American Institute of Mathematical Sciences

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February  2019, 39(2): 1019-1032. doi: 10.3934/dcds.2019042

Intermediate Lyapunov exponents for systems with periodic orbit gluing property

Xueting Tian 1, , Shirou Wang 2,3, and  Xiaodong Wang 4,,

1. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2. 

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G2G1, Canada
3. 

Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China
4. 

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Xiaodong Wang

Received  March 2018 Revised  August 2018 Published  November 2018

We prove that the average Lyapunov exponents of asymptotically additive functions have the intermediate value property provided the dynamical system has the periodic gluing orbit property. To be precise, consider a continuous map
$f \colon X\rightarrow X$
 over a compact metric space
$X$
 and an asymptotically additive sequence of functions
$\Phi = \{\phi_n\colon X\rightarrow \mathbb{R}\}_{n\geq 1}$
. If
$f$
 has the periodic gluing orbit property, then for any constant
$a$
 satisfying
$\inf\limits_{\mu\in \mathcal M_{inv} (f,X)} \chi_\Phi(\mu) <a<\sup\limits_{\mu\in\mathcal M_{inv} (f,X)} \chi_\Phi(\mu)$
where
$\chi_\Phi(\mu) = \liminf_{n\rightarrow \infty}\int\frac1n\phi_n d\mu$
, and the infimum and supremum are taken over the set of all
$f$
-invariant probability measures, there is an ergodic measure
$\mu_a\in \mathcal M_{inv} (f,X)$
 such that
$\chi_\Phi(\mu_a) = a$
 and
${\rm{supp}}(\mu)=X.$
Keywords: Shadowing, specification property, gluing orbit property, asymptotically additive functions, Lyapunov exponents.
Mathematics Subject Classification: 37A25, 37D25, 54H20.
Citation: Xueting Tian, Shirou Wang, Xiaodong Wang. Intermediate Lyapunov exponents for systems with periodic orbit gluing property. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1019-1032. doi: 10.3934/dcds.2019042
References:
[1]

L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics, Basel: Birkhäuser, 2008.

[2]

L. Barreira, Thermodynamic Formalism and Applications to Dimension Theory, Springer Science & Business Media, 2011. doi: 10.1007/978-3-0348-0206-2.

[3]

L. Barreira and P. Doutor, Almost additive multifractal analysis, Journal de Mathématiques Pures et Appliquées, 92 (2009), 1-17.  doi: 10.1016/j.matpur.2009.04.006.

[4]

L. Barreira and K. Gelfert, Multifractal analysis for Lyapunov exponents on nonconformal repellers, Comm. Math. Phys., 267 (2006), 393-418.  doi: 10.1007/s00220-006-0084-3.

[5]

L. Barreira and B. Saussol, Variational principles and mixed multifractal spectra, Trans. Amer. Math. Soc., 353 (2001), 3919-3944.  doi: 10.1090/S0002-9947-01-02844-6.

[6]

A. M. Blokh, Decomposition of dynamical systems on an interval, Uspekhi Mat. Nauk., 38 (1983), 179-180. 

[7]

T. Bomfim, M. J. Torres and P. Varandas, Topological features of flows with the reparameterized gluing orbit property, Journal of Differential Equations, 262 (2017), 4292-4313.  doi: 10.1016/j.jde.2017.01.008.

[8]

T. Bomfim and P. Varandas, The gluing orbit property, uniform hyperbolicity and large deviations principles for semiflows, arXiv: 1507.03905.

[9]

C. Bonatti, L. Díaz and A. Gorodetski, Non-hyperbolic ergodic measures with large support, Nonlinearity, 23 (2010), 687-705.  doi: 10.1088/0951-7715/23/3/015.

[10]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.2307/1996403.

[11]

J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754.  doi: 10.1090/S0002-9947-97-01873-4.

[12]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space, Lecture Notes in Mathematics, Vol. 527. Springer-Verlag, Berlin-New York, 1976.

[13]

L. Díaz and A. Gorodetski, Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes, Ergodic Theory and Dynam. Systems, 29 (2009), 1479-1513.  doi: 10.1017/S0143385708000849.

[14]

D. Feng, Lyapunov exponents for products of matrices and multifractal analysis. Ⅰ. Positive matrices, Israel J. Math., 138 (2003), 353-376.  doi: 10.1007/BF02783432.

[15]

D. Feng, Lyapunov exponents for products of matrices and multifractal analysis. Ⅱ. General matrices, Israel J. Math., 170 (2009), 355-394.  doi: 10.1007/s11856-009-0033-x.

[16]

D. Feng and W. Huang, Lyapunov spectrum of asymptotically sub-additive potentials, Comm. Math. Phys., 297 (2010), 1-43.  doi: 10.1007/s00220-010-1031-x.

[17]

D. Feng and K. S. Lau, The pressure function for products of non-negative matrices, Math. Res. Lett., 9 (2002), 363-378.  doi: 10.4310/MRL.2002.v9.n3.a10.

