American Institute of Mathematical Sciences

March  2007, 19(1): 67-87. doi: 10.3934/dcds.2007.19.67

Dynamical properties of singular-hyperbolic attractors

 1 UNAM - Instituto de Matemáticas, U. Cuernavaca, A.P. 273 Admon. de correos # 3, Cuernavaca, Morelos 62251, México, Mexico 2 IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil

Received  August 2005 Revised  April 2007 Published  June 2007

We provide a dynamical portrait of singular-hyperbolic transitive attractors of a flow on a 3-manifold. Our Main Theorem establishes the existence of unstable manifolds for a subset of the attractor which is visited infinitely many times by a residual subset. As a consequence, we prove that the set of periodic orbits is dense, that it is the closure of a unique homoclinic class of some periodic orbit, and that there is an SRB-measure supported on the attractor.
Citation: Aubin Arroyo, Enrique R. Pujals. Dynamical properties of singular-hyperbolic attractors. Discrete and Continuous Dynamical Systems, 2007, 19 (1) : 67-87. doi: 10.3934/dcds.2007.19.67
 [1] 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 [2] Andy Hammerlindl. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3641-3669. doi: 10.3934/dcds.2013.33.3641 [3] P.E. Kloeden, José A. Langa, José Real. Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations. Communications on Pure and Applied Analysis, 2007, 6 (4) : 937-955. doi: 10.3934/cpaa.2007.6.937 [4] Marcelo R. R. Alves. Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds. Journal of Modern Dynamics, 2016, 10: 497-509. doi: 10.3934/jmd.2016.10.497 [5] Nina Lebedeva, Vladimir Matveev, Anton Petrunin, Vsevolod Shevchishin. Smoothing 3-dimensional polyhedral spaces. Electronic Research Announcements, 2015, 22: 12-19. doi: 10.3934/era.2015.22.12 [6] Lingling Liu, Bo Gao, Dongmei Xiao, Weinian Zhang. Identification of focus and center in a 3-dimensional system. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 485-522. doi: 10.3934/dcdsb.2014.19.485 [7] Jaume Llibre, Claudio A. Buzzi, Paulo R. da Silva. 3-dimensional Hopf bifurcation via averaging theory. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 529-540. doi: 10.3934/dcds.2007.17.529 [8] Tiantian Wu, Xiao-Song Yang. A new class of 3-dimensional piecewise affine systems with homoclinic orbits. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5119-5129. doi: 10.3934/dcds.2016022 [9] Shinobu Hashimoto, Shin Kiriki, Teruhiko Soma. Moduli of 3-dimensional diffeomorphisms with saddle-foci. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5021-5037. doi: 10.3934/dcds.2018220 [10] Philippe Jouan, Ronald Manríquez. Solvable approximations of 3-dimensional almost-Riemannian structures. Mathematical Control and Related Fields, 2022, 12 (2) : 303-326. doi: 10.3934/mcrf.2021023 [11] Vadim Kaloshin, Maria Saprykina. Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 611-640. doi: 10.3934/dcds.2006.15.611 [12] Bochao Chen, Yixian Gao. Quasi-periodic travelling waves for beam equations with damping on 3-dimensional rectangular tori. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 921-944. doi: 10.3934/dcdsb.2021075 [13] Chiara Zanini. Singular perturbations of finite dimensional gradient flows. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 657-675. doi: 10.3934/dcds.2007.18.657 [14] Juan Dávila, Manuel Del Pino, Catalina Pesce, Juncheng Wei. Blow-up for the 3-dimensional axially symmetric harmonic map flow into $S^2$. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6913-6943. doi: 10.3934/dcds.2019237 [15] Jana Rodriguez Hertz. Genericity of nonuniform hyperbolicity in dimension 3. Journal of Modern Dynamics, 2012, 6 (1) : 121-138. doi: 10.3934/jmd.2012.6.121 [16] Keyan Wang, Yi Du. Stability of the two dimensional magnetohydrodynamic flows in $\mathbb{R}^3$. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 1061-1073. doi: 10.3934/dcdsb.2012.17.1061 [17] 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 [18] Tingyuan Deng. Three-dimensional sphere $S^3$-attractors in Rayleigh-Bénard convection. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 577-591. doi: 10.3934/dcdsb.2010.13.577 [19] Enrique R. Pujals. On the density of hyperbolicity and homoclinic bifurcations for 3D-diffeomorphisms in attracting regions. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 179-226. doi: 10.3934/dcds.2006.16.179 [20] Péter Bálint, Imre Péter Tóth. Hyperbolicity in multi-dimensional Hamiltonian systems with applications to soft billiards. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 37-59. doi: 10.3934/dcds.2006.15.37

2020 Impact Factor: 1.392

Metrics

• PDF downloads (124)
• HTML views (0)
• Cited by (14)

Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]