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The Fibonacci unimodal map. (English) Zbl 0778.58040
This paper studies topological, geometrical and measure-theoretical properties of the real Fibonacci map to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by Sullivan (1990). It turns out that the situation can be understood completely and is of quite regular nature. In particular, any Fibonacci map (with negative Schwarzian and nondegenerate critical point) has an absolutely continuous invariant measure. It turns out also that geometrical properties of the closure of the critical point are quite different from those of the Feigenbaum map; its Hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter.
Reviewer: Y.Kozai (Tokyo)

MSC:
37A99 Ergodic theory
37E99 Low-dimensional dynamical systems
37F99 Dynamical systems over complex numbers
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