真映射的拓扑熵
|
摘要
本文主要是针对一般的拓扑空间上的真映射,利用C-结构给出了它在空间的任意子集上的拓扑熵,上(下)容度拓扑熵的定义,其中上(下)容度拓扑熵是Patrao熵和AKM熵的推广.拓扑熵是Pesin-Pitskel熵和Bowen维数熵的推广.在此基础上,我们证明了新熵的一些基本性质,并计算了映射f(x)=x2,x∈R在实数集R的任意子集上的拓扑熵,最后我们研究了新定义的熵与几种经典拓扑熵的关系,本文的具体内容安排如下：
第一章是绪论,主要介绍了熵的发展历史以及拓扑熵目前的研究状况.
第二章是预备知识,介绍了本文需要用到的一些基本概念,记号以及相关基础知识.
在第三章中,我们先在拓扑动力系统(X,f)上定义了一个C-结构,然后利用C-结构的基本框架给出了真映射在任意子集上的拓扑熵,上(下)容度拓扑熵的定义,并给出了新熵的一些基本性质.在本章的最后,我们还给出了两个具体的例子.
在第四章中,我们研究了新定义的熵与几种经典拓扑熵的关系,即新定义的上(下)容度拓扑熵是Alder熵和Patrao熵的推广,同时还证明了上容度拓扑熵不大于Bowen用分离集(张成集)定义的拓扑熵,并指出当X是局部紧可分度量空间时,上容度拓扑熵等于Bowen用分离集(张成集)定义的拓扑熵的下确界.在证明的过程中,我们将Lebesgue覆盖引理从紧致度量空间推广到一般的度量空间.
By using the Caratheodory-structure(C-structure), the topological entropy on the whole topological space introduced for a proper map, is generalized to the cases of arbi-trary subset. The lower and upper capacities topological entropy are the extension of the Patrao entropy and Adler-Konheim-McAndrew entropy. The topological entropy is the extension of Pesin-Pitskel entropy and Bowen dimensional entropy. Some of the prop-erties of the new topological entropy and. the relationships between the new topological entropy and some classical topological entropies are provided. Two examples are given, one of them is that, for f:R→R,f(x)=x2, we give the topological entropy on any subset of R. This paper is organized as follows.
In Chapter 1, we introduce the development history and the investigation status of topological entropy.
Some general conceptions, knowledge and notations are given in preliminaries in Chapter 2.
In Chapter 3, we define a C-structure on topological dynamical system (X, f), where X is topological space. Then by applying the C-structure, we give definitions of topolog-ical entropy, upper (lower) capacity topological entropy of the proper map on any subset of X and some basic properties of the new entropy. Finally, we show some examples.
In Chapter 4, we give the relations between the new topological entropy and sev-eral classical topological entropies, i.e. upper(lower) capacity topological entropy is the generalized form of Alder entropy and Patrao entropy. At the same time, we also prove that upper (lower) capacity topological entropy is not larger than the topological entropy which is defined by separated sets of Bowen. Upper(Lower) capacity topological entropy equals infimum of the topological entropy defined by separated sets of Bowen when X is the local compact separable metric space. In our proof, we extend Lebesgue Covering Lemma from the compact metric space to general metric space.
第一章是绪论,主要介绍了熵的发展历史以及拓扑熵目前的研究状况.
第二章是预备知识,介绍了本文需要用到的一些基本概念,记号以及相关基础知识.
在第三章中,我们先在拓扑动力系统(X,f)上定义了一个C-结构,然后利用C-结构的基本框架给出了真映射在任意子集上的拓扑熵,上(下)容度拓扑熵的定义,并给出了新熵的一些基本性质.在本章的最后,我们还给出了两个具体的例子.
