"Fibonacci maps on the interval"Henk BruinThe family of quadratic maps $f_c(x) = x^2 + c$, or more generally $f_{c,r}(x) = |x|^r+c$ has been a prime example of a chaotic dynamical system since the 1970s. Studying its ergodic properties (behaviour of Lebesgue typical orbits) has been a long-term program, to which many mathematicians have contributed: M. Benedicks, A. Blokh, L. Carleson, J. Graczyk, F. Hofbauer, M. Jakobson, M. Lyubich, G. Keller, J. Milnor, M. Misiurewicz, T. Nowicki, W. Shen, S. van Strien, and others. Between maps with (i) a periodic attractor (open and dense in parameter space) and (ii) an invariant density (parameter set of positive Lebesgue measure), Fibonacci maps play a curious role. This map, for which important iterates are at Fibonacci numbers, has Viennese origins, and is an central test case in the theory, including Milnor's attractor conjecture. |
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