-
Previous Article
Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems
- DCDS Home
- This Issue
-
Next Article
Characterization of noncorrelated pattern sequences and correlation dimensions
Lyapunov exponents of cocycles over non-uniformly hyperbolic systems
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA |
We consider linear cocycles over non-uniformly hyperbolic dynamical systems. The base system is a diffeomorphism $f$ of a compact manifold $X$ preserving a hyperbolic ergodic probability measure $μ$. The cocycle $\mathcal{A}$ over $f$ is Hölder continuous and takes values in $GL(d, \mathbb{R})$ or, more generally, in the group of invertible bounded linear operators on a Banach space. For a $GL(d, \mathbb{R})$-valued cocycle $\mathcal{A}$ we prove that the Lyapunov exponents of $\mathcal{A}$ with respect to $μ$ can be approximated by the Lyapunov exponents of $\mathcal{A}$ with respect to measures on hyperbolic periodic orbits of $f$. In the infinite-dimensional setting one can define the upper and lower Lyapunov exponents of $\mathcal{A}$ with respect to $μ$, but they cannot always be approximated by the exponents of $\mathcal{A}$ on periodic orbits. We prove that they can be approximated in terms of the norms of the return values of $\mathcal{A}$ on hyperbolic periodic orbits of $f$.
References:
| [1] |
L. Backes, On the periodic approximation of Lyapunov exponents for semi-invertible cocycles,
Discrete Contin. Dyn. Syst., 37 (2017), 6353–6368, arXiv: 1612.04159.
doi: 10.3934/dcds.2017275. |
| [2] |
L. Barreira and Ya. Pesin,
Nonuniform Hyperbolicity: Dynamics Of Systems with Nonzero Lyapunov Exponents, Encyclopedia of Mathematics and Its Applications 115 Cambridge University Press, 2007.
doi: 10.1017/CBO9781107326026. |
| [3] |
S. Gouëzel and A. Karlsson, Subadditive and multiplicative ergodic theorems, Preprint. |
| [4] |
G. Grabarnik and M. Guysinsky,
Livšic theorem for Banach rings, Discrete and Continuous Dynamical Systems, 37 (2017), 4379-4390.
doi: 10.3934/dcds.2017187. |
| [5] |
B. Kalinin,
Livšic theorem for matrix cocycles, Annals of Math., 173 (2011), 1025-1042.
doi: 10.4007/annals.2011.173.2.11. |
| [6] |
B. Kalinin and V. Sadovskaya,
Cocycles with one exponent over partially hyperbolic systems, Geometriae Dedicata, 167 (2013), 167-188.
doi: 10.1007/s10711-012-9808-z. |
| [7] |
B. Kalinin and V. Sadovskaya, Periodic approximation of Lyapunov exponents for Banach cocycles,
Ergodic Theory and Dynamical Systems, 2017, arXiv: 1608.05757
doi: 10.1017/etds.2017.43. |
| [8] |
A. Karlsson and G. Margulis,
A multiplicative ergodic theorem and nonpositively curved spaces, Communications in Mathematical Physics, 208 (1999), 107-123.
doi: 10.1007/s002200050750. |
| [9] |
A. Katok,
Lyapunov exponents, entropy and periodic points of diffeomorphisms, Publ. Math. IHES, 51 (1980), 137-173.
|
| [10] |
V. Oseledets,
A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems, Trans. Mosc. Math. Soc., 19 (1968), 179-210.
|
| [11] |
Ya. Pesin,
Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv., 40 (1976), 1261-1305.
|
| [12] |
Ya. Pesin,
Characteristic Ljapunov exponents and smooth ergodic theory, Russ. Math. Surv., 32 (1977), 55-112,187.
|
| [13] |
M. Pollicott,
Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds, Cambridge University Press, 1993.
doi: 10.1017/CBO9780511752537. |
| [14] |
Z. Wang and W. Sun,
Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits, Trans. Amer. Math. Soc., 362 (2010), 4267-4282.
doi: 10.1090/S0002-9947-10-04947-0. |
show all references
References:
| [1] |
L. Backes, On the periodic approximation of Lyapunov exponents for semi-invertible cocycles,
Discrete Contin. Dyn. Syst., 37 (2017), 6353–6368, arXiv: 1612.04159.
doi: 10.3934/dcds.2017275. |
| [2] |
L. Barreira and Ya. Pesin,
Nonuniform Hyperbolicity: Dynamics Of Systems with Nonzero Lyapunov Exponents, Encyclopedia of Mathematics and Its Applications 115 Cambridge University Press, 2007.
doi: 10.1017/CBO9781107326026. |
| [3] |
S. Gouëzel and A. Karlsson, Subadditive and multiplicative ergodic theorems, Preprint. |
| [4] |
G. Grabarnik and M. Guysinsky,
Livšic theorem for Banach rings, Discrete and Continuous Dynamical Systems, 37 (2017), 4379-4390.
