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Measure theoretic pressure and dimension formula for non-ergodic measures
1. | Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, China |
2. | Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200062, China |
3. | Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou 215006, Jiangsu, China |
This paper studies the measure theoretic pressure of measures that are not necessarily ergodic. We define the measure theoretic pressure of an invariant measure (not necessarily ergodic) via the Carathéodory-Pesin structure described in [
References:
[1] |
J. C. Ban, Y. L. Cao and H. Y. Hu,
The dimensions of a non-conformal repeller and an average conformal repeller, Trans. Amer. Math. Soc., 362 (2010), 727-751.
doi: 10.1090/S0002-9947-09-04922-8. |
[2] |
L. Barreira and C. Wolf,
Pointwise dimension and ergodic decompositions, Ergodic Theory Dynam. Systems, 26 (2006), 653-671.
doi: 10.1017/S0143385705000672. |
[3] |
R. Bowen,
Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.
doi: 10.1090/S0002-9947-1973-0338317-X. |
[4] |
M. Brin and A. Katok.,
On local entropy, Geometric Dynamics, Lecture Notes in Mathematics, Spring-Verlag, Berlin, 1007 (1983), 30-38.
doi: 10.1007/BFb0061408. |
[5] |
Y. L. Cao,
Dimension spectrum of asymptotically additive potentials for $C^1$ average conformal repellers, Nonlinearity, 26 (2013), 2441-2468.
doi: 10.1088/0951-7715/26/9/2441. |
[6] |
Y. L. Cao, H. Y. Hu and Y. Zhao,
Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure, Ergodic Theory Dynam. Systems, 33 (2013), 831-850.
doi: 10.1017/S0143385712000090. |
[7] |
V. Climenhaga,
Bowen's equation in the non-uniform setting, Ergodic Theory Dynam. Systems, 31 (2011), 1163-1182.
doi: 10.1017/S0143385710000362. |
[8] |
L. F. He, J. F. Lv and L. N. Zhou,
Definition of measure-theoretic pressure using spanning sets, Acta Math. Sinica English Ser., 20 (2004), 709-718.
doi: 10.1007/s10114-004-0368-5. |
[9] |
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., (1980), 137–173. |
[10] |
J.-H. Ma and Z.-Y. Wen,
A Billingsley type theorem for Bowen entropy, C. R. Math. Acad. Sci. Paris, 346 (2008), 503-507.
doi: 10.1016/j.crma.2008.03.010. |
[11] |
V. I. Oseledec,
A multiplicative ergodic theorem: Lyapunov characteristic exponents for dynamical systems, Trans. Mosc. Math. Soc., 19 (1968), 197-231.
|
[12] |
Y. B. Pesin and B. S. Pitskel', Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. i Prilozhen., 18 (1984), 50–63, 96. |
[13] |
Y. B. Pesin, Dimension Theory in Dynamical Systems, Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.
doi: 10.7208/chicago/9780226662237.001.0001.![]() ![]() ![]() |
[14] |
C.-E. Pfister and W. G. Sullivan,
Large deviations estimates for dynamical systems without the specification property. Application to the $\beta$-shift, Nonlinearity, 18 (2005), 237-261.
doi: 10.1088/0951-7715/18/1/013. |
[15] |
C.-E. Pfister and W. G. Sullivan,
On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems, 27 (2007), 929-956.
doi: 10.1017/S0143385706000824. |
[16] |
X. J. Tang, W.-C. Cheng and Y. Zhao,
Variational principle for topological pressures on subsets, J. Math. Anal. Appl., 424 (2015), 1272-1285.
doi: 10.1016/j.jmaa.2014.11.066. |
[17] |
P. Walters,
A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971.
doi: 10.2307/2373682. |
[18] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. |
[19] |
J. Wang, Y. L. Cao and Y. Zhao,
Dimension estimate in non-conformal setting, Discrete Contin. Dynam. Systems, 34 (2014), 3847-3873.
doi: 10.3934/dcds.2014.34.3847. |
[20] |
J. Wang and Y. L. Cao,
The Hausdorff dimension estimation for an ergodic hyperbolic measure of $C^1$-diffeomorphism, Proceedings of the American Mathematical Society, 144 (2016), 119-128.
doi: 10.1090/proc/12696. |
[21] |
J. Wang, J. Wang, Y. L. Cao and Y. Zhao,
Dimensions of $C^1$-average conformal hyperbolic sets, Discrete Contin. Dynam. Systems, 40 (2020), 883-905.
doi: 10.3934/dcds.2020065. |
[22] |
L. S. Young,
Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems, 2 (1982), 109-124.
doi: 10.1017/S0143385700009615. |
[23] |
L. S. Young,
Some large deviations for dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543.
doi: 10.2307/2001318. |
[24] |
Y. Zhao,
Measure-theoretic pressure for amenable group actions, Colloquium Mathematicum, 148 (2017), 87-106.
doi: 10.4064/cm6784-6-2016. |
[25] |
Y. Zhao, Y. L. Cao and J. C. Ban,
The Hausdorff dimension of average conformal repellers under random perturbation, Nonlinearity, 22 (2009), 2405-2416.
