# American Institute of Mathematical Sciences

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

## Intermediate Lyapunov exponents for systems with periodic orbit gluing property

 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) 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 ${\rm{supp}}(\mu)=X.$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. 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