One-dimensional dynamics in the new millennium

Sebastian van Strien

doi:10.3934/dcds.2010.27.557

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2010, Volume 27, Issue 2: 557-588. Doi: 10.3934/dcds.2010.27.557

This issue Previous Article Mathematical theory of viscous fluids: Retrospective and future perspectives Next Article Open problems in the theory of non-uniform hyperbolicity

One-dimensional dynamics in the new millennium

Sebastian van Strien^1,

1.
Universiteit of Warwick, Math. Dept, Coventry CV4 7AL

Received: September 30, 2009

Revised: January 31, 2010

Published: June 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 37F10; Secondary: 30D05.

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