American Institute of Mathematical Sciences

Advanced
May  2003, 9(3): 617-632. doi: 10.3934/dcds.2003.9.617

Rotation sets for unimodal maps of the interval

Christopher Cleveland 1,

1. 

Department of Mathematics, Indiana University Purdue University - Indianapolis, 402 N. Blackford Street, Indianapolis, IN 46202, United States

Received  October 2001 Revised  November 2002 Published  February 2003

We relate the rotation interval $\rho(f)$ of a unimodal map $f$ of the interval with its kneading invariant $K(f)$. In particular, we show that for any $\mu \in (0,\frac{1}{2})$, there are kneading invariants $\nu_\mu$ and $\nu_{\mu, h o m}$ such that $\rho(f)=[\mu, \frac{1}{2}]$ if and only if $\nu_\mu \preceq K(f) \preceq \nu_{\mu, h o m}$.
Keywords: twist itinerary, twist homoclinic itinerary., pattern, Unimodal map, kneading invariant, rotation interval.
Mathematics Subject Classification: 26A18, 37B10, 37C25, 37C29, 37E05, 37E15, 37E4.
Citation: Christopher Cleveland. Rotation sets for unimodal maps of the interval. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 617-632. doi: 10.3934/dcds.2003.9.617
[1]

Salvador Addas-Zanata. Stability for the vertical rotation interval of twist mappings. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 631-642. doi: 10.3934/dcds.2006.14.631
[2]

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
[3]

Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343
[4]

Tifei Qian, Zhihong Xia. Heteroclinic orbits and chaotic invariant sets for monotone twist maps. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 69-95. doi: 10.3934/dcds.2003.9.69
[5]

Lianpeng Yang, Xiong Li. Existence of periodically invariant tori on resonant surfaces for twist mappings. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1389-1409. doi: 10.3934/dcds.2020081
[6]

Qiudong Wang. The diffusion time of the connecting orbit around rotation number zero for the monotone twist maps. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 255-274. doi: 10.3934/dcds.2000.6.255
[7]

Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo. Dynamics of continued fractions and kneading sequences of unimodal maps. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1313-1332. doi: 10.3934/dcds.2013.33.1313
[8]

Lei Wang, Quan Yuan, Jia Li. Persistence of the hyperbolic lower dimensional non-twist invariant torus in a class of Hamiltonian systems. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1233-1250. doi: 10.3934/cpaa.2016.15.1233
[9]

Yuri Chekanov, Felix Schlenk. Notes on monotone Lagrangian twist tori. Electronic Research Announcements, 2010, 17: 104-121. doi: 10.3934/era.2010.17.104
[10]

John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047
[11]

Zhihong Xia, Peizheng Yu. A fixed point theorem for twist maps. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4051-4059. doi: 10.3934/dcds.2022045
[12]

Michael C. Sullivan. Invariants of twist-wise flow equivalence. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 475-484. doi: 10.3934/dcds.1998.4.475
[13]

Daniel Núñez, Pedro J. Torres. Periodic solutions of twist type of an earth satellite equation. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 303-306. doi: 10.3934/dcds.2001.7.303
[14]

Michael C. Sullivan. Invariants of twist-wise flow equivalence. Electronic Research Announcements, 1997, 3: 126-130.

[15]

Thierry Champion, Luigi De Pascale. On the twist condition and $c$-monotone transport plans. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1339-1353. doi: 10.3934/dcds.2014.34.1339
[16]

Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295
[17]

Huiping Jin. Boundedness in a class of duffing equations with oscillating potentials via the twist theorem. Communications on Pure and Applied Analysis, 2011, 10 (1) : 179-192. doi: 10.3934/cpaa.2011.10.179
[18]

Yanmin Niu, Xiong Li. An application of Moser's twist theorem to superlinear impulsive differential equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 431-445. doi: 10.3934/dcds.2019017
[19]

Anna Belova. Rigorous enclosures of rotation numbers by interval methods. Journal of Computational Dynamics, 2016, 3 (1) : 81-91. doi: 10.3934/jcd.2016004
[20]

Julián López-Gómez, Eduardo Muñoz-Hernández, Fabio Zanolin. On the applicability of the poincaré–Birkhoff twist theorem to a class of planar periodic predator-prey models. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2393-2419. doi: 10.3934/dcds.2020119

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (77)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy