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.
Citation: |
R. L. Adler
, A. G. Konheim
and M. H. McAndrew
, Topological entropy, Trans. Amer. Math. Soc., 114 (1965)
, 309-319.
doi: 10.2307/1994177.![]() ![]() ![]() |
|
P. Billingsley,
Ergodic Theory and Information, John Wiley and Sons Inc., New York, 1965.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
R. Bowen,
Equilibrium States and the Ergodic Theorey of Anosov Diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, 1975.
![]() ![]() |
|
R. Bowen
, Hausdorff dimension of quasicircles, Publ. Math. Inst. Hautes Etudes Sci., 50 (1979)
, 11-25.
![]() ![]() |
|
M. Brin and A. Katok, On local entropy, Lecture Notes in Math., Springer, Berlin, 1007
(1983), 30–38.
doi: 10.1007/BFb0061408.![]() ![]() ![]() |
|
C. Carathéodory
, Über das lineare mass, Göttingen Nachr., (1914)
, 406-426.
![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
L. W. Goodwyn
, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc., 23 (1969)
, 679-688.
doi: 10.2307/2036610.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
D. Kerr and H. Li,
Ergodic Theory: Independence and Dichotomies, Springer, 2016.
doi: 10.1007/978-3-319-49847-8.![]() ![]() ![]() |
|
A. N. Kolmogorov
, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR, 119 (1958)
, 861-864.
![]() ![]() |
|
H. Li
, Sofic mean dimension, Adv. Math., 244 (2013)
, 570-604.
doi: 10.1016/j.aim.2013.05.005.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
I. Namioka
, Følner's conditions for amenable semi-groups, Math. Scand., 15 (1964)
, 18-28.
doi: 10.7146/math.scand.a-10723.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
Y. Pesin
and B. S. Pitskel
, Topological pressure and the variational principle for noncompact sets, Functional Anal. Appl., 18 (1984)
, 50-63, 96.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
D. Ruelle,
Thermodynamic Formalism, Vol.5 of Encyclopedia of Mathematics and its Applications. Reading, MA: Addision-Wesley, 1978.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
P. Walters
, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975)
, 937-971.
doi: 10.2307/2373682.![]() ![]() ![]() |