Newhouse phenomena in the Fibonacci trace map

Abstract

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 standard map, and can be suggested as another candidate for the simplest conservative system with highly non-trivial dynamics.

45 pages, 8 figures, 81 references

References in corpus (3)

• The Fibonacci Hamiltonian
• Absolutely Continuous Convolutions of Singular Measures and an Application to the Square Fibonacci Hamiltonian
• Properties of 1D classical and quantum Ising quasicrystals: rigorous results