在第四章中,我们研究了新定义的熵与几种经典拓扑熵的关系,即新定义的上(下)容度拓扑熵是Alder熵和Patrao熵的推广,同时还证明了上容度拓扑熵不大于Bowen用分离集(张成集)定义的拓扑熵,并指出当X是局部紧可分度量空间时,上容度拓扑熵等于Bowen用分离集(张成集)定义的拓扑熵的下确界.在证明的过程中,我们将Lebesgue覆盖引理从紧致度量空间推广到一般的度量空间.
By using the Caratheodory-structure(C-structure), the topological entropy on the whole topological space introduced for a proper map, is generalized to the cases of arbi-trary subset. The lower and upper capacities topological entropy are the extension of the Patrao entropy and Adler-Konheim-McAndrew entropy. The topological entropy is the extension of Pesin-Pitskel entropy and Bowen dimensional entropy. Some of the prop-erties of the new topological entropy and. the relationships between the new topological entropy and some classical topological entropies are provided. Two examples are given, one of them is that, for f:R→R,f(x)=x2, we give the topological entropy on any subset of R. This paper is organized as follows.
In Chapter 1, we introduce the development history and the investigation status of topological entropy.
Some general conceptions, knowledge and notations are given in preliminaries in Chapter 2.
In Chapter 3, we define a C-structure on topological dynamical system (X, f), where X is topological space. Then by applying the C-structure, we give definitions of topolog-ical entropy, upper (lower) capacity topological entropy of the proper map on any subset of X and some basic properties of the new entropy. Finally, we show some examples.
In Chapter 4, we give the relations between the new topological entropy and sev-eral classical topological entropies, i.e. upper(lower) capacity topological entropy is the generalized form of Alder entropy and Patrao entropy. At the same time, we also prove that upper (lower) capacity topological entropy is not larger than the topological entropy which is defined by separated sets of Bowen. Upper(Lower) capacity topological entropy equals infimum of the topological entropy defined by separated sets of Bowen when X is the local compact separable metric space. In our proof, we extend Lebesgue Covering Lemma from the compact metric space to general metric space.
引文
[1]Shannon C.E,Weaver W. The mathematical theory of communication[M].Urbana: The University of Illinois of Press,1949:1-68
[2]Komogorov A.N, A new metric invariant of transient dynamical systems and auto-morphisms of lebesgne spaces.Dokl.Akd.Sci.SSSR.119(1958),861-864(Russian)
[3]Cornfeld I.P, Fomin S.V, Sinai Y.G. Ergodic theory[M].Berlin:Springer,1982
[4]Eckmann J.P. Ruelle D. Ergodic theory of chaos and strangee attractors[J], Rev.Mod.Phys,1985,V57:617-656
[5]Ruelle D. Chaotic evohtion and strange attractors. New York, Cambridge University Press,1989.
[6]Van B.H, Dorfman J.R, Posch H.A, et al Kolomogorov-Sinai entropy for dilute gases in equuilibrium[J]. Phys.Rev E,1997, V56:5272-5277
[7]Sinai Y.G, Chernov N.I. Dynamical systems [A]. Sinai Y.G. Collection of papers[C]. Singapore:World Scientific,1991:373-389
[8]Adler R.L, Konheim A.G, and McAndrow.M.H, Topological Entropy[J]. Tran. Amer. Math, Sec.1965,114,309-319
[9]Bowen R. Topological Entropy and Axiom A, Global Analysis[J]. Proceedings of Symposia in Put Mathematics.1971,14,23-41
[10]Diuaburg E.I, The relation between topological entropy and metric entropy, Soviet math.Dokl.11.(1970).13-16.
[11]Bowen R. Topological Entropy for noncompact sets[J]. Tran. Amer. Math, Sec.,1973, 184,125-137
[12]Pesin Y, Pitskel B, Topological Pressure and the varistional principle for non-compact sets.Functional Anal, and Its Applications,18:4,(1984),50-63.
[13]Patrao M, Entropy and its variational pricciple for non-compact metric spaces. Er-god.Th. Dynam.sgs.30(2010).1529-1542
[14]魏征,王延庚,卫国,一种新的拓扑熵：余紧拓扑熵.西北大学学报,自然科学网路版2009,7(1):0380[2009-01-10]. http://jonline.nwu.edu.cn/wenzhang/209001.pdf.
[15]Liu Lei,Wang Yangeng, Wei Guo. Topological entropy of continuous functions on topological spaces[J]. Chaos, Solitons and Fractals.2009, V39(1):417-427
[16]Malziri M, Malaci M.R. An extension of the notion of the topological entropy [J]. Chaos, Solitons and Fractals.2008,V36(2):370-373
[17]Canovas J.S, Rodriguez J.M. Topological entropy of maps on the real line[J]. Top Appl,2005, V153:735-746
[18]Alessandro Fedeli. On two notions of topological entropy for noncompact spaces [J]. Chaos, Solitons and Fractals 40(2009) 432-435
[19]Block L. Non Continuity of topological entropy of map of the Cantor set and the interopy[J]. Proc.AMS.,1975,50(7):388-393
[20]Block L. Mappings of the interval with finitely many periodie points have zero en-tropy [J]. Proc.AMS.,1977,67(5):357-360
[21]Block L. Topological entropy at Q-explosion[J]. Trans.AMS,1978,235(8):323-330
[22]Bowen R. Entropy for group endomorphisms and homogeneous spaces [J]. Math. Phys,1971,153(3):401-414
[23]Bowen R. Periodic pionts and measures for Axiom A diffeomorphisms[J]. AMS., 1971,154(6):377-397
[24]Bowen R. Entropy versus homology for certain diffeomorphisms[J]. AMS,1974,13(1): 61-67
[25]Bowen R. Entropy and the fundamental group [J]. The structuer of Attractors in Dynamical Systems,1977,23(3):21-29
[26]Bowen R and Franks J. The periodic points of maps of the intervalTopology[J]. Math. Phys,1976,15(3):337-342
[27]Shub M and Sacksteder R.Entropy of a differentiable map Advonces in Math[J]. Phys,1978,28(9):161-185
[28]Shub M. Dynamical systms,filtrations and entropy [J]. Bull.AMS,1974,80(5):27-41
[29]Shub M and Bullivan D. A remark on the Lefschetz fixed point formula for differen-tiable maps[J]. Topology,1974,13(4):189-191
[30]Shub M and Bullivan. Homology theory and dynamical systems[J]. Tran. Amer. Math. Sec,1975,14(3):109-132
[31]Shub M and Williams R. Entropy and stability Topology [J]. AMS,1975,14(5): 329-338
[32]Manning A. Topological entropy and the first homology group [J]. Dynamical systems-warnick,1974,13(2):185-190
[33]Pugh C. On the entropy conjecture[J].Dyamical Systerms-warwick,1974,13(7):257-261
[34]Blanchard F. Fully positive topological entropy and topological mixing. Contempo-rary Mathematics,1992,135:95-105
[35]Blanchard F. A disjointness theorem involving topological entropy. Boll. Sor. Math. France,1993,121(4):465-478
[36]Blanchard F. Glanser E and Host B, A variation on the variational principle and applications to entropy pairs. Ergod. Th. and Dynam. Sys.1997,17(1):29-43
[37]Blanchard F and Lacroix Y. Zero-entropy factors of topological flows. Proc. Amer. Mtth. Soc.1993,119(3):85-992
[38]Huang W and Ye X. A local variational relation and applications. Isrnel Journal of Mathematics.2006,151:237-280
