December  2017, 37(12): 6353-6368. doi: 10.3934/dcds.2017275

On the periodic approximation of Lyapunov exponents for semi-invertible cocycles

Departamento de Matemática, Instituto de Matemática e Estatística, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500,91509-900, Porto Alegre, RS, Brazil

* Corresponding author

Received  January 2017 Revised  July 2017 Published  August 2017

We prove that, for semi-invertible linear cocycles, Lyapunov exponents of ergodic measures may be approximated by Lyapunov exponents on periodic points.

Citation: Lucas Backes. On the periodic approximation of Lyapunov exponents for semi-invertible cocycles. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6353-6368. doi: 10.3934/dcds.2017275
References:
[1]

L. Backes, Rigidity of fiber bunched cocycles, Bulletin of the Brazilian Mathematical Society, 46 (2015), 163-179.  doi: 10.1007/s00574-015-0089-7.  Google Scholar

[2]

L. Backes and A. Kocsard, Cohomology of dominated diffeomorphism-valued cocycles over hyperbolic systems, Ergodic Theory and Dynamical Systems, 36 (2016), 1703-1722.  doi: 10.1017/etds.2014.149.  Google Scholar

[3]

L. Barreira and Ya. Pesin, Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents, Cambridge University Press, 2007. doi: 10.1017/CBO9781107326026.  Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics 470. Springer-Verlag, 1975.  Google Scholar

[5]

Y. Cao, Non-zero Lyapunov exponents and uniform hyperbolicity, Nonlinearity, 16 (2003), 1473-1479.  doi: 10.1088/0951-7715/16/4/316.  Google Scholar

[6]

X. Dai, On the approximation of Lyapunov exponents and a question suggested by Anatole Katok, Nonlinearity, 23 (2010), 513-528.  doi: 10.1088/0951-7715/23/3/004.  Google Scholar

[7]

R. de la Llave and A. Windsor, Livšic theorems for non-commutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems, Ergodic Theory and Dynamical Systems, 30 (2010), 1055-1100.  doi: 10.1017/S014338570900039X.  Google Scholar

[8]

D. Dragičević and G. Froyland, Hölder continuity of Oseledets splittings for semi-invertible operator cocycles, Ergodic Theory and Dynamical Systems, 2016, 21pp. doi: 10.1017/etds.2016.55.  Google Scholar

[9]

G. FroylandS. LLoyd and A. Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory and Dynamical Systems, 30 (2010), 729-756.  doi: 10.1017/S0143385709000339.  Google Scholar

[10]

B. Kalinin, Livšic theorem for matrix cocycles, Annals of Mathematics, 173 (2011), 1025-1042.  doi: 10.4007/annals.2011.173.2.11.  Google Scholar

[11]

B. Kalinin and V. Sadovskaya, Periodic approximation of Lyapunov exponents for Banach cocycles, arXiv: 1608.05757. doi: 10.1017/etds.2017.43.  Google Scholar

[12]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, London-New York, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[13]

A. Kocsard and R. Potrie, Livišic theorem for low-dimensional diffeomorphism cocycles, Commentarii Mathematici Helvetici, 91 (2016), 39-64.  doi: 10.4171/CMH/377.  Google Scholar

[14]

A. Livšic, Homology properties of Y-systems, Math. Zametki, 10 (1971), 758-763.   Google Scholar

[15]

A. Livšic, Cohomology of dynamical systems, Math. USSR Izvestija, 6 (1972), 1278-1301.   Google Scholar

[16]

V. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 179-210.   Google Scholar

[17]

V. Sadovskaya, Cohomology of fiber bunched cocycles over hyperbolic systems, Ergodic Theory and Dynamical Systems, 35 (2015), 2669-2688.  doi: 10.1017/etds.2014.43.  Google Scholar

[18]

M. Viana, Lectures on Lyapunov Exponents, Cambridge University Press, 2014. doi: 10.1017/CBO9781139976602.  Google Scholar

[19]

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.  Google Scholar

show all references

References:
[1]

L. Backes, Rigidity of fiber bunched cocycles, Bulletin of the Brazilian Mathematical Society, 46 (2015), 163-179.  doi: 10.1007/s00574-015-0089-7.  Google Scholar

[2]

L. Backes and A. Kocsard, Cohomology of dominated diffeomorphism-valued cocycles over hyperbolic systems, Ergodic Theory and Dynamical Systems, 36 (2016), 1703-1722.  doi: 10.1017/etds.2014.149.  Google Scholar

[3]

L. Barreira and Ya. Pesin, Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents, Cambridge University Press, 2007. doi: 10.1017/CBO9781107326026.  Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics 470. Springer-Verlag, 1975.  Google Scholar

[5]

Y. Cao, Non-zero Lyapunov exponents and uniform hyperbolicity, Nonlinearity, 16 (2003), 1473-1479.  doi: 10.1088/0951-7715/16/4/316.  Google Scholar

[6]

X. Dai, On the approximation of Lyapunov exponents and a question suggested by Anatole Katok, Nonlinearity, 23 (2010), 513-528.  doi: 10.1088/0951-7715/23/3/004.  Google Scholar

[7]

R. de la Llave and A. Windsor, Livšic theorems for non-commutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems, Ergodic Theory and Dynamical Systems, 30 (2010), 1055-1100.  doi: 10.1017/S014338570900039X.  Google Scholar

[8]

D. Dragičević and G. Froyland, Hölder continuity of Oseledets splittings for semi-invertible operator cocycles, Ergodic Theory and Dynamical Systems, 2016, 21pp. doi: 10.1017/etds.2016.55.  Google Scholar

[9]

G. FroylandS. LLoyd and A. Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory and Dynamical Systems, 30 (2010), 729-756.  doi: 10.1017/S0143385709000339.  Google Scholar

[10]

B. Kalinin, Livšic theorem for matrix cocycles, Annals of Mathematics, 173 (2011), 1025-1042.  doi: 10.4007/annals.2011.173.2.11.  Google Scholar

[11]

B. Kalinin and V. Sadovskaya, Periodic approximation of Lyapunov exponents for Banach cocycles, arXiv: 1608.05757. doi: 10.1017/etds.2017.43.  Google Scholar

[12]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, London-New York, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[13]

A. Kocsard and R. Potrie, Livišic theorem for low-dimensional diffeomorphism cocycles, Commentarii Mathematici Helvetici, 91 (2016), 39-64.  doi: 10.4171/CMH/377.  Google Scholar

[14]

A. Livšic, Homology properties of Y-systems, Math. Zametki, 10 (1971), 758-763.   Google Scholar

[15]

A. Livšic, Cohomology of dynamical systems, Math. USSR Izvestija, 6 (1972), 1278-1301.   Google Scholar

[16]

V. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 179-210.   Google Scholar

[17]

V. Sadovskaya, Cohomology of fiber bunched cocycles over hyperbolic systems, Ergodic Theory and Dynamical Systems, 35 (2015), 2669-2688.  doi: 10.1017/etds.2014.43.  Google Scholar

[18]

M. Viana, Lectures on Lyapunov Exponents, Cambridge University Press, 2014. doi: 10.1017/CBO9781139976602.  Google Scholar

[19]

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.  Google Scholar

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