Fibonacci fixed point of renormalization

To study the geometry of a Fibonacci map f of even degree l greater than or equal to 4, Lyubich (Dynamics of quadratic polynomials, I-II. Acta Mathema tics 178 (1997), 185-297) defined a notion of generalized renormalization, so that f is renormalizable infinitely many times. van Strien and Nowicki ( Polynomial maps with a Julia set of positive Lebesgue measure: Fibonacci ma ps. Preprint, Institute for Mathematical Sciences, SUNY at Stony Brook, 199 4) proved that the generalized renormalizations R-on(f) converge to a cycle {f(1), f(2)} of order two depending only on l. We will explicitly relate f (1) and f(2) and show the convergence in shape of Fibonacci puzzle pieces t o the Julia set of an appropriate polynomial-like map.