Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-17T19:30:32.465Z Has data issue: false hasContentIssue false
  • English
  • Français

Gibbs-like measure for spectrum of a class of quasi-crystals

Published online by Cambridge University Press:  14 January 2011

SHEN FAN ,
QING-HUI LIU  and
ZHI-YING WEN
SHEN FAN
Affiliation:
Department of Mathematics, Tsinghua University, Beijing, 100084, PR China (email: wenzy@mail.tsinghua.edu.cn)
QING-HUI LIU
Affiliation:
Department of Computer Science and Engineering, Beijing Institute of Technology, Beijing, PR China (email: qhliu@bit.edu.cn)
ZHI-YING WEN
Affiliation:
Department of Mathematics, Tsinghua University, Beijing, 100084, PR China (email: wenzy@mail.tsinghua.edu.cn)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let α∈(0,1) be an irrational, and [0;a1,a2,…] the continued fraction expansion of α. Let Hα,V be the one-dimensional Schrödinger operator with Sturmian potential of frequency α. Suppose the potential strength V >20 and the sequence (ai)i≥1 is bounded. We proceed by developing some new ideas on dimensional theory of Cookie-cutter sets. We prove that the spectral generating bands satisfy the principles of bounded variation and bounded covariation, and then we show that there exists a Gibbs-like measure on the spectrum σ(Hα,V). As an application, we prove that where s* and s* are the lower and upper pre-dimensions. Moreover, if (an)n≥1 is ultimately periodic, then s* =s*.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 6 , December 2011 , pp. 1669 - 1695
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bellissard, J., Iochum, B., Scoppola, E. and Testart, D.. Spectral properties of one-dimensional quasi-crystals. Comm. Math. Phys. 125 (1989), 527–543.CrossRefGoogle Scholar
[2]Bovier, A. and Ghez, J.-M.. Spectrum properties of one-dimensional Schrödinger operators with potentials generated by substitutions. Comm. Math. Phys. 158 (1993), 45–66.CrossRefGoogle Scholar
[3]Casdagli, M.. Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation. Comm. Math. Phys. 107 (1986), 295–318.CrossRefGoogle Scholar
[4]Damanik, D., Killip, R. and Lenz, D.. Uniform spectral properties of one-dimensional quasicrystals, III. α-continuity. Comm. Math. Phys. 212 (2000), 191–204.CrossRefGoogle Scholar
[5]Damanik, D., Embree, M., Gorodetski, A. and Tcheremchantsev, S.. The fractal dimension of the spectrum of the Fibonacci Hamiltonian. Comm. Math. Phys. 280(2) (2008), 499–516.CrossRefGoogle Scholar
[6]Falconer, K.. Techniques in Fractal Geometry. John Wiley & Sons, New York, 1997.Google Scholar
[7]Feng, D.-J., Wen, Z.-Y. and Wu, J.. Some dimensional results for homogeneous Moran sets. Sci. China Ser. A 40(5) (1997), 475–482.CrossRefGoogle Scholar
[8]Hua, S., Rao, H., Wen, Z.-Y. and Wu, J.. On the structures and dimensions of Moran sets. Sci. China Ser. A 43(8) (1997), 836–852.CrossRefGoogle Scholar
[9]Lenz, D.. Singular spectrum of Lebesgue measure zero for one-dimensional quasicrystals. Comm. Math. Phys. 227 (2002), 119–130.CrossRefGoogle Scholar
[10]Liu, Q. H., Peyrière, J. and Wen, Z. Y.. Dimension of the spectrum of one-dimensional discrete Schrödinger operators with Sturmian potentials. C. R. Acad. Sci. Paris, Ser. I 345(12) (2007), 667–672.CrossRefGoogle Scholar
[11]Liu, Q. H., Tan, B., Wen, Z. X. and Wu, J.. Measure zero spectrum of a class of Schrödinger operator. J. Stat. Phys. 106(3–4) (2002), 681–691.CrossRefGoogle Scholar
[12]Liu, Q. H. and Wen, Z. Y.. Hausdorff dimension of spectrum of one-dimensional Schrödinger operator with Sturmian potentials. Potential Anal. 20(1) (2004), 33–59.CrossRefGoogle Scholar
[13]Liu, Q. H. and Wen, Z. Y.. On dimensions of multitype Moran sets. Math. Proc. Cambridge Philos. Soc. 139(3) (2005), 541–553.CrossRefGoogle Scholar
[14]Ma, J. H., Rao, H. and Wen, Z. Y.. Dimensions of Cookie-cutter-like sets. Sci. China Ser. A 44(11) (2001), 1400–1412.CrossRefGoogle Scholar
[15]Raymond, L.. A constructive gap labelling for the discrete Schrödinger operater on a quasiperiodic chain. Preprint, 1997.Google Scholar
[16]Schechtman, D., Blech, I., Gratias, D. and Cahn, J.. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53 (1984), 1951–1954.CrossRefGoogle Scholar
[17]Sütö, A.. Schrödinger difference equation with deterministic ergodic potentials. Beyond Quasicrystals. Eds. Alex, F. and Gratias, D.. Springer, Berlin, 1994.Google Scholar
[18]Toda, M.. Theory of Nonlinear Lattices (Solid-State Sciences, 20), second enlarged edition. Springer, Berlin, 1989, Ch. 4.CrossRefGoogle Scholar