[18]

A. Gogolev, Diffeomorphisms H¨older conjugate to Anosov diffeomorphisms, Ergodic Theory and Dynam. Systems, 30 (2010), 441-456.  doi: 10.1017/S0143385709000169.

[19]

A. Gorodetski, Yu. Ilyashenko, V. Kleptsyn and M. Nalsky, Nonremovable zero Lyapunov exponents, Funct. Anal. Appl., 39 (2005), 27-38.  doi: 10.1007/s10688-005-0014-8.

[20]

L. Guan, P. Sun and W. Wu, Measures of intermediate entropies and homogeneous dynamics, Nonlinearity, 30 (2017), 3349-3361.  doi: 10.1088/1361-6544/aa8040.

[21]

P. A. Guihéneuf and T. Lefeuvre, On the genericity of the shadowing property for conservative homeomorphisms, Proc. Amer. Math. Soc., 146 (2018), 4225-4237. 

[22]

M. V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one dimensional maps, Comm. Math. Phys., 81 (1981), 39-88. 

[23]

C. E. Pfister and W. G. Sullivan, Large deviations estimates for dynamical systems without the specification property. Application to the β-shifts, Nonlinearity, 18 (2005), 237-261.  doi: 10.1088/0951-7715/18/1/013.

[24]

C. E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets, Ergodic Theory and Dynam. Systems, 27 (2007), 929-956.  doi: 10.1017/S0143385706000824.

[25]

K. Sigmund, Generic properties of invariant measures for axiom A diffeomorphisms, Invention Math., 11 (1970), 99-109.  doi: 10.1007/BF01404606.

[26]

K. Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc., 190 (1974), 285-299.  doi: 10.2307/1996963.

[27]

P. Sun, Measures of intermediate entropies for skew product diffeomorphisms, Discrete Contin. Dyn. Syst., 27 (2010), 1219-1231.  doi: 10.3934/dcds.2010.27.1219.

[28]

P. Sun, Density of metric entropies for linear toral automorphisms, Dyn. Syst., 27 (2012), 197-204.  doi: 10.1080/14689367.2011.649246.

[29]

F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets, Ergodic theory and Dynam. Systems, 23 (2003), 317-348.  doi: 10.1017/S0143385702000913.

[30]

D. J. Thompson, A variational principle for topological pressure for certain non-compact sets, J. Lond. Math. Soc., 80 (2009), 585-602.  doi: 10.1112/jlms/jdp041.

[31]

D. J. Thompson, Irregular sets, the β-transformation and the almost specification property, Trans. Amer. Math. Soc., 364 (2012), 5395-5414.  doi: 10.1090/S0002-9947-2012-05540-1.

[32]

X. Tian, Different asymptotic behavior versus same dynamical complexity: Recurrence & (ir)regularity, Advances in Mathematics, 288 (2016), 464-526.  doi: 10.1016/j.aim.2015.11.006.

[33]

X. Tian and W. Sun, Diffeomorphisms with various C1 stable properties, Acta Mathematica Scientia, 32 (2012), 552-558.  doi: 10.1016/S0252-9602(12)60037-X.

[34]

P. Walters, Equilibrium states for β-transformations and related transformations, Mathematische Zeitschrift, 159 (1978), 65-88.  doi: 10.1007/BF01174569.

[35]

P. Walters, An Introduction to Ergodic Theory, Berlin-Heidelberg-New York: Springer-Verlag, 1982.

show all references

References:
[1]

L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics, Basel: Birkhäuser, 2008.

[2]

L. Barreira, Thermodynamic Formalism and Applications to Dimension Theory, Springer Science & Business Media, 2011. doi: 10.1007/978-3-0348-0206-2.

[3]

L. Barreira and P. Doutor, Almost additive multifractal analysis, Journal de Mathématiques Pures et Appliquées, 92 (2009), 1-17.  doi: 10.1016/j.matpur.2009.04.006.

[4]

L. Barreira and K. Gelfert, Multifractal analysis for Lyapunov exponents on nonconformal repellers, Comm. Math. Phys., 267 (2006), 393-418.  doi: 10.1007/s00220-006-0084-3.

[5]

L. Barreira and B. Saussol, Variational principles and mixed multifractal spectra, Trans. Amer. Math. Soc., 353 (2001), 3919-3944.  doi: 10.1090/S0002-9947-01-02844-6.

[6]

A. M. Blokh, Decomposition of dynamical systems on an interval, Uspekhi Mat. Nauk., 38 (1983), 179-180. 

[7]

T. Bomfim, M. J. Torres and P. Varandas, Topological features of flows with the reparameterized gluing orbit property, Journal of Differential Equations, 262 (2017), 4292-4313.  doi: 10.1016/j.jde.2017.01.008.

[8]

T. Bomfim and P. Varandas, The gluing orbit property, uniform hyperbolicity and large deviations principles for semiflows, arXiv: 1507.03905.

