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We study dynamical properties of the Fibonacci trace map - a polynomial mapthat 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 theFricke-Vogt invariant. The map has all the essential properties that wereobtained previously for the Taylor-Chirikov standard map, and can be suggestedas another candidate for the simplest conservative system with highlynon-trivial dynamics.

Newhouse phenomena in the Fibonacci trace map