Newhouse phenomena in the Fibonacci trace map

TL;DR

This paper investigates the complex dynamical behaviors of the Fibonacci trace map, revealing phenomena like homoclinic tangencies and elliptic islands, and positioning it as a fundamental example of chaotic conservative systems.

Contribution

It demonstrates that the Fibonacci trace map exhibits all key Newhouse phenomena, expanding understanding of conservative dynamical systems with rich chaotic behavior.

Findings

Existence of persistent homoclinic tangencies

Presence of a stochastic sea with full Hausdorff dimension

Infinitely many elliptic islands

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.