American Institute of Mathematical Sciences

Advanced
October  2007, 1(4): 615-648. doi: 10.3934/jmd.2007.1.615

Critical points for surface diffeomorphisms

Enrique R. Pujals 1, and  Federico Rodriguez Hertz 2,

1. 

IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil
2. 

IMERL-Facultad de Ingeniería, Universidad de la República, ulio Herrera y Reissig 565, CC 30, 11300 Montevideo, Uruguay

Received  February 2007 Revised  August 2007 Published  July 2007

Using the definition of dominated splitting, we introduce the notion of critical set for any dissipative surface diffeomorphism as an intrinsically well-defined object. We obtain a series of results related to this concept.
Keywords: partially hyperbolic systems, smooth mappings and diffeomorphisms, homeomorphisms and diffeomorphisms of planes and surfaces, dominated splitting., nonuniformly hyperbolic systems.
Mathematics Subject Classification: Primary: 37C05, 37E30, 37D25, 37D3.
Citation: Enrique R. Pujals, Federico Rodriguez Hertz. Critical points for surface diffeomorphisms. Journal of Modern Dynamics, 2007, 1 (4) : 615-648. doi: 10.3934/jmd.2007.1.615
[1]

Rafael Potrie. Partially hyperbolic diffeomorphisms with a trapping property. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 5037-5054. doi: 10.3934/dcds.2015.35.5037
[2]

Lorenzo J. Díaz, Todd Fisher. Symbolic extensions and partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1419-1441. doi: 10.3934/dcds.2011.29.1419
[3]

Andy Hammerlindl, Rafael Potrie, Mario Shannon. Seifert manifolds admitting partially hyperbolic diffeomorphisms. Journal of Modern Dynamics, 2018, 12: 193-222. doi: 10.3934/jmd.2018008
[4]

Lorenzo J. Díaz, Todd Fisher, M. J. Pacifico, José L. Vieitez. Entropy-expansiveness for partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4195-4207. doi: 10.3934/dcds.2012.32.4195
[5]

Boris Kalinin, Victoria Sadovskaya. Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 245-259. doi: 10.3934/dcds.2016.36.245
[6]

Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469
[7]

Andrey Gogolev. Partially hyperbolic diffeomorphisms with compact center foliations. Journal of Modern Dynamics, 2011, 5 (4) : 747-769. doi: 10.3934/jmd.2011.5.747
[8]

Thomas Barthelmé, Andrey Gogolev. Centralizers of partially hyperbolic diffeomorphisms in dimension 3. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4477-4484. doi: 10.3934/dcds.2021044
[9]

Dmitri Burago, Sergei Ivanov. Partially hyperbolic diffeomorphisms of 3-manifolds with Abelian fundamental groups. Journal of Modern Dynamics, 2008, 2 (4) : 541-580. doi: 10.3934/jmd.2008.2.541
[10]

Keith Burns, Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Anna Talitskaya, Raúl Ures. Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 75-88. doi: 10.3934/dcds.2008.22.75
[11]

Yujun Zhu. Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 869-882. doi: 10.3934/dcds.2014.34.869
[12]

Michael Brin, Dmitri Burago, Sergey Ivanov. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. Journal of Modern Dynamics, 2009, 3 (1) : 1-11. doi: 10.3934/jmd.2009.3.1
[13]

Doris Bohnet. Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation. Journal of Modern Dynamics, 2013, 7 (4) : 565-604. doi: 10.3934/jmd.2013.7.565
[14]

Radu Saghin. Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3789-3801. doi: 10.3934/dcds.2014.34.3789
[15]

Snir Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. Journal of Modern Dynamics, 2018, 13: 43-113. doi: 10.3934/jmd.2018013
[16]

Xinsheng Wang, Lin Wang, Yujun Zhu. Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2125-2140. doi: 10.3934/dcds.2018087
[17]

Zheng Yin, Ercai Chen. Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6581-6597. doi: 10.3934/dcds.2016085
[18]

Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435
[19]

Mauricio Poletti. Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5163-5188. doi: 10.3934/dcds.2018228
[20]

Jinhua Zhang. Partially hyperbolic diffeomorphisms with one-dimensional neutral center on 3-manifolds. Journal of Modern Dynamics, 2021, 17: 557-584. doi: 10.3934/jmd.2021019

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy