We study the topological complexities of relative entropy zero extensions acted upon by countable-infinite amenable groups. First, for a given Følner sequence $\{F_n\}_{n=0}^{+\infty}$, we define the relative entropy dimensions and the dimensions of the relative entropy generating sets to characterize the sub-exponential growth of the relative topological complexity. we also investigate the relations among these. Second, we introduce the notion of a relative dimension set. Moreover, using the method, we discuss the disjointness between the relative entropy zero extensions via the relative dimension sets of two extensions, which says that if the relative dimension sets of two extensions are different, then the extensions are disjoint.
Zubiao XIAO
,
Zhengyu YIN
. RELATIVE ENTROPY DIMENSION FOR COUNTABLE AMENABLE GROUP ACTIONS*[J]. Acta mathematica scientia, Series B, 2023
, 43(6)
: 2430
-2448
.
DOI: 10.1007/s10473-023-0607-4
[1] Auslander J.Minimal Flows and their Extensions. North-Holland Math Studies, Vol 153. Amsterdam: North-Holland, 1988
[2] Blanchard F. A disjointness theorem involving topological entropy. Bull Soc Math France, 1993, 121: 465-478
[3] Buzzi J. Piecewise isometries have zero topological entropy. Ergod Theor Dyn Syst, 2001, 21: 1371-1377
[4] Carvalho M D. Entropy dimension of dynamical systems. Portugaliae Math, 1997, 54: 19-40
[5] Cecherini-Silberstein T, Coornaert M.Cellular Automata and Groups. Berlin: Springer-Verlag, 2010
[6] Cao K, Dai X.Lifting the regionally proximal relation and characterizations of distal extensions. Discrete Contin Dyn Syst, 2022, 42: 2103-2174
[7] Dou D, Huang W, Park K K. Entropy dimension of topological dynamical systems. Trans Amer Math Soc, 2011, 363: 659-680
[8] Dou D, Huang W, Park K K. Entropy dimension of measure preserving systems. Trans Amer Math Soc, 2019, 371: 7029-7065
[9] Dai X, Liang H, Xiao Z. Characterizations of relativized distal points of topological dynamical systems. Topology and its Applications, 2021, 302: 107832
[10] Ellis R.Lectures on Topological Dynamics. New York: Benjamin, 1969
[11] Ferenczi S, Park K K. Enropy dimensions and a class of constructive examples. Discret Contin Dyn Sys, 2007, 17: 133-141
[12] Furstenberg H. Disjointness in ergodic theory, minimal sets,a problem in Diophantine approximation. Math Systems Theory, 1967, 1: 1-49
[13] Furstenberg H.Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton, New Jersey: Princeton University Press, 1981
[14] Gao J, Huang X, Zhu C. Sequence entropies and mean sequence dimension for amenable group actions. J Diff Equa, 2020, 269: 9404-9431
[15] Huang W, Park K K, Ye X. Topological disjointness from entropy zero systems. Bull Soc Math France, 2007, 135: 259-282
[16] Huang W, Ye X. Dynamical systems disjoint from any minmal system. Trans Amer Math Soc, 2004, 357: 669-694
[17] Huang W, Ye X, Zhang G. Local entropy theory for a countable discrete amenable group action. J Funct Anal, 2011, 261: 1028-1082
[18] Huang W, Ye X, Zhang G. Relative entropy tuples, relative U.P.E. and C.P.E. extensions. Israel J Math, 2007, 158: 249-283
[19] Lindenstrauss E. Pointwise theorems for amenable groups. Invent Math, 2001 146(2): 259-295
[20] Kuang R, Cheng W, Ma D, Li B. Different forms of entropy dimension for zero entropy systems. Dynamical Systems, 2014, 29(2): 239-254
[21] Kerr D, Li H. Ergodic Theory, Independence and Dicaotomies. Switzerland: Springer, 2016
[22] Kerr D, Li H. Independence in topological $C^*$-dynamics. Math Ann, 2007, 338: 869-926
[23] McMahon D C. Relativized weak disjointness and relatively invariant measures. Trans Amer Math Soc, 1978, 236: 225-237
[24] Pan J, Zhou Y. Some results on bundle systems for a countable discrete amenable group action. Acta Math Sci, 2023, 43B(3): 1382-1402
[25] Sun P. Unique ergodicity for zero-entropy dynamical systems with the approximate product property. Acta Math Sin, 2021, 37: 362-376
[26] Zhou X. Relative entropy dimension of topological dynamical systems. Discrete Cont Dyn Syst, 2019, 39: 6631-6642
