-
Previous Article
Periodic orbits and invariant cones in three-dimensional piecewise linear systems
- DCDS Home
- This Issue
-
Next Article
Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics
Smooth stabilizers for measures on the torus
1. | Department of Mathematics, The Pennsylvania State University, State College, PA 16802, United States |
References:
[1] |
S. K. Aranson, G. R. Belitsky and E. V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Systems on Surfaces, American Mathematical Society, Providence, RI, 1996. |
[2] |
M. Baake and J. A. G. Roberts, Reversing symmetry group of $ Gl(2,\mathbbZ)$ and $ PGl(2,\mathbbZ)$ matrices with connections to cat maps and trace maps, J. Phys. A, 30 (1997), 1549-1573.
doi: 10.1088/0305-4470/30/5/020. |
[3] |
L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Mathematics and its Applications, Cambridge University Press, Cambridge, 2007.
doi: 10.1017/CBO9781107326026. |
[4] |
L. Barreira, Y. Pesin and J. Schmeling, Dimension and product structure of hyperbolic measures, Ann. of Math. (2), 149 (1999), 755-783.
doi: 10.2307/121072. |
[5] |
A. W. Brown, Rigidity Properties of Measures on the Torus, Ph.D thesis, Tufts University, 2011. |
[6] |
M. Einsiedler and E. Lindenstrauss, Rigidity properties of $\mathbbZ^d$-actions on tori and solenoids, Electron. Res. Announc. Amer. Math. Soc., 9 (2003), 99-110.
doi: 10.1090/S1079-6762-03-00117-3. |
[7] |
J. Franks, Anosov diffeomorphisms, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 61-93. |
[8] |
B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory Dynam. Systems, 14 (1994), 645-666.
doi: 10.1017/S0143385700008105. |
[9] |
H. Hu, Some ergodic properties of commuting diffeomorphisms, Ergodic Theory Dynam. Systems, 13 (1993), 73-100.
doi: 10.1017/S0143385700007215. |
[10] |
B. Kalinin and A. Katok, Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori, J. Mod. Dyn., 1 (2007), 123-146. |
[11] |
B. Kalinin, A. Katok and F. Rodriguez Hertz, Nonuniform measure rigidity, Ann. of Math. (2), 174 (2011), 361-400.
doi: 10.4007/annals.2011.174.1.10. |
[12] |
A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems, 16 (1996), 751-778.
doi: 10.1017/S0143385700009081. |
[13] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511809187. |
[14] |
F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Math. (2), 122 (1985), 540-574.
doi: 10.2307/1971329. |
[15] |
R. Mañé, Ergodic Theory and Differentiable Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1987.
doi: 10.1007/978-3-642-70335-5. |
[16] |
A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429.
doi: 10.2307/2373551. |
[17] |
S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770.
doi: 10.2307/2373372. |
[18] |
V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210. |
[19] |
J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Annals of Mathematics. Second Series, 42 (1941), 874-920.
doi: 10.2307/1968772. |
[20] |
J. Palis, C. Pugh and R. C. Robinson, Nondifferentiability of invariant foliations, in Dynamical Systems-Warwick 1974, Springer, (1975), 234-240. |
[21] |
C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, Duke Math. J., 86 (1997), 517-546.
doi: 10.1215/S0012-7094-97-08616-6. |
[22] |
V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, 1952 (1952), 55 pp. |
[23] |
D. J. Rudolph, $\times 2$ and $\times 3$ invariant measures and entropy, Ergodic Theory Dynam. Systems, 10 (1990), 395-406.
doi: 10.1017/S0143385700005629. |
show all references
References:
[1] |
S. K. Aranson, G. R. Belitsky and E. V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Systems on Surfaces, American Mathematical Society, Providence, RI, 1996. |
[2] |
M. Baake and J. A. G. Roberts, Reversing symmetry group of $ Gl(2,\mathbbZ)$ and $ PGl(2,\mathbbZ)$ matrices with connections to cat maps and trace maps, J. Phys. A, 30 (1997), 1549-1573.
doi: 10.1088/0305-4470/30/5/020. |
[3] |
L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Mathematics and its Applications, Cambridge University Press, Cambridge, 2007.
doi: 10.1017/CBO9781107326026. |
[4] |
L. Barreira, Y. Pesin and J. Schmeling, Dimension and product structure of hyperbolic measures, Ann. of Math. (2), 149 (1999), 755-783.
doi: 10.2307/121072. |
[5] |
A. W. Brown, Rigidity Properties of Measures on the Torus, Ph.D thesis, Tufts University, 2011. |
[6] |
M. Einsiedler and E. Lindenstrauss, Rigidity properties of $\mathbbZ^d$-actions on tori and solenoids, Electron. Res. Announc. Amer. Math. Soc., 9 (2003), 99-110.