doi: 10.1088/0951-7715/22/10/005. |
show all references
References:
[1] |
J. C. Ban, Y. L. Cao and H. Y. Hu,
The dimensions of a non-conformal repeller and an average conformal repeller, Trans. Amer. Math. Soc., 362 (2010), 727-751.
doi: 10.1090/S0002-9947-09-04922-8. |
[2] |
L. Barreira and C. Wolf,
Pointwise dimension and ergodic decompositions, Ergodic Theory Dynam. Systems, 26 (2006), 653-671.
doi: 10.1017/S0143385705000672. |
[3] |
R. Bowen,
Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.
doi: 10.1090/S0002-9947-1973-0338317-X. |
[4] |
M. Brin and A. Katok.,
On local entropy, Geometric Dynamics, Lecture Notes in Mathematics, Spring-Verlag, Berlin, 1007 (1983), 30-38.
doi: 10.1007/BFb0061408. |
[5] |
Y. L. Cao,
Dimension spectrum of asymptotically additive potentials for $C^1$ average conformal repellers, Nonlinearity, 26 (2013), 2441-2468.
doi: 10.1088/0951-7715/26/9/2441. |
[6] |
Y. L. Cao, H. Y. Hu and Y. Zhao,
Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure, Ergodic Theory Dynam. Systems, 33 (2013), 831-850.
doi: 10.1017/S0143385712000090. |
[7] |
V. Climenhaga,
Bowen's equation in the non-uniform setting, Ergodic Theory Dynam. Systems, 31 (2011), 1163-1182.
doi: 10.1017/S0143385710000362. |
[8] |
L. F. He, J. F. Lv and L. N. Zhou,
Definition of measure-theoretic pressure using spanning sets, Acta Math. Sinica English Ser., 20 (2004), 709-718.
doi: 10.1007/s10114-004-0368-5. |
[9] |
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., (1980), 137–173. |
[10] |
J.-H. Ma and Z.-Y. Wen,
A Billingsley type theorem for Bowen entropy, C. R. Math. Acad. Sci. Paris, 346 (2008), 503-507.
doi: 10.1016/j.crma.2008.03.010. |
[11] |
V. I. Oseledec,
A multiplicative ergodic theorem: Lyapunov characteristic exponents for dynamical systems, Trans. Mosc. Math. Soc., 19 (1968), 197-231.
|
[12] |
Y. B. Pesin and B. S. Pitskel', Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. i Prilozhen., 18 (1984), 50–63, 96. |
[13] |
Y. B. Pesin, Dimension Theory in Dynamical Systems, Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.
doi: 10.7208/chicago/9780226662237.001.0001.![]() ![]() ![]() |
[14] |
C.-E. Pfister and W. G. Sullivan,
Large deviations estimates for dynamical systems without the specification property. Application to the $\beta$-shift, Nonlinearity, 18 (2005), 237-261.
doi: 10.1088/0951-7715/18/1/013. |
[15] |
C.-E. Pfister and W. G. Sullivan,
On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems, 27 (2007), 929-956.
doi: 10.1017/S0143385706000824. |
[16] |
X. J. Tang, W.-C. Cheng and Y. Zhao,
Variational principle for topological pressures on subsets, J. Math. Anal. Appl., 424 (2015), 1272-1285.
doi: 10.1016/j.jmaa.2014.11.066. |
[17] |
P. Walters,
A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971.
doi: 10.2307/2373682. |
[18] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. |
[19] |
J. Wang, Y. L. Cao and Y. Zhao,
Dimension estimate in non-conformal setting, Discrete Contin. Dynam. Systems, 34 (2014), 3847-3873.
doi: 10.3934/dcds.2014.34.3847. |
[20] |
J. Wang and Y. L. Cao,
The Hausdorff dimension estimation for an ergodic hyperbolic measure of $C^1$-diffeomorphism, Proceedings of the American Mathematical Society, 144 (2016), 119-128.
doi: 10.1090/proc/12696. |
[21] |
J. Wang, J. Wang, Y. L. Cao and Y. Zhao,
Dimensions of $C^1$-average conformal hyperbolic sets, Discrete Contin. Dynam. Systems, 40 (2020), 883-905.
doi: 10.3934/dcds.2020065. |
[22] |
L. S. Young,
Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems, 2 (1982), 109-124.
doi: 10.1017/S0143385700009615. |
[23] |
L. S. Young,
Some large deviations for dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543.
doi: 10.2307/2001318. |
[24] |
Y. Zhao,
Measure-theoretic pressure for amenable group actions, Colloquium Mathematicum, 148 (2017), 87-106.
doi: 10.4064/cm6784-6-2016. |
[25] |
Y. Zhao, Y. L. Cao and J. C. Ban,
The Hausdorff dimension of average conformal repellers under random perturbation, Nonlinearity, 22 (2009), 2405-2416.
doi: 10.1088/0951-7715/22/10/005. |
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