[39]Huang W, Maass A, Romagnoli P and Ye X. Entropy pairs and a local abramov formula for the mte of open covers. Ergod. Th and Dynam.Sys,2004b,24(4):1127-1153
[40]Kushnirenko A.G. On metric invariants of entropy type. Russion Math. Serveys. 1967,22:53-61
[41]Goodman T N T. Topological sequence entropy. Proc. London Math. Soc.1974, 29(3):331-350
[42]Saleski A. Sequence entropy and mixing. J. of Math. Anal. and Appli.1977,60: 58-66
[43]Hulse P. Sequence entropy and subsequence generators. J. London. Math. Soc. (2). 1982,26:441-450
[44]熊金城.线段映射的动力体系：非游荡集,拓扑熵以及混乱[J].数学进展,1988,17(1)：1-10
[45]熊金城.关于拓扑熵的一点注记[J].科学通报,1983,33(9)：1534-1536
[46]周作领.一维动力系统[J].数学季刊,1988,3(6)：45-47
[47]Zhou Zuoling. A proof of small conjecture on topological entropy. Scientia sini-ca(series A),1986, XXIX, NO,3:254-264
[48]Dongkui Ma, Min Wu and Cuijun Liu. The entropies and multifractal spectrum of some compact systems[J]. Chaos, Solitons and Fractals,2008,38 (1):840-851
[49]Dongkui Ma, Min Wu. On Hausdorff dimension and topological entropy [J]. Fractals Vol.18, No.3(2010):363-370
[50]Romagnoli, P. A local variational principle for the topological entropy. Ergod. Th.and Dynam. Sys. (2003),23,1601-1610.
[51]Dai X, Jiang Y. Distance entropy of dynamical systems on noncompact-phase spaces. Discrete and Continuous Dynamical Systems. (2008),20,313-333.
[52]Hasselblatt B, Nitecki Z, Propp J. Topological entropy for non-uniformly continuous maps. Discrete and Continuous Dynamical Systems. (2008),22,201-213
[53]Glasner E, Ye.X. Local entropy theory. Ergod. Theory Dyn. Syst.29:321-356(2009)
[54]周作领.符号动力系统[M].上海：科技教育出版社,1997
[55]叶向东,黄文,邵松.拓扑动力系统概论[M].北京：科学出版社,2008
[56]Bowen R, Walters P. Expansive one-parameter flows[J]. J Differ Equat,1972. V12: 180-193
[57]Thomas R. F. Some fundamental properties of continuous functions and topological entropy [J]. Pacific J Math,1981, V141:391-400
[58]Zhenzhen Yan and Ercai Chen. Multifractal analysis of local entropies for recurrence time[J]. Chaos, Solitons and Fractals,2007,33(6):1584-1591
[59]Takens F, Verbitski E. General multifractal analysis of local entropies[J]. Fundam Math,2000; 16(5):203-37
[60]Takens F, Verbitski E. Multifractal analysis of local entropies for expansive homeo-morphisms with specification. Commun.Math. Phys.203,593-612(1999)
[61]Pesin Y, Dimension theory in dynamical systems, Chicago:The university of Chicago Ppess, (1997).
[62]Walters P. An introduce to ergodic theory[M].New York:Spring-Verlag,1982.
[63]Carathe'odory C, Uber das lineare mass. Gottingen Nachr,1914,406-426.
[2]Komogorov A.N, A new metric invariant of transient dynamical systems and auto-morphisms of lebesgne spaces.Dokl.Akd.Sci.SSSR.119(1958),861-864(Russian)
[3]Cornfeld I.P, Fomin S.V, Sinai Y.G. Ergodic theory[M].Berlin:Springer,1982
[4]Eckmann J.P. Ruelle D. Ergodic theory of chaos and strangee attractors[J], Rev.Mod.Phys,1985,V57:617-656
[5]Ruelle D. Chaotic evohtion and strange attractors. New York, Cambridge University Press,1989.
