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 - A, 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]

Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380
[2]

Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269
[3]

Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011
[4]

Álvaro Castañeda, Pablo González, Gonzalo Robledo. Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020278
[5]

Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350
[6]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464
[7]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108
[8]

Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320
[9]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048
[10]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002
[11]

Haodong Yu, Jie Sun. Robust stochastic optimization with convex risk measures: A discretized subgradient scheme. Journal of Industrial & Management Optimization, 2021, 17 (1) : 81-99. doi: 10.3934/jimo.2019100
[12]

Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127
[13]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448
[14]

Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061
[15]

Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems & Imaging, 2021, 15 (1) : 1-25. doi: 10.3934/ipi.2020048
[16]

Claudio Bonanno, Marco Lenci. Pomeau-Manneville maps are global-local mixing. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1051-1069. doi: 10.3934/dcds.2020309
[17]

Azmy S. Ackleh, Nicolas Saintier. Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1469-1497. doi: 10.3934/dcdsb.2020169
[18]

Hongyan Guo. Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra. Electronic Research Archive, , () : -. doi: 10.3934/era.2021008
[19]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319
[20]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (133)
  • HTML views (159)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences