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May  2020, 40(5): 2767-2789. doi: 10.3934/dcds.2020149

Measure theoretic pressure and dimension formula for non-ergodic measures

Jialu Fang 1, , Yongluo Cao 1,2,3, and  Yun Zhao 1,2,3,,

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

* Corresponding author: Yun Zhao

Received  June 2019 Published  March 2020

Fund Project: The first and third author are partially supported by NSFC (11871361, 11790274). The second author is partially supported by NSFC (11790274, 11771317) and Science and Technology Commission of Shanghai Municipality (No. 18dz2271000)

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 [13], and show that this quantity is equal to the essential supremum of the absolute value of free energy of the measures in an ergodic decomposition. Meanwhile, we define the measure theoretic pressure in another way by using separated sets, it is showed that this quantity is exactly the absolute value of free energy if the measure is ergodic. Particularly, if the dynamical system satisfies the uniform separation condition and the ergodic measures are entropy dense, this quantity is still equal to the the absolute value of free energy even if the measure is non-ergodic. As an application of the main results, we find that the Hausdorff dimension of an invariant measure supported on an average conformal repeller is given by the zero of the measure theoretic pressure of this measure. Furthermore, if a hyperbolic diffeomorphism is average conformal and volume-preserving, the Hausdorff dimension of any invariant measure on the hyperbolic set is equal to the sum of the zeros of measure theoretic pressure restricted to stable and unstable directions.

Keywords: Dimension theory, ergodic measures, thermodynamic formalism.
Mathematics Subject Classification: Primary: 37D35, 37C45; Secondary: 37A30.
Citation: Jialu Fang, Yongluo Cao, Yun Zhao. Measure theoretic pressure and dimension formula for non-ergodic measures. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2767-2789. doi: 10.3934/dcds.2020149
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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