American Institute of Mathematical Sciences

Advanced
February  2019, 39(2): 959-993. doi: 10.3934/dcds.2019040

A Billingsley-type theorem for the pressure of an action of an amenable group

Xiaojun Huang , , Yuan Lian and  Changrong Zhu

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

* Corresponding author

Received  March 2018 Revised  August 2018 Published  November 2018

Fund Project: The research was supported by NSF of China (No. 11671057, No. 11471318, No. 11671058) and the Fundamental Research Funds for the Central Universities (No. 2018CDQYST0023).

This paper extends the definition of Bowen topological entropy of subsets to Pesin-Pitskel topological pressure for the continuous action of amenable groups on a compact metric space. We introduce the local measure theoretic pressure of subsets and investigate the relation between local measure theoretic pressure of Borel probability measures and Pesin-Pitskel topological pressure on an arbitrary subset of a compact metric space.

Keywords: Bowen topological pressure, local pressure of measures, amenable group.
Mathematics Subject Classification: 37A35, 37B40.
Citation: Xiaojun Huang, Yuan Lian, Changrong Zhu. A Billingsley-type theorem for the pressure of an action of an amenable group. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 959-993. doi: 10.3934/dcds.2019040
References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.2307/1994177.  Google Scholar

[2]

P. Billingsley, Ergodic Theory and Information, John Wiley and Sons Inc., New York, 1965.  Google Scholar

[3]

A. Bis, An analogue of the variational principle for group and pseudogroup actions, Ann. Inst. Fourier (Grenoble), 63 (2013), 839-863.  doi: 10.5802/aif.2778.  Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theorey of Anosov Diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, 1975.  Google Scholar

[5]

R. Bowen, Hausdorff dimension of quasicircles, Publ. Math. Inst. Hautes Etudes Sci., 50 (1979), 11-25.   Google Scholar

[6]

M. Brin and A. Katok, On local entropy, Lecture Notes in Math., Springer, Berlin, 1007 (1983), 30–38. doi: 10.1007/BFb0061408.  Google Scholar

[7]

C. Carathéodory, Über das lineare mass, Göttingen Nachr., (1914), 406-426.   Google Scholar

[8]

D. Feng and W. Huang, Variational principles for topological entropies of subsets, J. Funct. Anal., 263 (2012), 2228-2254.  doi: 10.1016/j.jfa.2012.07.010.  Google Scholar

[9]

T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc., 3 (1971), 176-180.  doi: 10.1112/blms/3.2.176.  Google Scholar

[10]

L. W. Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc., 23 (1969), 679-688.  doi: 10.2307/2036610.  Google Scholar

[11]

X. Huang, J. Liu and C. Zhu, The Bowen topological entropy of subsets for amenable group actions, Discrete Contin. Dyn. Syst., 38 (2018), 4467-4482.  doi: 10.3934/dcds.2018195.  Google Scholar

[12]

D. Kerr and H. Li, Ergodic Theory: Independence and Dichotomies, Springer, 2016. doi: 10.1007/978-3-319-49847-8.  Google Scholar

[13]

A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR, 119 (1958), 861-864.   Google Scholar

[14]

H. Li, Sofic mean dimension, Adv. Math., 244 (2013), 570-604.  doi: 10.1016/j.aim.2013.05.005.  Google Scholar

[15]

B. Liang and K. Yan, Topological pressure for sub-additive potentials of amenable group actions, J. Funct. Anal., 262 (2012), 584-601.  doi: 10.1016/j.jfa.2011.09.020.  Google Scholar

[16]

J. Ma and Z. 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

[17]

I. Namioka, Følner's conditions for amenable semi-groups, Math. Scand., 15 (1964), 18-28.  doi: 10.7146/math.scand.a-10723.  Google Scholar

[18]

D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.  Google Scholar

[19]

Y. B. Pesin, Dimension Theory in Dynamical Systems Contemporary Views and Applications, Chicago lecture in Mathematics. University of Chicago Press, Chicago, IL, 1997. doi: 10.7208/chicago/9780226662237.001.0001.  Google Scholar

[20]

Y. Pesin and B. S. Pitskel, Topological pressure and the variational principle for noncompact sets, Functional Anal. Appl., 18 (1984), 50-63, 96.   Google Scholar

[21]

D. Ruelle, Statistical mechanics on compact set with Zv action satisfying expansiveness and specification, Trans. Amer. Math. Soc., 187 (1973), 237-251.  doi: 10.2307/1996437.  Google Scholar

[22]

D. Ruelle, Thermodynamic Formalism, Vol.5 of Encyclopedia of Mathematics and its Applications. Reading, MA: Addision-Wesley, 1978.  Google Scholar

[23]

X. Tang, W. 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

[24]

P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971.  doi: 10.2307/2373682.  Google Scholar

show all references

References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.2307/1994177.  Google Scholar

[2]

P. Billingsley, Ergodic Theory and Information, John Wiley and Sons Inc., New York, 1965.  Google Scholar

[3]

A. Bis, An analogue of the variational principle for group and pseudogroup actions, Ann. Inst. Fourier (Grenoble), 63 (2013), 839-863.  doi: 10.5802/aif.2778.  Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theorey of Anosov Diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, 1975.  Google Scholar

[5]

R. Bowen, Hausdorff dimension of quasicircles, Publ. Math. Inst. Hautes Etudes Sci., 50 (1979), 11-25.   Google Scholar

[6]

M. Brin and A. Katok, On local entropy, Lecture Notes in Math., Springer, Berlin, 1007 (1983), 30–38. doi: 10.1007/BFb0061408.  Google Scholar

