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Topological entropy of free semigroup actions for noncompact sets
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 |
$f \colon X\rightarrow X$ |
$X$ |
$\Phi = \{\phi_n\colon X\rightarrow \mathbb{R}\}_{n\geq 1}$ |
$f$ |
$a$ |
$\inf\limits_{\mu\in \mathcal M_{inv} (f,X)} \chi_\Phi(\mu) <a<\sup\limits_{\mu\in\mathcal M_{inv} (f,X)} \chi_\Phi(\mu)$ |
$\chi_\Phi(\mu) = \liminf_{n\rightarrow \infty}\int\frac1n\phi_n d\mu$ |
$f$ |
$\mu_a\in \mathcal M_{inv} (f,X)$ |
$\chi_\Phi(\mu_a) = a$ |
${\rm{supp}}(\mu)=X.$ |
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. 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. |
[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. 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. |
[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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