[9]

C. Bonatti, L. Díaz and A. Gorodetski, Non-hyperbolic ergodic measures with large support, Nonlinearity, 23 (2010), 687-705.  doi: 10.1088/0951-7715/23/3/015.

[10]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.2307/1996403.

[11]

J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754.  doi: 10.1090/S0002-9947-97-01873-4.

[12]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space, Lecture Notes in Mathematics, Vol. 527. Springer-Verlag, Berlin-New York, 1976.

[13]

L. Díaz and A. Gorodetski, Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes, Ergodic Theory and Dynam. Systems, 29 (2009), 1479-1513.  doi: 10.1017/S0143385708000849.

[14]

D. Feng, Lyapunov exponents for products of matrices and multifractal analysis. Ⅰ. Positive matrices, Israel J. Math., 138 (2003), 353-376.  doi: 10.1007/BF02783432.

[15]

D. Feng, Lyapunov exponents for products of matrices and multifractal analysis. Ⅱ. General matrices, Israel J. Math., 170 (2009), 355-394.  doi: 10.1007/s11856-009-0033-x.

[16]

D. Feng and W. Huang, Lyapunov spectrum of asymptotically sub-additive potentials, Comm. Math. Phys., 297 (2010), 1-43.  doi: 10.1007/s00220-010-1031-x.

[17]

D. Feng and K. S. Lau, The pressure function for products of non-negative matrices, Math. Res. Lett., 9 (2002), 363-378.  doi: 10.4310/MRL.2002.v9.n3.a10.

[18]

A. Gogolev, Diffeomorphisms H¨older conjugate to Anosov diffeomorphisms, Ergodic Theory and Dynam. Systems, 30 (2010), 441-456.  doi: 10.1017/S0143385709000169.

[19]

A. Gorodetski, Yu. Ilyashenko, V. Kleptsyn and M. Nalsky, Nonremovable zero Lyapunov exponents, Funct. Anal. Appl., 39 (2005), 27-38.  doi: 10.1007/s10688-005-0014-8.

[20]

L. Guan, P. Sun and W. Wu, Measures of intermediate entropies and homogeneous dynamics, Nonlinearity, 30 (2017), 3349-3361.  doi: 10.1088/1361-6544/aa8040.

[21]

P. A. Guihéneuf and T. Lefeuvre, On the genericity of the shadowing property for conservative homeomorphisms, Proc. Amer. Math. Soc., 146 (2018), 4225-4237. 

[22]

M. V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one dimensional maps, Comm. Math. Phys., 81 (1981), 39-88. 

[23]

C. E. Pfister and W. G. Sullivan, Large deviations estimates for dynamical systems without the specification property. Application to the β-shifts, Nonlinearity, 18 (2005), 237-261.  doi: 10.1088/0951-7715/18/1/013.

[24]

C. E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets, Ergodic Theory and Dynam. Systems, 27 (2007), 929-956.  doi: 10.1017/S0143385706000824.

[25]

K. Sigmund, Generic properties of invariant measures for axiom A diffeomorphisms, Invention Math., 11 (1970), 99-109.  doi: 10.1007/BF01404606.

[26]

K. Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc., 190 (1974), 285-299.  doi: 10.2307/1996963.

[27]

P. Sun, Measures of intermediate entropies for skew product diffeomorphisms, Discrete Contin. Dyn. Syst., 27 (2010), 1219-1231.  doi: 10.3934/dcds.2010.27.1219.

[28]

P. Sun, Density of metric entropies for linear toral automorphisms, Dyn. Syst., 27 (2012), 197-204.  doi: 10.1080/14689367.2011.649246.

[29]

F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets, Ergodic theory and Dynam. Systems, 23 (2003), 317-348.  doi: 10.1017/S0143385702000913.

[30]

D. J. Thompson, A variational principle for topological pressure for certain non-compact sets, J. Lond. Math. Soc., 80 (2009), 585-602.  doi: 10.1112/jlms/jdp041.

[31]

D. J. Thompson, Irregular sets, the β-transformation and the almost specification property, Trans. Amer. Math. Soc., 364 (2012), 5395-5414.  doi: 10.1090/S0002-9947-2012-05540-1.

[32]

X. Tian, Different asymptotic behavior versus same dynamical complexity: Recurrence & (ir)regularity, Advances in Mathematics, 288 (2016), 464-526.  doi: 10.1016/j.aim.2015.11.006.

[33]

X. Tian and W. Sun, Diffeomorphisms with various C1 stable properties, Acta Mathematica Scientia, 32 (2012), 552-558.  doi: 10.1016/S0252-9602(12)60037-X.

[34]

P. Walters, Equilibrium states for β-transformations and related transformations, Mathematische Zeitschrift, 159 (1978), 65-88.  doi: 10.1007/BF01174569.

[35]

P. Walters, An Introduction to Ergodic Theory, Berlin-Heidelberg-New York: Springer-Verlag, 1982.

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