
-
Previous Article
Fluctuations of ergodic sums on periodic orbits under specification
- DCDS Home
- This Issue
-
Next Article
Ruelle operator for continuous potentials and DLR-Gibbs measures
Structure of accessibility classes
1. | Department of Mathematics, Southern University of Science and Technology of China, No 1088, xueyuan Rd., Xili, Nanshan District, Shenzhen, Guangdong 518055, China |
2. | SUSTech International Center for Mathematics |
3. | Instituto de Matemática, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile |
In this work we deal with dynamically coherent partially hyperbolic diffeomorphisms whose central direction is two dimensional. We prove that in general the accessibility classes are topologically immersed manifolds. If, furthermore, the diffeomorphism satisfies certain bunching condition, then the accessibility classes are immersed $ C^{1} $-manifolds.
References:
[1] |
A. Avila and M. Viana,
Stable accessibility with 2-dimensional center, Astérisque, 416 (2020), 299-318.
|
[2] |
M. I. Brin and J. B. Pesin,
Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.
doi: 10.1070/IM1974v008n01ABEH002101. |
[3] |
A. Brown, Smoothness of stable holonomies inside center-stable manifolds and the $C^{2}$ hypothesis in Pugh-Shub and Ledrappier-Young theory, preprint, arXiv: 1608.05886. |
[4] |
K. Burns, F. R. Hertz, M. A. R Hertz, A. Talitskaya and R. Ures,
Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center, Discrete Contin. Dyn. Syst., 22 (2008), 75-88.
doi: 10.3934/dcds.2008.22.75. |
[5] |
P. Didier,
Stability of accessibility, Ergodic Theory Dynam. Systems, 23 (2003), 1717-1731.
doi: 10.1017/S0143385702001785. |
[6] |
D. Dolgopyat and A. Wilkinson,
Stable accessibility is $C^1$ dense. Geometric methods in dynamics. Ⅱ, Astérisque, 287 (2003), 33-60.
|
[7] |
M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.
doi: 10.1007/BFb0092042. |
[8] |
V. Horita and M. Sambarino,
Stable ergodicity and accessibility for certain partially hyperbolic diffeomorphisms with bidimensional center leaves, Comment. Math. Helv., 92 (2017), 467-512.
doi: 10.4171/CMH/417. |
[9] |
J.-L. Journé,
A regularity lemma for functions of several variables, Rev. Mat. Iberoamericana, 4 (1988), 187-193.
doi: 10.4171/RMI/69. |
[10] |
A. Katok and V. Niţică, Rigidity in Higher Rank Abelian Group Actions. Volume I. Introduction and Cocycle Problem, Cambridge Tracts in Mathematics, 185, Cambridge University
Press, Cambridge, 2011.
doi: 10.1017/CBO9780511803550. |
[11] |
C. Pugh and M. Shub, Stable Ergodicity and Partial Hyperbolicity, in International Conference on Dynamical Systems (Montevideo, 1995), Pitman Res. Notes Math. Ser., 362, Longman, Harlow, 1996,182–187. |
[12] |
C. Pugh and M. Shub,
Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. (JEMS), 2 (2000), 1-52.
doi: 10.1007/s100970050013. |
[13] |
C. Pugh, M. Shub and A. Wilkinson,
Hölder foliations, Duke Math. J., 86 (1997), 517-546.
doi: 10.1215/S0012-7094-97-08616-6. |
[14] |
D. Repovš, A. B. Skopenkov and E. V. Ščepin,
$C^1$-homogeneous compacta in $\Bbb{R}^n$ are $C^1$-submanifolds of $\Bbb{R}^n$, Proc. Amer. Math. Soc., 124 (1996), 1219-1226.
doi: 10.1090/S0002-9939-96-03157-7. |
[15] |
F. Rodriguez Hertz, Stable ergodicity of certain linear automorphisms of the torus, Ann. of
Math. (2), 162 (2005), 65–107.
doi: 10.4007/annals.2005.162.65. |
[16] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures,
Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math., 172 (2008), 353-381.
doi: 10.1007/s00222-007-0100-z. |
[17] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures,
A non-dynamically coherent example on $\Bbb{T}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1023-1032.
doi: 10.1016/j.anihpc.2015.03.003. |
[18] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. A. Ures, A survey of partially hyperbolic dynamics, in Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007, 35–87. |
[19] |
A. Wilkinson,
The cohomological equation for partially hyperbolic diffeomorphisms, Astérisque, 358 (2013), 75-165.
|
show all references
References:
[1] |
A. Avila and M. Viana,
Stable accessibility with 2-dimensional center, Astérisque, 416 (2020), 299-318.
|
[2] |
M. I. Brin and J. B. Pesin,
Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.
doi: 10.1070/IM1974v008n01ABEH002101. |
[3] |
A. Brown, Smoothness of stable holonomies inside center-stable manifolds and the $C^{2}$ hypothesis in Pugh-Shub and Ledrappier-Young theory, preprint, arXiv: 1608.05886. |
[4] |
K. Burns, F. R. Hertz, M. A. R Hertz, A. Talitskaya and R. Ures,
Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center, Discrete Contin. Dyn. Syst., 22 (2008), 75-88.
doi: 10.3934/dcds.2008.22.75. |
[5] |
P. Didier,
Stability of accessibility, Ergodic Theory Dynam. Systems, 23 (2003), 1717-1731.
doi: 10.1017/S0143385702001785. |
[6] |
D. Dolgopyat and A. Wilkinson,
Stable accessibility is $C^1$ dense. Geometric methods in dynamics. Ⅱ, Astérisque, 287 (2003), 33-60.
|
[7] |
M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.
doi: 10.1007/BFb0092042. |
[8] |
V. Horita and M. Sambarino,
Stable ergodicity and accessibility for certain partially hyperbolic diffeomorphisms with bidimensional center leaves, Comment. Math. Helv., 92 (2017), 467-512.
doi: 10.4171/CMH/417. |
[9] |
J.-L. Journé,
A regularity lemma for functions of several variables, Rev. Mat. Iberoamericana, 4 (1988), 187-193.
doi: 10.4171/RMI/69. |
[10] |
A. Katok and V. Niţică, Rigidity in Higher Rank Abelian Group Actions. Volume I. Introduction and Cocycle Problem, Cambridge Tracts in Mathematics, 185, Cambridge University
Press, Cambridge, 2011.
doi: 10.1017/CBO9780511803550. |
[11] |
C. Pugh and M. Shub, Stable Ergodicity and Partial Hyperbolicity, in International Conference on Dynamical Systems (Montevideo, 1995), Pitman Res. Notes Math. Ser., 362, Longman, Harlow, 1996,182–187. |
[12] |
C. Pugh and M. Shub,
Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. (JEMS), 2 (2000), 1-52.
doi: 10.1007/s100970050013. |
[13] |
C. Pugh, M. Shub and A. Wilkinson,
Hölder foliations, Duke Math. J., 86 (1997), 517-546.
doi: 10.1215/S0012-7094-97-08616-6. |
[14] |
D. Repovš, A. B. Skopenkov and E. V. Ščepin,
$C^1$-homogeneous compacta in $\Bbb{R}^n$ are $C^1$-submanifolds of $\Bbb{R}^n$, Proc. Amer. Math. Soc., 124 (1996), 1219-1226.
doi: 10.1090/S0002-9939-96-03157-7. |
[15] |
F. Rodriguez Hertz, Stable ergodicity of certain linear automorphisms of the torus, Ann. of
Math. (2), 162 (2005), 65–107.
doi: 10.4007/annals.2005.162.65. |
[16] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures,
Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math., 172 (2008), 353-381.
doi: 10.1007/s00222-007-0100-z. |
[17] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures,
A non-dynamically coherent example on $\Bbb{T}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1023-1032.
doi: 10.1016/j.anihpc.2015.03.003. |
[18] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. A. Ures, A survey of partially hyperbolic dynamics, in Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007, 35–87. |
[19] |
A. Wilkinson,
The cohomological equation for partially hyperbolic diffeomorphisms, Astérisque, 358 (2013), 75-165.
|
[1] |
Keith Burns, Dmitry Dolgopyat, Yakov Pesin, Mark Pollicott. Stable ergodicity for partially hyperbolic attractors with negative central exponents. Journal of Modern Dynamics, 2008, 2 (1) : 63-81. doi: 10.3934/jmd.2008.2.63 |
[2] |
Carlos H. Vásquez. Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents. Journal of Modern Dynamics, 2009, 3 (2) : 233-251. doi: 10.3934/jmd.2009.3.233 |
[3] |
Charles Pugh, Michael Shub, Alexander Starkov. Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 845-855. doi: 10.3934/dcds.2006.14.845 |
[4] |
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 |
[5] |
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 |
[6] |
F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures. A criterion for ergodicity for non-uniformly hyperbolic diffeomorphisms. Electronic Research Announcements, 2007, 14: 74-81. doi: 10.3934/era.2007.14.74 |
[7] |
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 |
[8] |
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 |
[9] |
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 |
[10] |
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 |
[11] |
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 |
[12] |
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 |
[13] |
C.P. Walkden. Stable ergodicity of skew products of one-dimensional hyperbolic flows. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 897-904. doi: 10.3934/dcds.1999.5.897 |
[14] |
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 |
[15] |
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 |
[16] |
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 |
[17] |
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 |
[18] |
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 |
[19] |
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 |
[20] |
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 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]