• Corpus ID: 117771266

Newhouse phenomena in the Fibonacci trace map

@article{Yessen2015NewhousePI,
  title={Newhouse phenomena in the Fibonacci trace map},
  author={William N. Yessen},
  journal={arXiv: Dynamical Systems},
  year={2015}
}
  • W. Yessen
  • Published 28 July 2015
  • Mathematics
  • arXiv: Dynamical Systems
We study dynamical properties of the Fibonacci trace map - a polynomial map that is related to numerous problems in geometry, algebra, analysis, mathematical physics, and number theory. Persistent homoclinic tangencies, stochastic sea of full Hausdorff dimension, infinitely many elliptic islands - all the conservative Newhouse phenomena are obtained for many values of the Fricke-Vogt invariant. The map has all the essential properties that were obtained previously for the Taylor-Chirikov… 
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References

SHOWING 1-10 OF 69 REFERENCES

Hyperbolicity of the trace map for the weakly coupled Fibonacci Hamiltonian

We consider the trace map associated with the Fibonacci Hamiltonian as a diffeomorphism on the invariant surface associated with a given coupling constant and prove that the non-wandering set of this…

The Fibonacci Hamiltonian

We consider the Fibonacci Hamiltonian, the central model in the study of electronic properties of one-dimensional quasicrystals, and establish relations between its spectrum and spectral…

THE SPECTRUM OF THE WEAKLY COUPLED FIBONACCI HAMILTONIAN

We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the…

ESCAPING ORBITS IN TRACE MAPS

Markoff triples and quasifuchsian groups

We study the global behaviour of trees of Markoff triples over the complex numbers. We relate this to the space of type‐preserving representations of the punctured torus group into SL(2,C). In…

Spectral and Quantum Dynamical Properties of the Weakly Coupled Fibonacci Hamiltonian

We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the…

Mixed spectral regimes for square Fibonacci Hamiltonians

For the square tridiagonal Fibonacci Hamiltonian, we prove existence of an open set of parameters which yield mixed interval-Cantor spectra (i.e. spectra containing an interval as well as a Cantor…

Spectral properties of Schrödinger operators arising in the study of quasicrystals

We survey results that have been obtained for self-adjoint operators, and especially Schrodinger operators, associated with mathematical models of quasicrystals. After presenting general results that…

Tridiagonal substitution Hamiltonians

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate…

Spectral analysis of tridiagonal Fibonacci Hamiltonians

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice, with the diagonal and the off-diagonal entries given by two sequences generated by the Fibonacci substitution…
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