American Institute of Mathematical Sciences

Advanced
February  2005, 13(2): 239-269. doi: 10.3934/dcds.2005.13.239

Robustly transitive singular sets via approach of an extended linear Poincaré flow

Ming Li 1, , Shaobo Gan 1, and  Lan Wen 2,

1. 

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China, China
2. 

School of Mathematic Sciences, Peking University, Beijing, 100871

Received  August 2004 Revised  December 2004 Published  April 2005

Morales, Pacifico and Pujals proved recently that every robustly transitive singular set for a 3-dimensional flow must be partially hyperbolic. In this paper we generalize the result to higher dimensions. By definition, an isolated invariant set $\Lambda$ of a $C^1$ vector field $S$ is called robustly transitive if there exist an isolating neighborhood $U$ of $\Lambda$ in $M$ and a $C^1$ neighborhood $\mathcal U$ of $S$ in $\mathcal X^1(M)$ such that for every $X\in\mathcal U$, the maximal invariant set of $X$ in $U$ is non-trivially transitive. Such a set $\Lambda$ is called singular if it contains a singularity. The set $\Lambda$ is called strongly homogeneous of index $i$, if there exist an isolating neighborhood $U$ of $\Lambda$ in $M$ and a $C^1$ neighborhood $\mathcal U$ of $S$ in $\mathcal X^1(M)$ such that all periodic orbits of all $X\in\mathcal U$ contained in $U$ have the same index $i$. We prove in this paper that any robustly transitive singular set that is strongly homogeneous must be partially hyperbolic, as long as the indices of singularities and periodic orbits fit in certain way. As corollaries we obtain that every robust singular attractor (or repeller) that is strongly homogeneous must be partially hyperbolic and, if dim$M\le 4$, every robustly transitive singular set that is strongly homogeneous must be partially hyperbolic. The main novelty of the proofs in this paper is an extension of the usual linear Poincaré flow "to singularities".
Keywords: star flow., Robustly transitive set, extended linear poincaré flow, partial hyperbolicity.
Mathematics Subject Classification: 37D30.
Citation: Ming Li, Shaobo Gan, Lan Wen. Robustly transitive singular sets via approach of an extended linear Poincaré flow. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 239-269. doi: 10.3934/dcds.2005.13.239
[1]

Shengzhi Zhu, Shaobo Gan, Lan Wen. Indices of singularities of robustly transitive sets. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 945-957. doi: 10.3934/dcds.2008.21.945
[2]

Cheng Cheng, Shaobo Gan, Yi Shi. A robustly transitive diffeomorphism of Kan's type. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 867-888. doi: 10.3934/dcds.2018037
[3]

Cristina Lizana, Vilton Pinheiro, Paulo Varandas. Contribution to the ergodic theory of robustly transitive maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 353-365. doi: 10.3934/dcds.2015.35.353
[4]

Pablo G. Barrientos, Artem Raibekas. Robustly non-hyperbolic transitive symplectic dynamics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 5993-6013. doi: 10.3934/dcds.2018259
[5]

Yi Shi, Shaobo Gan, Lan Wen. On the singular-hyperbolicity of star flows. Journal of Modern Dynamics, 2014, 8 (2) : 191-219. doi: 10.3934/jmd.2014.8.191
[6]

Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901
[7]

Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Linear model of traffic flow in an isolated network. Conference Publications, 2015, 2015 (special) : 670-677. doi: 10.3934/proc.2015.0670
[8]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9
[9]

Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351
[10]

Fanghua Lin, Chun Liu. Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 1-22. doi: 10.3934/dcds.1996.2.1
[11]

Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541
[12]

Yakov Pesin. On the work of Dolgopyat on partial and nonuniform hyperbolicity. Journal of Modern Dynamics, 2010, 4 (2) : 227-241. doi: 10.3934/jmd.2010.4.227
[13]

Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Partial hyperbolicity and ergodicity in dimension three. Journal of Modern Dynamics, 2008, 2 (2) : 187-208. doi: 10.3934/jmd.2008.2.187
[14]

Jérôme Buzzi, Todd Fisher. Entropic stability beyond partial hyperbolicity. Journal of Modern Dynamics, 2013, 7 (4) : 527-552. doi: 10.3934/jmd.2013.7.527
[15]

Artyom Nahapetyan, Panos M. Pardalos. A bilinear relaxation based algorithm for concave piecewise linear network flow problems. Journal of Industrial & Management Optimization, 2007, 3 (1) : 71-85. doi: 10.3934/jimo.2007.3.71
[16]

K.H. Wong, C. Myburgh, L. Omari. A gradient flow approach for computing jump linear quadratic optimal feedback gains. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 803-808. doi: 10.3934/dcds.2000.6.803
[17]

Andy Hammerlindl. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3641-3669. doi: 10.3934/dcds.2013.33.3641
[18]

Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006
[19]

Rafael Potrie. Partial hyperbolicity and foliations in $\mathbb{T}^3$. Journal of Modern Dynamics, 2015, 9: 81-121. doi: 10.3934/jmd.2015.9.81
[20]

Yoshikazu Giga, Hiroyoshi Mitake, Hung V. Tran. Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019228

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (17)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences