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Published online by Cambridge University Press: 10 November 2000
To study the geometry of a Fibonacci map $f$ of even degree $\ell\geq 4$, Lyubich (Dynamics of quadratic polynomials, I–II. Acta Mathematica178 (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 maps. Preprint, Institute for Mathematical Sciences, SUNY at Stony Brook, 1994) proved that the generalized renormalizations ${\cal R}^{\circ n}(f)$ converge to a cycle $\{f_1,f_2\}$ of order two depending only on $\ell$. We will explicitly relate $f_1$ and $f_2$ and show the convergence in shape of Fibonacci puzzle pieces to the Julia set of an appropriate polynomial-like map.
