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August  2020, 40(8): 4653-4664. doi: 10.3934/dcds.2020196

Structure of accessibility classes

Jana Rodriguez Hertz 1,2, and  Carlos H. Vásquez 3,

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

CV was supported by Proyecto Fondecyt 1171427.
JRH was partially supported by NSFC 11871262 and NSFC 11871394.

Received  April 2019 Revised  February 2020 Published  May 2020

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.

Keywords: Accesibility, partially hyperbolic diffeomorphisms, stable ergodicity.
Mathematics Subject Classification: Primary: 37D30; Secondary: 37D10, 37C14.
Citation: Jana Rodriguez Hertz, Carlos H. Vásquez. Structure of accessibility classes. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4653-4664. doi: 10.3934/dcds.2020196
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. 

Figure 1.  A simple triod in $ AC(x)\cap D $
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Figure 2.  $ AC^{D}(\xi) $ separates $ D $ into at least 3 connected components
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