ∗-algebras. I. Representations by bounded operators, Reviews in Mathematics and Mathematical Physics, vol. 11, pt. 1 (1999), Amsterdam: Amsterdam Harwood Academic Publishers, iv+261pp · Zbl 0947.46037
[8] Popovych Stanislav V, Maistrenko Tatyana Yu. \(C^{∗;}\); Popovych Stanislav V, Maistrenko Tatyana Yu. \(C^{∗;}\) · Zbl 0966.46039
[9] Sharkovsky AN. Coexistence of cycles of a continuous map of the line into itself. Translated from the Russian [Ukrain Mat Zh. 16 (1964), no. 1, 61-71] by J. Tolosa. In: Proceedings of the conference “Thirty Years after Sharkovsky Theorem: New Perspectives” (Murcia, 1994). Internat J Bifur Chaos Appl Sci Engrg. 5 (1995), no. 5, 1263-1273.; Sharkovsky AN. Coexistence of cycles of a continuous map of the line into itself. Translated from the Russian [Ukrain Mat Zh. 16 (1964), no. 1, 61-71] by J. Tolosa. In: Proceedings of the conference “Thirty Years after Sharkovsky Theorem: New Perspectives” (Murcia, 1994). Internat J Bifur Chaos Appl Sci Engrg. 5 (1995), no. 5, 1263-1273.
[10] Sharkovsky, A. N.; Kolyada, S. F.; Sivak, A. S.; Fedorenko, V. V., Dynamics of one-dimensional maps (1997), Kluwer Academic Publishers. · Zbl 0881.58020
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