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May  2010, 27(2): 557-588. doi: 10.3934/dcds.2010.27.557

One-dimensional dynamics in the new millennium

Sebastian van Strien 1,

1. 

Universiteit of Warwick, Math. Dept, Coventry CV4 7AL, United Kingdom

Received  October 2009 Revised  February 2010 Published  February 2010

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.
Keywords: Palis Conjecture., Quadratic map, Hyperbolicity, Logistic map, Invariant Measures, One-dimensional dynamics.
Mathematics Subject Classification: Primary: 37F10; Secondary: 30D0.
Citation: Sebastian van Strien. One-dimensional dynamics in the new millennium. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 557-588. doi: 10.3934/dcds.2010.27.557
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