Packing topological entropy for amenable group actions
Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system $(X,G)$ , where X is a compact...
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| Published in | Ergodic theory and dynamical systems Vol. 43; no. 2; pp. 480 - 514 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Cambridge, UK
Cambridge University Press
01.02.2023
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| Abstract | Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system
$(X,G)$
, where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure
$\mu $
coincides with the metric entropy if either
$\mu $
is ergodic or the system satisfies a kind of specification property. |
|---|---|
| AbstractList | Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system $(X,G)$, where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure $\mu $ coincides with the metric entropy if either $\mu $ is ergodic or the system satisfies a kind of specification property. Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system $(X,G)$ , where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure $\mu $ coincides with the metric entropy if either $\mu $ is ergodic or the system satisfies a kind of specification property. Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G -action dynamical system $(X,G)$ , where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X , the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure $\mu $ coincides with the metric entropy if either $\mu $ is ergodic or the system satisfies a kind of specification property. |
| Author | ZHENG, DONGMEI DOU, DOU ZHOU, XIAOMIN |
| Author_xml | – sequence: 1 givenname: DOU surname: DOU fullname: DOU, DOU email: doumath@163.com organization: †Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, P. R. China – sequence: 2 givenname: DONGMEI surname: ZHENG fullname: ZHENG, DONGMEI email: dongmzheng@163.com organization: ‡School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing, Jiangsu 211816, P. R. China (e-mail: dongmzheng@163.com) – sequence: 3 givenname: XIAOMIN orcidid: 0000-0002-4019-1345 surname: ZHOU fullname: ZHOU, XIAOMIN email: zxm12@mail.ustc.edu.cn organization: §School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P. R. China (e-mail: zxm12@mail.ustc.edu.cn) |
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| Cites_doi | 10.1007/s11425-016-9050-0 10.1112/S002557930001130X 10.1007/s11856-016-1312-y 10.1007/s002220100162 10.3390/e19100526 10.1016/j.jfa.2012.07.010 10.1007/BF02790325 10.1007/978-3-642-14034-1 10.1007/978-3-319-49847-8 10.1214/lnms/1196285808 10.4064/sm-72-2-151-159 10.1017/S0143385706000824 10.1090/S0002-9947-1973-0338317-X 10.1080/14689367.2012.701605 10.1007/s00222-014-0524-1 10.2307/1971397 10.1214/11-AOP688 10.1017/CBO9780511623813 10.1007/s10884-017-9610-6 10.1090/S0002-9947-1971-0274707-X 10.1017/S0305004100059119 10.7208/chicago/9780226662237.001.0001 10.1006/aima.1995.1050 10.1007/BFb0061408 |
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| Keywords | packing topological entropy variational principle 37B40 generic point 28D20 37A15 amenable group |
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| Title | Packing topological entropy for amenable group actions |
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