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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

 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.

Citation: Jialu Fang, Yongluo Cao, Yun Zhao. Measure theoretic pressure and dimension formula for non-ergodic measures. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar [2] L. Barreira and C. Wolf, Pointwise dimension and ergodic decompositions, Ergodic Theory Dynam. Systems, 26 (2006), 653-671.  doi: 10.1017/S0143385705000672.  Google Scholar [3] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] V. Climenhaga, Bowen's equation in the non-uniform setting, Ergodic Theory Dynam. Systems, 31 (2011), 1163-1182.  doi: 10.1017/S0143385710000362.  Google Scholar [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.  Google Scholar [9] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., (1980), 137–173.  Google Scholar [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.  Google Scholar [11] V. I. Oseledec, A multiplicative ergodic theorem: Lyapunov characteristic exponents for dynamical systems, Trans. Mosc. Math. Soc., 19 (1968), 197-231.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971.  doi: 10.2307/2373682.  Google Scholar [18] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] L. S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems, 2 (1982), 109-124.  doi: 10.1017/S0143385700009615.  Google Scholar [23] L. S. Young, Some large deviations for dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543.  doi: 10.2307/2001318.  Google Scholar [24] Y. Zhao, Measure-theoretic pressure for amenable group actions, Colloquium Mathematicum, 148 (2017), 87-106.  doi: 10.4064/cm6784-6-2016.  Google Scholar [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.  Google Scholar

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.  Google Scholar [2] L. Barreira and C. Wolf, Pointwise dimension and ergodic decompositions, Ergodic Theory Dynam. Systems, 26 (2006), 653-671.  doi: 10.1017/S0143385705000672.  Google Scholar [3] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] V. Climenhaga, Bowen's equation in the non-uniform setting, Ergodic Theory Dynam. Systems, 31 (2011), 1163-1182.  doi: 10.1017/S0143385710000362.  Google Scholar [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.  Google Scholar [9] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., (1980), 137–173.  Google Scholar [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.  Google Scholar [11] V. I. Oseledec, A multiplicative ergodic theorem: Lyapunov characteristic exponents for dynamical systems, Trans. Mosc. Math. Soc., 19 (1968), 197-231.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971.  doi: 10.2307/2373682.  Google Scholar [18] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] L. S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems, 2 (1982), 109-124.  doi: 10.1017/S0143385700009615.  Google Scholar [23] L. S. Young, Some large deviations for dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543.  doi: 10.2307/2001318.  Google Scholar [24] Y. Zhao, Measure-theoretic pressure for amenable group actions, Colloquium Mathematicum, 148 (2017), 87-106.  doi: 10.4064/cm6784-6-2016.  Google Scholar [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.  Google Scholar
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