doi: 10.3934/dcds.2017187. |
| [5] |
B. Kalinin,
Livšic theorem for matrix cocycles, Annals of Math., 173 (2011), 1025-1042.
doi: 10.4007/annals.2011.173.2.11. |
| [6] |
B. Kalinin and V. Sadovskaya,
Cocycles with one exponent over partially hyperbolic systems, Geometriae Dedicata, 167 (2013), 167-188.
doi: 10.1007/s10711-012-9808-z. |
| [7] |
B. Kalinin and V. Sadovskaya, Periodic approximation of Lyapunov exponents for Banach cocycles,
Ergodic Theory and Dynamical Systems, 2017, arXiv: 1608.05757
doi: 10.1017/etds.2017.43. |
| [8] |
A. Karlsson and G. Margulis,
A multiplicative ergodic theorem and nonpositively curved spaces, Communications in Mathematical Physics, 208 (1999), 107-123.
doi: 10.1007/s002200050750. |
| [9] |
A. Katok,
Lyapunov exponents, entropy and periodic points of diffeomorphisms, Publ. Math. IHES, 51 (1980), 137-173.
|
| [10] |
V. Oseledets,
A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems, Trans. Mosc. Math. Soc., 19 (1968), 179-210.
|
| [11] |
Ya. Pesin,
Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv., 40 (1976), 1261-1305.
|
| [12] |
Ya. Pesin,
Characteristic Ljapunov exponents and smooth ergodic theory, Russ. Math. Surv., 32 (1977), 55-112,187.
|
| [13] |
M. Pollicott,
Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds, Cambridge University Press, 1993.
doi: 10.1017/CBO9780511752537. |
| [14] |
Z. Wang and W. Sun,
Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits, Trans. Amer. Math. Soc., 362 (2010), 4267-4282.
doi: 10.1090/S0002-9947-10-04947-0. |
| [1] |
Suzete Maria Afonso, Vanessa Ramos, Jaqueline Siqueira. Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4485-4513. doi: 10.3934/dcds.2021045 |
| [2] |
Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505 |
| [3] |
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 |
| [4] |
Marzie Zaj, Abbas Fakhari, Fatemeh Helen Ghane, Azam Ehsani. Physical measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1777-1807. doi: 10.3934/dcds.2018073 |
| [5] |
Yongluo Cao, Stefano Luzzatto, Isabel Rios. Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 61-71. doi: 10.3934/dcds.2006.15.61 |
| [6] |
Nicolai T. A. Haydn, Kasia Wasilewska. Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2585-2611. doi: 10.3934/dcds.2016.36.2585 |
| [7] |
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 |
| [8] |
Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020 |
| [9] |
Mark Pollicott. Local Hölder regularity of densities and Livsic theorems for non-uniformly hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1247-1256. doi: 10.3934/dcds.2005.13.1247 |
| [10] |
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 |
| [11] |
Andrey Gogolev, Ali Tahzibi. Center Lyapunov exponents in partially hyperbolic dynamics. Journal of Modern Dynamics, 2014, 8 (3&4) : 549-576. doi: 10.3934/jmd.2014.8.549 |
| [12] |
Lucas Backes. On the periodic approximation of Lyapunov exponents for semi-invertible cocycles. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6353-6368. doi: 10.3934/dcds.2017275 |
| [13] |
Yuan-Ling Ye. Non-uniformly expanding dynamical systems: Multi-dimension. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2511-2553. doi: 10.3934/dcds.2019106 |
| [14] |
Boris Kalinin, Victoria Sadovskaya. Linear cocycles over hyperbolic systems and criteria of conformality. Journal of Modern Dynamics, 2010, 4 (3) : 419-441. doi: 10.3934/jmd.2010.4.419 |
| [15] |
Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235 |
| [16] |
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 |
| [17] |
Carlos Arnoldo Morales. A note on periodic orbits for singular-hyperbolic flows. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 615-619. doi: 10.3934/dcds.2004.11.615 |
| [18] |
Francois Ledrappier and Omri Sarig. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Electronic Research Announcements, 2005, 11: 89-94. |
| [19] |
Vaughn Climenhaga, Yakov Pesin, Agnieszka Zelerowicz. Equilibrium measures for some partially hyperbolic systems. Journal of Modern Dynamics, 2020, 16: 155-205. doi: 10.3934/jmd.2020006 |
| [20] |
Lucas Backes, Aaron Brown, Clark Butler. Continuity of Lyapunov exponents for cocycles with invariant holonomies. Journal of Modern Dynamics, 2018, 12: 223-260. doi: 10.3934/jmd.2018009 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]