American Institute of Mathematical Sciences

Advanced
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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[35]

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

show all references

References:
[1]

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

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[35]

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

[1]

Peng Sun. Minimality and gluing orbit property. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 4041-4056. doi: 10.3934/dcds.2019162
[2]

Kazumine Moriyasu, Kazuhiro Sakai, Kenichiro Yamamoto. Regular maps with the specification property. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2991-3009. doi: 10.3934/dcds.2013.33.2991
[3]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91
[4]

Manseob Lee, Jumi Oh, Xiao Wen. Diffeomorphisms with a generalized Lipschitz shadowing property. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1913-1927. doi: 10.3934/dcds.2020346
[5]

Fang Zhang, Yunhua Zhou. On the limit quasi-shadowing property. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2861-2879. doi: 10.3934/dcds.2017123
[6]

Jinjun Li, Min Wu. Generic property of irregular sets in systems satisfying the specification property. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 635-645. doi: 10.3934/dcds.2014.34.635
[7]

Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469
[8]

Jinjun Li, Min Wu. Divergence points in systems satisfying the specification property. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 905-920. doi: 10.3934/dcds.2013.33.905
[9]

Manfred G. Madritsch, Izabela Petrykiewicz. Non-normal numbers in dynamical systems fulfilling the specification property. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4751-4764. doi: 10.3934/dcds.2014.34.4751
[10]

Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231
[11]

Jihoon Lee, Ngocthach Nguyen. Flows with the weak two-sided limit shadowing property. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4375-4395. doi: 10.3934/dcds.2021040
[12]

Zheng Yin, Ercai Chen. The conditional variational principle for maps with the pseudo-orbit tracing property. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 463-481. doi: 10.3934/dcds.2019019
[13]

Shingo Takeuchi. The basis property of generalized Jacobian elliptic functions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2675-2692. doi: 10.3934/cpaa.2014.13.2675
[14]

Krzysztof Frączek, M. Lemańczyk, E. Lesigne. Mild mixing property for special flows under piecewise constant functions. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 691-710. doi: 10.3934/dcds.2007.19.691
[15]

Thomas Bartsch, Anna Maria Micheletti, Angela Pistoia. The Morse property for functions of Kirchhoff-Routh path type. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1867-1877. doi: 10.3934/dcdss.2019123
[16]

Magdalena Foryś-Krawiec, Jiří Kupka, Piotr Oprocha, Xueting Tian. On entropy of $ \Phi $-irregular and $ \Phi $-level sets in maps with the shadowing property. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1271-1296. doi: 10.3934/dcds.2020317
[17]

Kazuhiro Sakai. The oe-property of diffeomorphisms. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 581-591. doi: 10.3934/dcds.1998.4.581
[18]

Pablo Sánchez, Jaume Sempere. Conflict, private and communal property. Journal of Dynamics & Games, 2016, 3 (4) : 355-369. doi: 10.3934/jdg.2016019
[19]

Konstantinos Drakakis, Scott Rickard. On the generalization of the Costas property in the continuum. Advances in Mathematics of Communications, 2008, 2 (2) : 113-130. doi: 10.3934/amc.2008.2.113
[20]

Bo Su. Doubling property of elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (1) : 143-147. doi: 10.3934/cpaa.2008.7.143

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (208)
  • HTML views (157)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences