American Institute of Mathematical Sciences

Advanced
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 - A, 2010, 27 (2) : 557-588. doi: 10.3934/dcds.2010.27.557
[1]

Zenonas Navickas, Rasa Smidtaite, Alfonsas Vainoras, Minvydas Ragulskis. The logistic map of matrices. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 927-944. doi: 10.3934/dcdsb.2011.16.927
[2]

Anatoli F. Ivanov. On global dynamics in a multi-dimensional discrete map. Conference Publications, 2015, 2015 (special) : 652-659. doi: 10.3934/proc.2015.0652
[3]

Dyi-Shing Ou, Kenneth James Palmer. A constructive proof of the existence of a semi-conjugacy for a one dimensional map. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 977-992. doi: 10.3934/dcdsb.2012.17.977
[4]

Francisco J. López-Hernández. Dynamics of induced homeomorphisms of one-dimensional solenoids. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4243-4257. doi: 10.3934/dcds.2018185
[5]

C. Bonanno, G. Menconi. Computational information for the logistic map at the chaos threshold. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 415-431. doi: 10.3934/dcdsb.2002.2.415
[6]

Zhi-An Wang, Kun Zhao. Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model. Communications on Pure & Applied Analysis, 2013, 12 (6) : 3027-3046. doi: 10.3934/cpaa.2013.12.3027
[7]

Charles Nguyen, Stephen Pankavich. A one-dimensional kinetic model of plasma dynamics with a transport field. Evolution Equations & Control Theory, 2014, 3 (4) : 681-698. doi: 10.3934/eect.2014.3.681
[8]

Hsuan-Wen Su. Finding invariant tori with Poincare's map. Communications on Pure & Applied Analysis, 2008, 7 (2) : 433-443. doi: 10.3934/cpaa.2008.7.433
[9]

Jacopo De Simoi. On cyclicity-one elliptic islands of the standard map. Journal of Modern Dynamics, 2013, 7 (2) : 153-208. doi: 10.3934/jmd.2013.7.153
[10]

Flávia M. Branco. Sub-actions and maximizing measures for one-dimensional transformations with a critical point. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 271-280. doi: 10.3934/dcds.2007.17.271
[11]

Steffen Klassert, Daniel Lenz, Peter Stollmann. Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1553-1571. doi: 10.3934/dcds.2011.29.1553
[12]

Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Period doubling and reducibility in the quasi-periodically forced logistic map. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1507-1535. doi: 10.3934/dcdsb.2012.17.1507
[13]

Luigi Ambrosio, Federico Glaudo, Dario Trevisan. On the optimal map in the $ 2 $-dimensional random matching problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7291-7308. doi: 10.3934/dcds.2019304
[14]

Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123
[15]

Gamaliel Blé. External arguments and invariant measures for the quadratic family. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 241-260. doi: 10.3934/dcds.2004.11.241
[16]

Marie-Claude Arnaud. A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map. Journal of Modern Dynamics, 2011, 5 (3) : 583-591. doi: 10.3934/jmd.2011.5.583
[17]

Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089
[18]

Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics. Conference Publications, 2013, 2013 (special) : 515-524. doi: 10.3934/proc.2013.2013.515
[19]

Ben Duan, Zhen Luo. Dynamics of vacuum states for one-dimensional full compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2543-2564. doi: 10.3934/cpaa.2013.12.2543
[20]

Maria do Carmo Pacheco de Toledo, Sergio Muniz Oliva. A discretization scheme for an one-dimensional reaction-diffusion equation with delay and its dynamics. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1041-1060. doi: 10.3934/dcds.2009.23.1041

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences