Ergodic theory for the one-dimensional Jacobi operator

Carmen Núñez; Rafael Obaya

Studia Mathematica (1996)

  • Volume: 117, Issue: 2, page 149-171
  • ISSN: 0039-3223

Abstract

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We determine the number and properties of the invariant measures under the projective flow defined by a family of one-dimensional Jacobi operators. We calculate the derivative of the Floquet coefficient on the absolutely continuous spectrum and deduce the existence of the non-tangential limit of Weyl m-functions in the L 1 -topology.

How to cite

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Núñez, Carmen, and Obaya, Rafael. "Ergodic theory for the one-dimensional Jacobi operator." Studia Mathematica 117.2 (1996): 149-171. <http://eudml.org/doc/216249>.

@article{Núñez1996,
abstract = {We determine the number and properties of the invariant measures under the projective flow defined by a family of one-dimensional Jacobi operators. We calculate the derivative of the Floquet coefficient on the absolutely continuous spectrum and deduce the existence of the non-tangential limit of Weyl m-functions in the $L^1$-topology.},
author = {Núñez, Carmen, Obaya, Rafael},
journal = {Studia Mathematica},
keywords = {invariant measures; projective flow; Jacobi operators; Floquet coefficient; Weyl -functions},
language = {eng},
number = {2},
pages = {149-171},
title = {Ergodic theory for the one-dimensional Jacobi operator},
url = {http://eudml.org/doc/216249},
volume = {117},
year = {1996},
}

TY - JOUR
AU - Núñez, Carmen
AU - Obaya, Rafael
TI - Ergodic theory for the one-dimensional Jacobi operator
JO - Studia Mathematica
PY - 1996
VL - 117
IS - 2
SP - 149
EP - 171
AB - We determine the number and properties of the invariant measures under the projective flow defined by a family of one-dimensional Jacobi operators. We calculate the derivative of the Floquet coefficient on the absolutely continuous spectrum and deduce the existence of the non-tangential limit of Weyl m-functions in the $L^1$-topology.
LA - eng
KW - invariant measures; projective flow; Jacobi operators; Floquet coefficient; Weyl -functions
UR - http://eudml.org/doc/216249
ER -

References

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  3. [3] R. Johnson, Analyticity of spectral subbundles, J. Differential Equations 35 (1980), 366-387. Zbl0458.34017
  4. [4] R. Johnson, Exponential dichotomy, rotation number, and linear differential equations with bounded coefficients, ibid. 61 (1986), 54-78. 
  5. [5] S. Kotani, Lyapunov indices determine absolutely continuous spectrum of stationary random 1-dimensional Schrödinger equations, in: Stochastic Analysis, K. Ito (ed.), North-Holland, Amsterdam, 1984, 225-248. 
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  7. [7] S. Novo and R. Obaya, An ergodic classification of bidimensional linear systems, preprint, University of Valladolid, 1994. Zbl0869.28009
  8. [8] C. Núñez and R. Obaya, Non-tangential limit of the Weyl m-functions for the ergodic Schrödinger equation, preprint, University of Valladolid, 1994. 
  9. [9] C. Núñez and R. Obaya, Semicontinuity of the derivative of the rotation number, C. R. Acad. Sci. Paris Sér. I 320 (1995), 1243-1248. Zbl0846.58049
  10. [10] R. Obaya and M. Paramio, Directional differentiability of the rotation number for the almost periodic Schrödinger equation, Duke Math. J. 66 (1992), 521-552. Zbl0763.34060
  11. [11] V. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197-231. 
  12. [12] L. Pastur, Spectral properties of disordered systems in the one body approximation, Comm. Math. Phys. 75 (1980), 179-196. Zbl0429.60099
  13. [13] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations 27 (1978), 320-358. Zbl0372.34027
  14. [14] B. Simon, Kotani theory for one dimensional stochastic Jacobi matrices, Comm. Math. Phys. 89 (1983), 227-234. Zbl0534.60057

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