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The Density of States Measure of the Weakly Coupled Fibonacci Hamiltonian

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Abstract

We consider the density of states measure of the Fibonacci Hamiltonian and show that, for small values of the coupling constant V, this measure is exact-dimensional and the almost everywhere value d V of the local scaling exponent is a smooth function of V, is strictly smaller than the Hausdorff dimension of the spectrum, and converges to one as V tends to zero. The proof relies on a new connection between the density of states measure of the Fibonacci Hamiltonian and the measure of maximal entropy for the Fibonacci trace map on the non-wandering set in the V-dependent invariant surface. This allows us to make a connection between the spectral problem at hand and the dimension theory of dynamical systems.

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References

  1. Avron J., Simon B.: Almost periodic Schrödinger operators. II. The integrated density of states. Duke Mathematical Journal 50, 369–391 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bellissard J., Bessis D., Moussa P: Chaotic states of almost periodic Schrödinger operators. Physical Review Letters 49, 701–704 (1982)

    Article  MathSciNet  Google Scholar 

  3. Bellissard J., Guarneri I., Schulz-Baldes H.: Phase-averaged transport for quasi-periodic Hamiltonians. Communications in Mathematical Physics 227, 515–539 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Cantat. Bers and Hénon, Painlevé and Schrödinger. Duke Mathematical Journal, 149 (2009), 411–460.

    Google Scholar 

  5. Casdagli M.: Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation. Communications in Mathematical Physics 107, 295–318 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Damanik. Gordon-type arguments in the spectral theory of one-dimensional quasicrystals. In: Directions in Mathematical Quasicrystals, CRM Monograph Series 13. American Mathematical Society, Providence, RI (2000), pp. 277–305.

  7. D. Damanik. Strictly ergodic subshifts and associated operators. In: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday. Proceedings of Symposium in Pure Mathematics, Vol. 76, Part 2, American Mathematical Society, Providence, RI (2007), pp. 505–538.

  8. D. Damanik. Almost everything about the Fibonacci operator. In: New Trends in Mathematical Physics. Selected contributions of the XVth International Congress on Mathematical Physics. Springer, Berlin (2009), pp. 149–159.

  9. Damanik D., Embree M., Gorodetski A., Tcheremchantsev S.: The fractal dimension of the spectrum of the Fibonacci Hamiltonian. Communications in Mathematical Physics 280, 499–516 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Damanik D., Gorodetski A.: Hyperbolicity of the trace map for the weakly coupled Fibonacci Hamiltonian. Nonlinearity 22, 123–143 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Damanik D., Gorodetski A.: The spectrum of the weakly coupled Fibonacci Hamiltonian. Electronic Research Announcements 16, 23–29 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Damanik D., Gorodetski A.: Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian. Communications in Mathematical Physics 305, 221–277 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Damanik D., Tcheremchantsev S.: Power-law bounds on transfer matrices and quantum dynamics in one dimension. Communications in Mathematical Physics 236, 513–534 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Damanik D., Tcheremchantsev S.: Upper bounds in quantum dynamics. Journal of American Mathematical Society 20, 799–827 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Damanik D., Tcheremchantsev S.: Quantum dynamics via complex analysis methods: general upper bounds without time-averaging and tight lower bounds for the strongly coupled Fibonacci Hamiltonian. Journal of Functional Analysis 255, 2872–2887 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Falconer K.: Techniques in Fractal Geometry. Wiley, Chichester (1997)

    MATH  Google Scholar 

  17. Garnett J., Marshall D.: Harmonic Measure. New Mathematical Monographs, Vol. 2. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  18. M. Hirsch, C. Pugh, and M. Shub. Invariant manifolds. Lecture Notes on Mathematics, # 583. Springer, Heidelberg (1977).

  19. A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and its Applications, Vol. 54. Cambridge University Press, Cambridge (1995).

  20. Killip R., Kiselev A., Last Y.: Dynamical upper bounds on wavepacket spreading. American Journal of Mathematics 125, 1165–1198 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Manning A.: A relation between Lyapunov exponents, Hausdorff dimension and entropy. Ergodic Theory and Dynamical Systems 1, 451–459 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  22. Makarov N.: Fine structure of harmonic measure. St. Petersburg Mathematical Journal 10, 217–268 (1999)

    MathSciNet  Google Scholar 

  23. N. Makarov and A. Volberg. On the harmonic measure of discontinuous fractals. Preprint (1986).

  24. McCluskey H., Manning A.: Hausdorff dimension for horseshoes. Ergodic Theory and Dynamical Systems 3, 251–260 (1983)

    Article  MathSciNet  Google Scholar 

  25. Ya. Pesin. Dimension Theory In: Dynamical Systems. Contemporary Views and Applications. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL (1997).

  26. L. Raymond. A constructive gap labelling for the discrete Schrödinger operator on a quasiperiodic chain. Preprint (1997).

  27. Roberts J.: Escaping orbits in trace maps. Journal of Physics A 228, 295–325 (1996)

    Google Scholar 

  28. Simon B.: Equilibrium measures and capacities in spectral theory. Image Processing in Inverse Problems 1, 713–772 (2007)

    Article  MATH  Google Scholar 

  29. Sütő A.: The spectrum of a quasiperiodic Schrödinger operator. Communications in Mathematical Physics 111, 409–415 (1987)

    Article  MathSciNet  Google Scholar 

  30. A. Sütő. Schrödinger difference equation with deterministic ergodic potentials. In: Beyond Quasicrystals (Les Houches, 1994). Springer, Berlin (1995), pp. 481–549.

  31. Volberg A.: On the dimension of harmonic measure of Cantor repellers. The Michigan Mathematical Journal 40, 239–258 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Rice University, Houston, TX, 77005, USA

    David Damanik

  2. Department of Mathematics, University of California, Irvine, CA, 92697, USA

    Anton Gorodetski

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  1. David Damanik
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  2. Anton Gorodetski
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Corresponding author

Correspondence to Anton Gorodetski.

Additional information

D. D. was supported in part by NSF grants DMS–0800100 and DMS–1067988.

A. G. was supported in part by NSF grants DMS–0901627 and IIS-1018433.

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Damanik, D., Gorodetski, A. The Density of States Measure of the Weakly Coupled Fibonacci Hamiltonian. Geom. Funct. Anal. 22, 976–989 (2012). https://doi.org/10.1007/s00039-012-0173-8

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  • DOI: https://doi.org/10.1007/s00039-012-0173-8

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