[6]Van B.H, Dorfman J.R, Posch H.A, et al Kolomogorov-Sinai entropy for dilute gases in equuilibrium[J]. Phys.Rev E,1997, V56:5272-5277
[7]Sinai Y.G, Chernov N.I. Dynamical systems [A]. Sinai Y.G. Collection of papers[C]. Singapore:World Scientific,1991:373-389
[8]Adler R.L, Konheim A.G, and McAndrow.M.H, Topological Entropy[J]. Tran. Amer. Math, Sec.1965,114,309-319
[9]Bowen R. Topological Entropy and Axiom A, Global Analysis[J]. Proceedings of Symposia in Put Mathematics.1971,14,23-41
[10]Diuaburg E.I, The relation between topological entropy and metric entropy, Soviet math.Dokl.11.(1970).13-16.
[11]Bowen R. Topological Entropy for noncompact sets[J]. Tran. Amer. Math, Sec.,1973, 184,125-137
[12]Pesin Y, Pitskel B, Topological Pressure and the varistional principle for non-compact sets.Functional Anal, and Its Applications,18:4,(1984),50-63.
[13]Patrao M, Entropy and its variational pricciple for non-compact metric spaces. Er-god.Th. Dynam.sgs.30(2010).1529-1542
[14]魏征,王延庚,卫国,一种新的拓扑熵：余紧拓扑熵.西北大学学报,自然科学网路版2009,7(1):0380[2009-01-10]. http://jonline.nwu.edu.cn/wenzhang/209001.pdf.
[15]Liu Lei,Wang Yangeng, Wei Guo. Topological entropy of continuous functions on topological spaces[J]. Chaos, Solitons and Fractals.2009, V39(1):417-427
[16]Malziri M, Malaci M.R. An extension of the notion of the topological entropy [J]. Chaos, Solitons and Fractals.2008,V36(2):370-373
[17]Canovas J.S, Rodriguez J.M. Topological entropy of maps on the real line[J]. Top Appl,2005, V153:735-746
[18]Alessandro Fedeli. On two notions of topological entropy for noncompact spaces [J]. Chaos, Solitons and Fractals 40(2009) 432-435
[19]Block L. Non Continuity of topological entropy of map of the Cantor set and the interopy[J]. Proc.AMS.,1975,50(7):388-393
[20]Block L. Mappings of the interval with finitely many periodie points have zero en-tropy [J]. Proc.AMS.,1977,67(5):357-360
[21]Block L. Topological entropy at Q-explosion[J]. Trans.AMS,1978,235(8):323-330
[22]Bowen R. Entropy for group endomorphisms and homogeneous spaces [J]. Math. Phys,1971,153(3):401-414
[23]Bowen R. Periodic pionts and measures for Axiom A diffeomorphisms[J]. AMS., 1971,154(6):377-397
[24]Bowen R. Entropy versus homology for certain diffeomorphisms[J]. AMS,1974,13(1): 61-67
[25]Bowen R. Entropy and the fundamental group [J]. The structuer of Attractors in Dynamical Systems,1977,23(3):21-29
[26]Bowen R and Franks J. The periodic points of maps of the intervalTopology[J]. Math. Phys,1976,15(3):337-342
[27]Shub M and Sacksteder R.Entropy of a differentiable map Advonces in Math[J]. Phys,1978,28(9):161-185
[28]Shub M. Dynamical systms,filtrations and entropy [J]. Bull.AMS,1974,80(5):27-41
[29]Shub M and Bullivan D. A remark on the Lefschetz fixed point formula for differen-tiable maps[J]. Topology,1974,13(4):189-191
[30]Shub M and Bullivan. Homology theory and dynamical systems[J]. Tran. Amer. Math. Sec,1975,14(3):109-132
[31]Shub M and Williams R. Entropy and stability Topology [J]. AMS,1975,14(5): 329-338
[32]Manning A. Topological entropy and the first homology group [J]. Dynamical systems-warnick,1974,13(2):185-190
[33]Pugh C. On the entropy conjecture[J].Dyamical Systerms-warwick,1974,13(7):257-261
[34]Blanchard F. Fully positive topological entropy and topological mixing. Contempo-rary Mathematics,1992,135:95-105
[35]Blanchard F. A disjointness theorem involving topological entropy. Boll. Sor. Math. France,1993,121(4):465-478
[36]Blanchard F. Glanser E and Host B, A variation on the variational principle and applications to entropy pairs. Ergod. Th. and Dynam. Sys.1997,17(1):29-43
[37]Blanchard F and Lacroix Y. Zero-entropy factors of topological flows. Proc. Amer. Mtth. Soc.1993,119(3):85-992
[38]Huang W and Ye X. A local variational relation and applications. Isrnel Journal of Mathematics.2006,151:237-280
[39]Huang W, Maass A, Romagnoli P and Ye X. Entropy pairs and a local abramov formula for the mte of open covers. Ergod. Th and Dynam.Sys,2004b,24(4):1127-1153
[40]Kushnirenko A.G. On metric invariants of entropy type. Russion Math. Serveys. 1967,22:53-61
[41]Goodman T N T. Topological sequence entropy. Proc. London Math. Soc.1974, 29(3):331-350
[42]Saleski A. Sequence entropy and mixing. J. of Math. Anal. and Appli.1977,60: 58-66
[43]Hulse P. Sequence entropy and subsequence generators. J. London. Math. Soc. (2). 1982,26:441-450
[44]熊金城.线段映射的动力体系：非游荡集,拓扑熵以及混乱[J].数学进展,1988,17(1)：1-10
[45]熊金城.关于拓扑熵的一点注记[J].科学通报,1983,33(9)：1534-1536
[46]周作领.一维动力系统[J].数学季刊,1988,3(6)：45-47
[47]Zhou Zuoling. A proof of small conjecture on topological entropy. Scientia sini-ca(series A),1986, XXIX, NO,3:254-264
[48]Dongkui Ma, Min Wu and Cuijun Liu. The entropies and multifractal spectrum of some compact systems[J]. Chaos, Solitons and Fractals,2008,38 (1):840-851
[49]Dongkui Ma, Min Wu. On Hausdorff dimension and topological entropy [J]. Fractals Vol.18, No.3(2010):363-370
[50]Romagnoli, P. A local variational principle for the topological entropy. Ergod. Th.and Dynam. Sys. (2003),23,1601-1610.
[51]Dai X, Jiang Y. Distance entropy of dynamical systems on noncompact-phase spaces. Discrete and Continuous Dynamical Systems. (2008),20,313-333.
[52]Hasselblatt B, Nitecki Z, Propp J. Topological entropy for non-uniformly continuous maps. Discrete and Continuous Dynamical Systems. (2008),22,201-213
[53]Glasner E, Ye.X. Local entropy theory. Ergod. Theory Dyn. Syst.29:321-356(2009)
[54]周作领.符号动力系统[M].上海：科技教育出版社,1997
[55]叶向东,黄文,邵松.拓扑动力系统概论[M].北京：科学出版社,2008
[56]Bowen R, Walters P. Expansive one-parameter flows[J]. J Differ Equat,1972. V12: 180-193
[57]Thomas R. F. Some fundamental properties of continuous functions and topological entropy [J]. Pacific J Math,1981, V141:391-400
[58]Zhenzhen Yan and Ercai Chen. Multifractal analysis of local entropies for recurrence time[J]. Chaos, Solitons and Fractals,2007,33(6):1584-1591
[59]Takens F, Verbitski E. General multifractal analysis of local entropies[J]. Fundam Math,2000; 16(5):203-37
[60]Takens F, Verbitski E. Multifractal analysis of local entropies for expansive homeo-morphisms with specification. Commun.Math. Phys.203,593-612(1999)
[61]Pesin Y, Dimension theory in dynamical systems, Chicago:The university of Chicago Ppess, (1997).
[62]Walters P. An introduce to ergodic theory[M].New York:Spring-Verlag,1982.
[63]Carathe'odory C, Uber das lineare mass. Gottingen Nachr,1914,406-426.