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One-dimensional dynamics in the new millennium

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  • In the early 60's Sarkovskii discovered his famous theorem on the coexistence of periodic orbits for interval maps. Then, in the mid 70's, Li & Yorke rediscovered this result and somewhat later the papers by Feigenbaum and Coullet & Tresser on renormalisation and by Guckenheimer and Misiurewicz on sensitive dependence and existence of invariant measures, kicked off one of the most exciting areas within dynamical systems: iterations in dimension one. The purpose of this paper is to survey some of the recent developments, and pose some of the challenges and questions that keep this subject so intriguing.
    Mathematics Subject Classification: Primary: 37F10; Secondary: 30D05.


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