doi: 10.1090/S1079-6762-03-00117-3. |
[7] |
J. Franks, Anosov diffeomorphisms, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 61-93. |
[8] |
B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory Dynam. Systems, 14 (1994), 645-666.
doi: 10.1017/S0143385700008105. |
[9] |
H. Hu, Some ergodic properties of commuting diffeomorphisms, Ergodic Theory Dynam. Systems, 13 (1993), 73-100.
doi: 10.1017/S0143385700007215. |
[10] |
B. Kalinin and A. Katok, Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori, J. Mod. Dyn., 1 (2007), 123-146. |
[11] |
B. Kalinin, A. Katok and F. Rodriguez Hertz, Nonuniform measure rigidity, Ann. of Math. (2), 174 (2011), 361-400.
doi: 10.4007/annals.2011.174.1.10. |
[12] |
A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems, 16 (1996), 751-778.
doi: 10.1017/S0143385700009081. |
[13] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511809187. |
[14] |
F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Math. (2), 122 (1985), 540-574.
doi: 10.2307/1971329. |
[15] |
R. Mañé, Ergodic Theory and Differentiable Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1987.
doi: 10.1007/978-3-642-70335-5. |
[16] |
A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429.
doi: 10.2307/2373551. |
[17] |
S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770.
doi: 10.2307/2373372. |
[18] |
V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210. |
[19] |
J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Annals of Mathematics. Second Series, 42 (1941), 874-920.
doi: 10.2307/1968772. |
[20] |
J. Palis, C. Pugh and R. C. Robinson, Nondifferentiability of invariant foliations, in Dynamical Systems-Warwick 1974, Springer, (1975), 234-240. |
[21] |
C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, Duke Math. J., 86 (1997), 517-546.
doi: 10.1215/S0012-7094-97-08616-6. |
[22] |
V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, 1952 (1952), 55 pp. |
[23] |
D. J. Rudolph, $\times 2$ and $\times 3$ invariant measures and entropy, Ergodic Theory Dynam. Systems, 10 (1990), 395-406.
doi: 10.1017/S0143385700005629. |
[1] |
Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123 |
[2] |
Woochul Jung, Keonhee Lee, Carlos Morales, Jumi Oh. Rigidity of random group actions. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6845-6854. doi: 10.3934/dcds.2020130 |
[3] |
Boris Kalinin, Anatole Katok, Federico Rodriguez Hertz. Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\Zk$ actions with Cartan homotopy data". Journal of Modern Dynamics, 2010, 4 (1) : 207-209. doi: 10.3934/jmd.2010.4.207 |
[4] |
Xiankun Ren. Periodic measures are dense in invariant measures for residually finite amenable group actions with specification. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1657-1667. doi: 10.3934/dcds.2018068 |
[5] |
Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Journal of Modern Dynamics, 2010, 4 (2) : 271-327. doi: 10.3934/jmd.2010.4.271 |
[6] |
Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68 |
[7] |
Maik Gröger, Olga Lukina. Measures and stabilizers of group Cantor actions. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2001-2029. doi: 10.3934/dcds.2020350 |
[8] |
Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269 |
[9] |
A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133. |
[10] |
Andrei Török. Rigidity of partially hyperbolic actions of property (T) groups. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 193-208. doi: 10.3934/dcds.2003.9.193 |
[11] |
João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837 |
[12] |
Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811 |
[13] |
Eleonora Catsigeras, Heber Enrich. SRB measures of certain almost hyperbolic diffeomorphisms with a tangency. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 177-202. doi: 10.3934/dcds.2001.7.177 |
[14] |
Francois Ledrappier and Omri Sarig. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Electronic Research Announcements, 2005, 11: 89-94. |
[15] |
Lennard F. Bakker, Pedro Martins Rodrigues. A profinite group invariant for hyperbolic toral automorphisms. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 1965-1976. doi: 10.3934/dcds.2012.32.1965 |
[16] |
Shigenori Matsumoto. A generic-dimensional property of the invariant measures for circle diffeomorphisms. Journal of Modern Dynamics, 2013, 7 (4) : 553-563. doi: 10.3934/jmd.2013.7.553 |
[17] |
Yong Fang, Patrick Foulon, Boris Hasselblatt. Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows. Electronic Research Announcements, 2010, 17: 80-89. doi: 10.3934/era.2010.17.80 |
[18] |
Stefano Galatolo, Mathieu Hoyrup, Cristóbal Rojas. Dynamics and abstract computability: Computing invariant measures. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 193-212. doi: 10.3934/dcds.2011.29.193 |
[19] |
Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185 |
[20] |
Zhenqi Jenny Wang. New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions. Journal of Modern Dynamics, 2010, 4 (4) : 585-608. doi: 10.3934/jmd.2010.4.585 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]