# Ergodic theory for the one-dimensional Jacobi operator

Studia Mathematica (1996)

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

## Access Full Article

top## Abstract

top## How to cite

topNúñez, 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

top- [1] P. Deift and B. Simon, Almost periodic Schrödinger operators III, Comm. Math. Phys. 90 (1983), 389-411. Zbl0562.35026
- [2] F. Delyon and B. Souillard, The rotation number for finite difference operators and its properties, ibid. 89 (1983), 415-426. Zbl0525.39003
- [3] R. Johnson, Analyticity of spectral subbundles, J. Differential Equations 35 (1980), 366-387. Zbl0458.34017
- [4] R. Johnson, Exponential dichotomy, rotation number, and linear differential equations with bounded coefficients, ibid. 61 (1986), 54-78.
- [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.
- [6] Y. Last, A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants, Comm. Math. Phys. 151 (1993), 183-192. Zbl0782.34084
- [7] S. Novo and R. Obaya, An ergodic classification of bidimensional linear systems, preprint, University of Valladolid, 1994. Zbl0869.28009
- [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] 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] 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] V. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197-231.
- [12] L. Pastur, Spectral properties of disordered systems in the one body approximation, Comm. Math. Phys. 75 (1980), 179-196. Zbl0429.60099
- [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] B. Simon, Kotani theory for one dimensional stochastic Jacobi matrices, Comm. Math. Phys. 89 (1983), 227-234. Zbl0534.60057

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.