November  2019, 39(11): 6631-6642. doi: 10.3934/dcds.2019288

Relative entropy dimension of topological dynamical systems

Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

Received  February 2019 Revised  April 2019 Published  August 2019

Fund Project: (*) This research is supported by NSFC(11801193).

We introduce the notion of relative topological entropy dimension to classify the different intermediate levels of relative complexity for factor maps. By considering the dimension or ''density" of special class of sequences along which the entropy is encountered, we provide equivalent definitions of relative entropy dimension. As applications, we investigate the corresponding localization theory and obtain a disjointness theorem involving relative entropy dimension.

Citation: Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288
References:
[1]

F. Blanchard, A disjointness theorem involving topological entropy, Bull. Soc. Math. France, 121 (1993), 465-478.  doi: 10.24033/bsmf.2216.  Google Scholar

[2]

M. de Carvalho, Entropy dimension of dynamical systems, Portugal. Math., 54 (1997), 19-40.   Google Scholar

[3]

D. DouW. Huang and K. Park, Entropy dimension of topological dynamical systems, Trans. Amer. Math. Soc., 363 (2011), 659-680.  doi: 10.1090/S0002-9947-2010-04906-2.  Google Scholar

[4]

D. DouW. Huang and K. Park, Entropy dimension of measure preserving systems, Trans. Amer. Math. Soc., 371 (2019), 7029-7065.  doi: 10.1090/tran/7542.  Google Scholar

[5]

D. Dou and K. K. Park, Examples of entropy generating sequence, Sci. China Math., 54 (2011), 531-538.  doi: 10.1007/s11425-010-4152-y.  Google Scholar

[6]

S. Ferenczi and K. K. Park, Entropy dimensions and a class of constructive examples, Discrete Cont. Dyn. Syst., 17 (2007), 133-141.  doi: 10.3934/dcds.2007.17.133.  Google Scholar

[7]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49.  doi: 10.1007/BF01692494.  Google Scholar

[8] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981.   Google Scholar
[9]

T. N. T. Goodman, Topological sequence entropy, Proc. London Math. Soc., 29 (1974), 331-350.  doi: 10.1112/plms/s3-29.2.331.  Google Scholar

[10]

W. HuangS. M. LiS. Shao and X. D. Ye, Null systems and sequence entropy pairs, Ergodic Theory Dynam. Systems, 23 (2003), 1505-1523.  doi: 10.1017/S0143385702001724.  Google Scholar

[11]

W. HuangK. K. Park and X. D. Ye, Topological disjointness for entropy zero systems, Bull. Soc. Math. France, 135 (2007), 259-282.  doi: 10.24033/bsmf.2534.  Google Scholar

[12]

W. Huang and X. D. Ye, Dynamical systems disjoint from any minimal system, Trans. Amer. Math. Soc., 357 (2005), 669-694.  doi: 10.1090/S0002-9947-04-03540-8.  Google Scholar

[13]

W. Huang and X. D. Ye, Combinatorial lemmas and applications to dynamics, Adv. Math., 220 (2009), 1689-1716.  doi: 10.1016/j.aim.2008.11.009.  Google Scholar

[14]

W. HuangX. D. Ye and G. H. Zhang, Relative entropy tuples, relative U.P.E. and C.P.E. extensions, Israel J. Math., 158 (2007), 249-283.  doi: 10.1007/s11856-007-0013-y.  Google Scholar

[15]

P. Hulse, Sequence entropy and subsequence generators, J. London Math. Soc., 26 (1982), 441-450.  doi: 10.1112/jlms/s2-26.3.441.  Google Scholar

[16]

T. Kamae and L. Zamboni, Sequence entropy and the maximal pattern complexity of infinite words, Ergodic Theory Dynam. Systems, 22 (2002), 1191-1199.  doi: 10.1017/S0143385702000585.  Google Scholar

[17]

A. Katok and J.-P. Thouvenot, Slow entropy type invanriants and smooth realization of commuting measure-preserving transformations, Ann. Inst. Henri Poincare Probab. Statist., 33 (1997), 323-338.  doi: 10.1016/S0246-0203(97)80094-5.  Google Scholar

[18]

D. Kerr and H. Li, Independence in topological C*-dynamics, Math. Ann., 338 (2007), 869-926.   Google Scholar

[19]

A. G. Kušhnirenkov, Metric invariants of entropy type, Uspekhi Mat. Nauk, 22 (1967), 57-65.   Google Scholar

show all references

References:
[1]

F. Blanchard, A disjointness theorem involving topological entropy, Bull. Soc. Math. France, 121 (1993), 465-478.  doi: 10.24033/bsmf.2216.  Google Scholar

[2]

M. de Carvalho, Entropy dimension of dynamical systems, Portugal. Math., 54 (1997), 19-40.   Google Scholar

[3]

D. DouW. Huang and K. Park, Entropy dimension of topological dynamical systems, Trans. Amer. Math. Soc., 363 (2011), 659-680.  doi: 10.1090/S0002-9947-2010-04906-2.  Google Scholar

[4]

D. DouW. Huang and K. Park, Entropy dimension of measure preserving systems, Trans. Amer. Math. Soc., 371 (2019), 7029-7065.  doi: 10.1090/tran/7542.  Google Scholar

[5]

D. Dou and K. K. Park, Examples of entropy generating sequence, Sci. China Math., 54 (2011), 531-538.  doi: 10.1007/s11425-010-4152-y.  Google Scholar

[6]

S. Ferenczi and K. K. Park, Entropy dimensions and a class of constructive examples, Discrete Cont. Dyn. Syst., 17 (2007), 133-141.  doi: 10.3934/dcds.2007.17.133.  Google Scholar

[7]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49.  doi: 10.1007/BF01692494.  Google Scholar

[8] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981.   Google Scholar
[9]

T. N. T. Goodman, Topological sequence entropy, Proc. London Math. Soc., 29 (1974), 331-350.  doi: 10.1112/plms/s3-29.2.331.  Google Scholar

[10]

W. HuangS. M. LiS. Shao and X. D. Ye, Null systems and sequence entropy pairs, Ergodic Theory Dynam. Systems, 23 (2003), 1505-1523.  doi: 10.1017/S0143385702001724.  Google Scholar

[11]

W. HuangK. K. Park and X. D. Ye, Topological disjointness for entropy zero systems, Bull. Soc. Math. France, 135 (2007), 259-282.  doi: 10.24033/bsmf.2534.  Google Scholar

[12]

W. Huang and X. D. Ye, Dynamical systems disjoint from any minimal system, Trans. Amer. Math. Soc., 357 (2005), 669-694.  doi: 10.1090/S0002-9947-04-03540-8.  Google Scholar

[13]

W. Huang and X. D. Ye, Combinatorial lemmas and applications to dynamics, Adv. Math., 220 (2009), 1689-1716.  doi: 10.1016/j.aim.2008.11.009.  Google Scholar

[14]

W. HuangX. D. Ye and G. H. Zhang, Relative entropy tuples, relative U.P.E. and C.P.E. extensions, Israel J. Math., 158 (2007), 249-283.  doi: 10.1007/s11856-007-0013-y.  Google Scholar

[15]

P. Hulse, Sequence entropy and subsequence generators, J. London Math. Soc., 26 (1982), 441-450.  doi: 10.1112/jlms/s2-26.3.441.  Google Scholar

[16]

T. Kamae and L. Zamboni, Sequence entropy and the maximal pattern complexity of infinite words, Ergodic Theory Dynam. Systems, 22 (2002), 1191-1199.  doi: 10.1017/S0143385702000585.  Google Scholar

[17]

A. Katok and J.-P. Thouvenot, Slow entropy type invanriants and smooth realization of commuting measure-preserving transformations, Ann. Inst. Henri Poincare Probab. Statist., 33 (1997), 323-338.  doi: 10.1016/S0246-0203(97)80094-5.  Google Scholar

[18]

D. Kerr and H. Li, Independence in topological C*-dynamics, Math. Ann., 338 (2007), 869-926.   Google Scholar

[19]

A. G. Kušhnirenkov, Metric invariants of entropy type, Uspekhi Mat. Nauk, 22 (1967), 57-65.   Google Scholar

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