They are traditionally taken to be infinite sequences on the alphabet of the two symbols 0 and 1.Ĭombinatorial definitions Sequences of low complexity įor an infinite sequence of symbols w, let σ( n) be the complexity function of w i.e., σ( n) = the number of distinct contiguous subwords (factors) in w of length n. Sturmian sequences can be defined strictly in terms of their combinatoric properties or geometrically as cutting sequences for lines of irrational slope or codings for irrational rotations. The struck ball will successively hit the vertical and horizontal edges labelled 0 and 1 generating a sequence of letters. Such a sequence can be generated by considering a game of English billiards on a square table. In mathematics, a Sturmian word ( Sturmian sequence or billiard sequence ), named after Jacques Charles François Sturm, is a certain kind of infinitely long sequence of characters. The start of the cutting sequence shown here illustrates the start of the word 0100101001. Spectr.The Fibonacci word is an example of a Sturmian word. Yessen, W.: Spectral analysis of tridiagonal Fibonacci Hamiltonians. Sűto, A.: Schrödinger difference equation with deterministic ergodic potentials. Sűto, A.: Singular continuous spectrum on a set of zero Lebesgue measure for the Fibonacci Hamiltonian. Sűto, A.: The spectrum of a quasiperiodic Schrödinger operator. American Mathematical Society, Providence (2004) Simon, B.: Orthogonal Polynomaials on the Unit Circle. Ong, D.C.: Limit-periodic Verblunsky coefficients for orthogonal polynomials on the unit circle. Proceedings of the 9th AIMS Conference Special Issue, pp. Discrete and Continuous Dynamical Systems. Ong, D.C.: Orthogonal polynomials on the unit circle with quasiperiodic Verblunsky coefficients have generic purely singular continuous spectrum. Kaminaga, M.: Absence of point spectrum for a class of discrete Schrödinger operators with quasiperiodic potential. Gordon, A.: The point spectrum of the one-dimensional Schrödinger operator. Gesztesy, F., Zinchenko, M.: Weyl–Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle. Theory 173, 56–88 (2013)ĭamanik, David, Munger, Paul, Yessen, William: Orthogonal polynomials on the unit circle with Fibonacci Verblunsky coefficients, II. 305, 221–277 (2011)ĭamanik, D., Munger, P., Yessen, W.: Orthogonal polynomials on the unit circle with Fibonacci Verblunsky coefficients, I. Theory 144, 133–138 (2007)ĭamanik, D., Gorodetski, A.: Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian. Henri Poincaré 2, 101–108 (2001)ĭamanik, D., Lenz, D.: Uniform Szegő cocycles over strictly ergodic subshifts. 212, 191–204 (2000)ĭamanik, D.: Uniform singular continuous spectrum for the period doubling Hamiltonian. American Mathematical Society, Providence (2007)ĭamanik, D., Killip, R., Lenz, D.: Uniform spectral properties of one-dimensional quasicrystals, III. Proceedings of Symposia in Pure Mathematics, vol. (eds.) Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday. In: Gesztesy, F., Deift, P., Galvez, C., Perry, P., Schlag, W. American Mathematical Society, Providence (2000)ĭamanik, D.: Strictly ergodic subshifts and associated operators. (eds.) Directions in Mathematical Quasicrystals. arXiv:1210.5753 (2012)ĭamanik, D.: Gordon-type arguments in the spectral theory of one-dimensional quasicrystals. thesis, Rice University (2011)ĭamanik, D., Embree, M., Gorodetski, A.: Spectral properties of Schrödinger operators arising in the study of quasicrystals. Springer, Berlin (2008)ĭahl, J.: The spectrum of the off-diagonal Fibonacci operator. LXIII, 0464–0507 (2010)Ĭycon, H., Froese, R., Kirsch, W., Simon, B.: Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry. 362, 29–56 (2003)Ĭantero, M.-J., Grünbaum, F.A., Moral, L., Velázquez, L.: Matrix-valued Szegö polynomials and quantum random walks. 125, 527–543 (1989)Ĭantero, M.-J., Moral, L., Velázquez, L.: Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. 135, 379–399 (1991)īellissard, J., Iochum, B., Scoppola, E., Testard, D.: Spectral properties of one-dimensional quasicrystals. Springer, Berlin (1990)īellisard, J., Bovier, A., Ghez, J.-M.: Spectral properties of a tight-binding Hamiltonian with period doubling potential. In: Waldshmidt, P., Luck, M., Moussa, J.M. arXiv:1304.0519 (2013)īellisard, J.: Spectral properties of Schrödinger’s operator with a Thue-Morse potential. Avila, A., Damanik, D., Zhang, Z.: Singular density of states measure for subshift and quasi-periodic Schrödinger operators.
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