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Parameter regularity of dynamical determinants of expanding maps of the circle and an application to linear response
A Billingsley-type theorem for the pressure of an action of an amenable group
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
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.
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. |
[2] |
P. Billingsley,
Ergodic Theory and Information, John Wiley and Sons Inc., New York, 1965. |
[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. |
[4] |
R. Bowen,
Equilibrium States and the Ergodic Theorey of Anosov Diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, 1975. |
[5] |
R. Bowen,
Hausdorff dimension of quasicircles, Publ. Math. Inst. Hautes Etudes Sci., 50 (1979), 11-25.
|
[6] |
M. Brin and A. Katok, On local entropy, Lecture Notes in Math., Springer, Berlin, 1007
(1983), 30–38.
doi: 10.1007/BFb0061408. |
[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. |
[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. |
[10] |
L. W. Goodwyn,
Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc., 23 (1969), 679-688.
doi: 10.2307/2036610. |
[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. |
[12] |
D. Kerr and H. Li,
Ergodic Theory: Independence and Dichotomies, Springer, 2016.
doi: 10.1007/978-3-319-49847-8. |
[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.
|
[14] |
H. Li,
Sofic mean dimension, Adv. Math., 244 (2013), 570-604.
doi: 10.1016/j.aim.2013.05.005. |
[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. |
[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. |
[17] |
I. Namioka,
Følner's conditions for amenable semi-groups, Math. Scand., 15 (1964), 18-28.
doi: 10.7146/math.scand.a-10723. |
[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. |
[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. |
[20] |
Y. Pesin and B. S. Pitskel,
Topological pressure and the variational principle for noncompact sets, Functional Anal. Appl., 18 (1984), 50-63, 96.
|
[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. |
[22] |
D. Ruelle,
Thermodynamic Formalism, Vol.5 of Encyclopedia of Mathematics and its Applications. Reading, MA: Addision-Wesley, 1978. |
[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. |
[24] |
P. Walters,
A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971.
doi: 10.2307/2373682. |
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. |
[2] |
P. Billingsley,
Ergodic Theory and Information, John Wiley and Sons Inc., New York, 1965. |
[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. |
[4] |
R. Bowen,
Equilibrium States and the Ergodic Theorey of Anosov Diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, 1975. |
[5] |
R. Bowen,
Hausdorff dimension of quasicircles, Publ. Math. Inst. Hautes Etudes Sci., 50 (1979), 11-25.
|
[6] |
M. Brin and A. Katok, On local entropy, Lecture Notes in Math., Springer, Berlin, 1007
(1983), 30–38.
doi: 10.1007/BFb0061408. |
[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. |
[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. |
[10] |
L. W. Goodwyn,
Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc., 23 (1969), 679-688.
doi: 10.2307/2036610. |
[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. |
[12] |
D. Kerr and H. Li,
Ergodic Theory: Independence and Dichotomies, Springer, 2016.
doi: 10.1007/978-3-319-49847-8. |
[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.
|
[14] |
H. Li,
Sofic mean dimension, Adv. Math., 244 (2013), 570-604.
doi: 10.1016/j.aim.2013.05.005. |
[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. |
[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. |
[17] |
I. Namioka,
Følner's conditions for amenable semi-groups, Math. Scand., 15 (1964), 18-28.
doi: 10.7146/math.scand.a-10723. |
[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. |
[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. |
[20] |
Y. Pesin and B. S. Pitskel,
Topological pressure and the variational principle for noncompact sets, Functional Anal. Appl., 18 (1984), 50-63, 96.
|
[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. |
[22] |
D. Ruelle,
Thermodynamic Formalism, Vol.5 of Encyclopedia of Mathematics and its Applications. Reading, MA: Addision-Wesley, 1978. |
[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. |
[24] |
P. Walters,
A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971.
doi: 10.2307/2373682. |
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