非自治动力系统Bowen估计熵的变分原理(英文)A variational principle for Bowen estimation entropy of nonautonomous dynamical systems
刘磊;彭冬梅;
摘要(Abstract):
推广了经典动力系统的Bowen拓扑熵,给出了非自治动力系统的Bowen估计熵的定义,讨论了非自治动力系统的Bowen估计熵和局部估计熵的关系,推广了自治动力系统Bowen拓扑熵的Billingsley型定理,研究了非自治动力系统的Bowenα-估计熵的Billingsley型定理.此外,给出了非自治动力系统Bowen估计熵的变分原理.
关键词(KeyWords): Bowen估计熵;测度理论下α-估计熵;非自治动力系统;变分原理
基金项目(Foundation): 国家自然科学基金(11401363);; 河南省高等学校青年骨干教师培养计划(2018GGJS134)
作者(Authors): 刘磊;彭冬梅;
参考文献(References):
- [1]Adler R,Konheim A,McAndrew M.Topological entropy[J].Trans.Amer.Math.Soc.,1965,114(2):309-319.
- [2]Bowen R.Topological entropy and axiom A[J].Proc.Symp.Pure Math,Amer.Math.Soc.,1970,14(1):23-42.
- [3]Bowen R.Entropy for group endomorphisms and homogeneous spaces[J].Trans.Amer.Math.Soc.,1971,153(1):401-414.
- [4]Walters P.An Introduction to Ergodic Theory[M].Berlin:Grad.Texts in Math.,1982.
- [5]Cornfeld I,Fomin S,Sinai Y.Ergodic Theory[M].Berlin:Springer,1982.
- [6]Kolmogorov A,Tihomiorov Y.?-Entropy and?-capacity of sets in function spaces[J].Trans.Amer.Math.Soc.,1961,17(2):277-364.
- [7]Bowen R,Walters P.Expansive one-parameter flows[J].J.Differ.Equat.,1972,12(1):180-193.
- [8]Keynes H,Sears M.Real-expansive flows and topological dimensions[J].Ergod.Th.Dynam.Sys.,1981,1(2):179-195.
- [9]Thomas R.Some fundamental properties of continuous functions and topological entropy[J].Pacific.J.Math.,1990,141(2):391-400.
- [10]Block L,Coppel W.Dynamics in One Dimension[M].Berlin:Lect.Notes Math.,1992.
- [11]Bowen R.Topological entropy for noncompact sets[J].Trans.Amer.Math.Soc.,1973,184(1):125-136.
- [12]Pesin Y.Dimension Theory in Dynamical Systems:Contemporary Views and Applications[M].Chicago and London:The University of Chicago Press,1997.
- [13]Feng D J,Huang W.Variational principles for topological entropies of subsets[J].J.Funct.Anal.,2012,263(8):2228-2254.
- [14]Brin M,Katok A.On Local Entropy,Geometric Dynamics (Rio de Janeiro,1981)[M].Berlin:In:Lecture Notes in Mathematics,1983.
- [15]Liberzon D,Mitra S.Entropy and minimal data rates for state estimation and model detection[C]//Proceedings of the 19th International Conference on Hybrid Systems:Computation and Control (ACM).Qingdao,2016:247-256
- [16]Liberzon D,Mitra S.Entropy notions for state estimation and model detection with finite-datarate measurements[C]//2016 IEEE 55th Conference on Decision and Control (CDC).Las Vegas,2016:7335-7340.
- [17]Liberzon D,Mitra S.Entropy and minimal bit rates for state estimation and model detection[J].IEEE Trans.Automat.Control,2018,63(10):3330-3344.
- [18]Kawan Ch.Exponential state estimation,entropy and Lyapunov exponents[J].Systems Control Lett.,2018,113(1):78-85.
- [19]钟兴富.自由半群的估计熵和?-弱混合集[J].数学学报,2019,62(6):889-902.
- [20]钟兴富,陈志景.Bowen估计熵的变分原理[J].中国科学:数学,2020,51(8):1213-1224.
- [21]Kolyada S,Snoha L.Topological entropy of nonautonomous dynamical systems[J].Random Comput.Dynam.,1996,4(2):205-233.
- [22]Bi′s A.Topological and measure-theoretical entropies of nonautonomous dynamical systems[J].J.Dyn.Diff.Equat.,2018,30(1):273-285.
- [23]Canovas J S.Some results on (X,f,A) nonautonomous systems[J].Grazer Math.Ber.,2004,346(1):53-60.
- [24]Huang X J,Wen X,Zeng F P.Topological pressure of nonautonomous dynamical systems[J].Nonlinear Dyn.Syst.Theory,2008,8(1):43-48.
- [25]Kawan Ch.Metric entropy of nonautonomous dynamical systems[J].Nonauton.Dyn.Syst.,2013,1(1):26-52.
- [26]Kolyada S,Misiurewicz M,Snoha L.Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval[J].Fund.Math.,1999,160(2):161-181.
- [27]Kuang R,Cheng W,Li B.Fractal entropy of nonautonomous systems[J].Pacific J.Math.,2013,262(2):421-436.
- [28]Liu K R,Qiao Y X,Xu L Y.Topological entropy of nonautonomous dynamical systems[J].J.Diff.Equat.,2020,268(9):5353-5365.
- [29]Zhang J L,Chen L X.Lower bounds of the topological entropy for nonautonomous dynamical systems[J].Appl.Math.J.Chin.Univ.Ser.B.,2009,24(1):76-82.
- [30]Zhu Y J,Liu Z F,Xue X L,et al.Entropy of nonautonomous dynamical systems[J].J.Korean Math.Soc.,2012,49(1):165-185.
- [31]Xu L Y,Zhou X M.Variational principles for entropies of nonautonomous dynamical systems[J].J.Dyn.Diff.Equat.,2018,30(3):1053-1062.
- [32]Billingsley P.Ergodic Theory and Information[M].New York:John Wiley and Sons Inc.,1965.
- [33]Ma J H,Wen Z Y.A Billingsley type theorem for Bowen entropy[J].C.R.Math.Acad.Sci.Paris,Ser.I.,2008,346(2):503-507.
- [34]Matveev A.S,Savkin A.V.Estimation and Control over Communication Networks[M].Boston:Birkh¨auser Boston,2009.
- [35]Mattilla P.Geometry of Sets and Measures in Euclidean Spaces[M].Cambridge:Cambridge University Press,1995.
- [36]Engelking R.General Topology[M].Warszawa:PWN,1977.