# American Institute of Mathematical Sciences

• 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
October  2018, 38(10): 5105-5118. doi: 10.3934/dcds.2018224

## Lyapunov exponents of cocycles over non-uniformly hyperbolic systems

 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Received  December 2017 Published  July 2018

Fund Project: The first author was supported in part by Simons Foundation grant 426243, the second author was supported in part by NSF grant DMS-1301693.

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 $f$.

Citation: Boris Kalinin, Victoria Sadovskaya. Lyapunov exponents of cocycles over non-uniformly hyperbolic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5105-5118. doi: 10.3934/dcds.2018224
##### References:

show all references

##### References:
 [1] Suzete Maria Afonso, Vanessa Ramos, Jaqueline Siqueira. Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4485-4513. doi: 10.3934/dcds.2021045 [2] Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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] 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 [20] Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619

2020 Impact Factor: 1.392