[7]

C. Carathéodory, Über das lineare mass, Göttingen Nachr., (1914), 406-426.   Google Scholar

[8]

D. Feng and W. Huang, Variational principles for topological entropies of subsets, J. Funct. Anal., 263 (2012), 2228-2254.  doi: 10.1016/j.jfa.2012.07.010.  Google Scholar

[9]

T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc., 3 (1971), 176-180.  doi: 10.1112/blms/3.2.176.  Google Scholar

[10]

L. W. Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc., 23 (1969), 679-688.  doi: 10.2307/2036610.  Google Scholar

[11]

X. Huang, J. Liu and C. Zhu, The Bowen topological entropy of subsets for amenable group actions, Discrete Contin. Dyn. Syst., 38 (2018), 4467-4482.  doi: 10.3934/dcds.2018195.  Google Scholar

[12]

D. Kerr and H. Li, Ergodic Theory: Independence and Dichotomies, Springer, 2016. doi: 10.1007/978-3-319-49847-8.  Google Scholar

[13]

A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR, 119 (1958), 861-864.   Google Scholar

[14]

H. Li, Sofic mean dimension, Adv. Math., 244 (2013), 570-604.  doi: 10.1016/j.aim.2013.05.005.  Google Scholar

[15]

B. Liang and K. Yan, Topological pressure for sub-additive potentials of amenable group actions, J. Funct. Anal., 262 (2012), 584-601.  doi: 10.1016/j.jfa.2011.09.020.  Google Scholar

[16]

J. Ma and Z. 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

[17]

I. Namioka, Følner's conditions for amenable semi-groups, Math. Scand., 15 (1964), 18-28.  doi: 10.7146/math.scand.a-10723.  Google Scholar

[18]

D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.  Google Scholar

[19]

Y. B. Pesin, Dimension Theory in Dynamical Systems Contemporary Views and Applications, Chicago lecture in Mathematics. University of Chicago Press, Chicago, IL, 1997. doi: 10.7208/chicago/9780226662237.001.0001.  Google Scholar

[20]

Y. Pesin and B. S. Pitskel, Topological pressure and the variational principle for noncompact sets, Functional Anal. Appl., 18 (1984), 50-63, 96.   Google Scholar

[21]

D. Ruelle, Statistical mechanics on compact set with Zv action satisfying expansiveness and specification, Trans. Amer. Math. Soc., 187 (1973), 237-251.  doi: 10.2307/1996437.  Google Scholar

[22]

D. Ruelle, Thermodynamic Formalism, Vol.5 of Encyclopedia of Mathematics and its Applications. Reading, MA: Addision-Wesley, 1978.  Google Scholar

[23]

X. Tang, W. 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

[24]

P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971.  doi: 10.2307/2373682.  Google Scholar

[1]

Xiaojun Huang, Zhiqiang Li, Yunhua Zhou. A variational principle of topological pressure on subsets for amenable group actions. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2687-2703. doi: 10.3934/dcds.2020146
[2]

Dandan Cheng, Qian Hao, Zhiming Li. Scale pressure for amenable group actions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1091-1102. doi: 10.3934/cpaa.2021008
[3]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545
[4]

Jean-Paul Thouvenot. The work of Lewis Bowen on the entropy theory of non-amenable group actions. Journal of Modern Dynamics, 2019, 15: 133-141. doi: 10.3934/jmd.2019016
[5]

Marc Rauch. Variational principles for the topological pressure of measurable potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 367-394. doi: 10.3934/dcdss.2017018
[6]

Xueting Tian. Topological pressure for the completely irregular set of birkhoff averages. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2745-2763. doi: 10.3934/dcds.2017118
[7]

Jialu Fang, Yongluo Cao, Yun Zhao. Measure theoretic pressure and dimension formula for non-ergodic measures. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2767-2789. doi: 10.3934/dcds.2020149
[8]

Xiankun Ren. Periodic measures are dense in invariant measures for residually finite amenable group actions with specification. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1657-1667. doi: 10.3934/dcds.2018068
[9]

Xinsheng Wang, Weisheng Wu, Yujun Zhu. Local unstable entropy and local unstable pressure for random partially hyperbolic dynamical systems. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 81-105. doi: 10.3934/dcds.2020004
[10]

Guohua Zhang. Variational principles of pressure. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1409-1435. doi: 10.3934/dcds.2009.24.1409
[11]

Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417
[12]

Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827
[13]

Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113
[14]

Ruxandra Stavre. Optimization of the blood pressure with the control in coefficients. Evolution Equations & Control Theory, 2020, 9 (1) : 131-151. doi: 10.3934/eect.2020019
[15]

Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207
[16]

Maria Pires De Carvalho. Persistence of Bowen-Ruelle-Sinai measures. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 213-221. doi: 10.3934/dcds.2007.17.213
[17]

Michel Coornaert, Fabrice Krieger. Mean topological dimension for actions of discrete amenable groups. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 779-793. doi: 10.3934/dcds.2005.13.779
[18]

Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195
[19]

M. Bulíček, Josef Málek, Dalibor Pražák. On the dimension of the attractor for a class of fluids with pressure dependent viscosities. Communications on Pure & Applied Analysis, 2005, 4 (4) : 805-822. doi: 10.3934/cpaa.2005.4.805
[20]

Vishal Vasan, Katie Oliveras. Pressure beneath a traveling wave with constant vorticity. Discrete & Continuous Dynamical Systems, 2014, 34 (8) : 3219-3239. doi: 10.3934/dcds.2014.34.3219

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (149)
  • HTML views (163)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences