American Institute of Mathematical Sciences

Advanced
January  2001, 7(1): 35-50. doi: 10.3934/dcds.2001.7.35

Invariant manifolds for delay endomorphisms

Rovella Alvaro 1, , Vilamajó Francesc 2, and  Romero Neptalí 3,

1. 

Universidad de La República, Facultad de Ciencias. Centro de Matemática, Iguá 4225. Montevideo 11400, Uruguay
2. 

Universidad Politéecnica de Cataluña, Departament de Matemática Aplicada 2, Escola Técnica Superior d'Enginyers Industrials. Colom 11, 08222. Terrasa, Barcelona, Spain
3. 

Universidad Centro Occidental Lisandro Alvarado, Departamento de Matemática, Decanato de Ciencias. Apartado Postal 400. Barquisimeto, Venezuela

Received  October 1999 Revised  September 2000 Published  November 2000

Let $F_\mu(x_1,\cdots,x_k)=(x_2,\cdots,x_k,-x_1^2+\mu x_1)$. For any $G$ in a $C^2$ neighborhood $\mathcal{U}$ of the family $F_\mu$, the point at $\infty$ is an attractor (with basin denoted by $B_\infty$), and there exists a repelling fixed point in the boundary of $B_\infty$. This gives the initial step to the study of the whole boundary of $B_\infty$ and the changes it suffers: for perturbations of $F_\mu$ with $\mu$ small, the boundary of $B_\infty$ is an invariant codimension one manifold, while for large values of $\mu$ the basin $B_\infty$ is dense and its complementary set an expanding Cantor set. The techniques developed will be applied to delay endomorphisms.
Keywords: Expanding Cantor sets., Invariant manifolds.
Mathematics Subject Classification: 37C05, 37D10, 37D2.
Citation: Rovella Alvaro, Vilamajó Francesc, Romero Neptalí. Invariant manifolds for delay endomorphisms. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 35-50. doi: 10.3934/dcds.2001.7.35
[1]

Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3367-3387. doi: 10.3934/dcds.2020409
[2]

S. Astels. Thickness measures for Cantor sets. Electronic Research Announcements, 1999, 5: 108-111.

[3]

Mehdi Pourbarat. On the arithmetic difference of middle Cantor sets. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4259-4278. doi: 10.3934/dcds.2018186
[4]

Weiyuan Qiu, Fei Yang, Yongcheng Yin. Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3375-3416. doi: 10.3934/dcds.2016.36.3375
[5]

Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417
[6]

Silvére Gangloff, Alonso Herrera, Cristobal Rojas, Mathieu Sablik. Computability of topological entropy: From general systems to transformations on Cantor sets and the interval. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4259-4286. doi: 10.3934/dcds.2020180
[7]

Frank D. Grosshans, Jürgen Scheurle, Sebastian Walcher. Invariant sets forced by symmetry. Journal of Geometric Mechanics, 2012, 4 (3) : 271-296. doi: 10.3934/jgm.2012.4.271
[8]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[9]

Michael Hochman. Smooth symmetries of $\times a$-invariant sets. Journal of Modern Dynamics, 2018, 13: 187-197. doi: 10.3934/jmd.2018017
[10]

Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811
[11]

Fawwaz Batayneh, Cecilia González-Tokman. On the number of invariant measures for random expanding maps in higher dimensions. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5887-5914. doi: 10.3934/dcds.2021100
[12]

José F. Alves, Davide Azevedo. Statistical properties of diffeomorphisms with weak invariant manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 1-41. doi: 10.3934/dcds.2016.36.1
[13]

George Osipenko. Indestructibility of invariant locally non-unique manifolds. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 203-219. doi: 10.3934/dcds.1996.2.203
[14]

Henk Broer, Aaron Hagen, Gert Vegter. Numerical approximation of normally hyperbolic invariant manifolds. Conference Publications, 2003, 2003 (Special) : 133-140. doi: 10.3934/proc.2003.2003.133
[15]

Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967
[16]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579
[17]

Arturo Echeverría-Enríquez, Alberto Ibort, Miguel C. Muñoz-Lecanda, Narciso Román-Roy. Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (4) : 397-419. doi: 10.3934/jgm.2012.4.397
[18]

Roberto Castelli. Efficient representation of invariant manifolds of periodic orbits in the CRTBP. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 563-586. doi: 10.3934/dcdsb.2018197
[19]

Pablo Aguirre, Bernd Krauskopf, Hinke M. Osinga. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics, 2014, 1 (1) : 1-38. doi: 10.3934/jcd.2014.1.1
[20]

Clara Cufí-Cabré, Ernest Fontich. Differentiable invariant manifolds of nilpotent parabolic points. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4667-4704. doi: 10.3934/dcds.2021053

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (74)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy