Amenable群作用的一维动力系统
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摘要
本文主要讨论了一维空间上amenable群作用的动力实现问题,即:对于给定的拓扑空间X,离散群G和动力性质P,考虑G在X上的作用是否可以具有性质P。
第一章介绍了拓扑动力系统理论、连续统理论和群论中的一些基本概念和定义。
第二章考虑连续统上群作用的可扩性与几何熵。首先我们证明了含自由dendrite的Peano连续统上的可扩群作用必存在一个ping-pong game。通过这一结论,我们推出任一有限生成群在含自由dendrite的Peano连续统上的可扩作用都有正几何熵,且含自由dendrite的Peano连续统上不存在可扩幂零群作用。其次,我们证明了正则曲线上有限生成群作用的几何熵以作用群的增长率为上界,并由此推出正则曲线上有限生成幂零群作用的几何熵为零。
第三章考虑直线R上各种传递群作用的存在性问题。首先我们分析了直线上拓扑传递幂零群作用的结构,并对每个有限生成无挠幂零群G,在直线上构造了一个拓扑传递的G×Z~2-作用。更一般地,我们证明了每个非循环的poly-infinite-cyclic群在直线上存在一个忠实的拓扑传递的保向作用。其次,我们引入了pseudo-k-传递性的定义,并证明:直线上的多循环可解群作用是至多pseudo-2-传递的;如果一个可解群的可导长度为n,则该群在直线上的作用是至多pseudo-(4~n-1)-传递的;直线上不存在pseudo-2-传递的幂零群作用。
第四章考虑dendrite上的拓扑k-传递群作用和极小群作用。我们证明了dendrite上弱混合的有限生成群作用必有正几何熵;dendrite上不存在弱混合的幂零群作用;并且dendrite上不存在拓扑4-传递群作用。随后,我们对dendrite上的极小群作用进行了讨论。我们证明了如果群G在非退化dendrite X上的作用是极小的,则必存在一个ping-pong game。进而,G包含二元生成自由子半群,且X上不存在G-不变有限测度。特别地,G不是amenable群。
第五章考虑连续统上混沌与敏感群作用的存在性问题。我们证明了dendrite上不存在Devaney意义下的混沌群作用;含自由dendrite的Peano连续统上不存在敏感交换群作用。
In this paper,we mainly study the realization of amenable group actions on onedimensional space.That is,given a topological space X,a discrete group G and a dynamical property P,can G act on X with the property P?
In Chapter one,we introduce some basic notions and definitions in topological dynamical systems,continuum theory and group theory.
In Chapter two,we consider the expansiveness and geometric entropy of group actions on continua.First,we show that each expansive group action on a Peano continuum having a free dendrite must have a ping-pong game.By this conclusion,we prove that each expansive finitely generated group action on a Peano continuum having a free dendrite has positive geometric entropy,and each Peano continuum having a free dendrite admits no expansive nilpotent group actions.Secondly,we show that the geometric entropy of a finitely generated group action on a regular curve is bounded above by the growth rate of the acting group.As a corollary,the geometric entropy of each finitely generated nilpotent group action on any regular curve is zero.
In Chapter three,we study the existence of kinds of transitive group actions on the line R.First,we analyse the structure of topologically transitive nilpotent group actions on the line.Then for each finitely generated torsion free nilpotent group G,a topologically transitive G×Z~2- action on the line is constructed.More generally,we show that every noncyclic poly-infinite-cyclic group possesses a faithful topologically transitive orientation preserving action on the line.Next,the definition of pseudo-k-transitivity is introduced in this chapter.It is shown that each polycyclic solvable group action on the line is at most pseudo-2-transitive,and if the derived length of a solvable group G is n,then the action of G is at most pseudo-(4~n-1)-transitive.Also,it is shown that no nilpotent group action on the line is pseudo-2-transitive.
In Chapter four,we consider topologically k-transitive group actions and minimal group actions on dendrites.We show that each weakly mixing group action on a dendrite has positive geometric entropy when the acting group is finitely generated,and dendrites admit no weakly mixing nilpotent group actions.Also,it is shown that there are no topologically 4- transitive group actions on dendrites.Next,we study minimal group actions on dendrites.We prove that if a group G acts on a nondegenerate dendrite X minimally,then there must be a ping-pong game for the action.Moreover,G contains a free sub-semigroup on two generators, and X admits no G-invariant finite measure.In particular,G can not be amenable.
In Chapter five,the existence of chaotic group actions and sensitive group actions on continua are considered.It is shown that dendrites admit no chaotic group actions in the sense of Devaney,and each Peano continuum having a free dendrite admits no sensitive commutative group actions.
第一章介绍了拓扑动力系统理论、连续统理论和群论中的一些基本概念和定义。
第二章考虑连续统上群作用的可扩性与几何熵。首先我们证明了含自由dendrite的Peano连续统上的可扩群作用必存在一个ping-pong game。通过这一结论,我们推出任一有限生成群在含自由dendrite的Peano连续统上的可扩作用都有正几何熵,且含自由dendrite的Peano连续统上不存在可扩幂零群作用。其次,我们证明了正则曲线上有限生成群作用的几何熵以作用群的增长率为上界,并由此推出正则曲线上有限生成幂零群作用的几何熵为零。
第三章考虑直线R上各种传递群作用的存在性问题。首先我们分析了直线上拓扑传递幂零群作用的结构,并对每个有限生成无挠幂零群G,在直线上构造了一个拓扑传递的G×Z~2-作用。更一般地,我们证明了每个非循环的poly-infinite-cyclic群在直线上存在一个忠实的拓扑传递的保向作用。其次,我们引入了pseudo-k-传递性的定义,并证明:直线上的多循环可解群作用是至多pseudo-2-传递的;如果一个可解群的可导长度为n,则该群在直线上的作用是至多pseudo-(4~n-1)-传递的;直线上不存在pseudo-2-传递的幂零群作用。
第四章考虑dendrite上的拓扑k-传递群作用和极小群作用。我们证明了dendrite上弱混合的有限生成群作用必有正几何熵;dendrite上不存在弱混合的幂零群作用;并且dendrite上不存在拓扑4-传递群作用。随后,我们对dendrite上的极小群作用进行了讨论。我们证明了如果群G在非退化dendrite X上的作用是极小的,则必存在一个ping-pong game。进而,G包含二元生成自由子半群,且X上不存在G-不变有限测度。特别地,G不是amenable群。
第五章考虑连续统上混沌与敏感群作用的存在性问题。我们证明了dendrite上不存在Devaney意义下的混沌群作用;含自由dendrite的Peano连续统上不存在敏感交换群作用。
In this paper,we mainly study the realization of amenable group actions on onedimensional space.That is,given a topological space X,a discrete group G and a dynamical property P,can G act on X with the property P?
In Chapter one,we introduce some basic notions and definitions in topological dynamical systems,continuum theory and group theory.
In Chapter two,we consider the expansiveness and geometric entropy of group actions on continua.First,we show that each expansive group action on a Peano continuum having a free dendrite must have a ping-pong game.By this conclusion,we prove that each expansive finitely generated group action on a Peano continuum having a free dendrite has positive geometric entropy,and each Peano continuum having a free dendrite admits no expansive nilpotent group actions.Secondly,we show that the geometric entropy of a finitely generated group action on a regular curve is bounded above by the growth rate of the acting group.As a corollary,the geometric entropy of each finitely generated nilpotent group action on any regular curve is zero.
In Chapter three,we study the existence of kinds of transitive group actions on the line R.First,we analyse the structure of topologically transitive nilpotent group actions on the line.Then for each finitely generated torsion free nilpotent group G,a topologically transitive G×Z~2- action on the line is constructed.More generally,we show that every noncyclic poly-infinite-cyclic group possesses a faithful topologically transitive orientation preserving action on the line.Next,the definition of pseudo-k-transitivity is introduced in this chapter.It is shown that each polycyclic solvable group action on the line is at most pseudo-2-transitive,and if the derived length of a solvable group G is n,then the action of G is at most pseudo-(4~n-1)-transitive.Also,it is shown that no nilpotent group action on the line is pseudo-2-transitive.
In Chapter four,we consider topologically k-transitive group actions and minimal group actions on dendrites.We show that each weakly mixing group action on a dendrite has positive geometric entropy when the acting group is finitely generated,and dendrites admit no weakly mixing nilpotent group actions.Also,it is shown that there are no topologically 4- transitive group actions on dendrites.Next,we study minimal group actions on dendrites.We prove that if a group G acts on a nondegenerate dendrite X minimally,then there must be a ping-pong game for the action.Moreover,G contains a free sub-semigroup on two generators, and X admits no G-invariant finite measure.In particular,G can not be amenable.
In Chapter five,the existence of chaotic group actions and sensitive group actions on continua are considered.It is shown that dendrites admit no chaotic group actions in the sense of Devaney,and each Peano continuum having a free dendrite admits no sensitive commutative group actions.
引文
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[2]Auslander J.,Minimal Flows and Their Extensions,North-Holland Mathematics Studies 153,North-Holland,1988.
[3]Baldwin S.,Toward a theory of forcing on maps of trees,Internat.J.Bifur.Chaos Appl.Sci.Engrg.,5(1995),1307-1318.
[4]Baldwin S.,Continuous itinerary functions and dendrite maps,Topology Appl.,154(2007),2889-2938.
[5]Balibrea F.,Hric R.and Snoha L'.,Minimal sets on graphs and dendrites,Internat.J.Bifur.Chaos Appl.Sci.Engrg.,13(2003),1721-1725.
[6]Banks J.,Brooks J.,Cairns G.,Davis G.and Stacey P.,On Devaney's definition of chaos,Amer.Math.Monthly,99(1992),332-334.
[7]Beardon A.F.,Iteration of Rational Functions,Springer-Verlag,NY,1991.
[8]Beklaryan L.A.,Groups of homeomorphisms of the line and the circle.Topological characteristics and metric invariants,Uspekhi Mat.Nauk,59(2004),no.4(358),3-68(Russian);translation in Russian Math.Surveys,59(2004),no.4,599-660.
[9]Bowditch B.H.,Hausdorff dimension and dendritic limit sets,Math.Ann.,332(2005),667-676.
[10]Brin Matthew G.and Squier Craig C.,Groups of piecewise linear homeomorphisms of the real line,Invent.Math.,79(1985),no.3,485-498.
[11]Bryant B.F.,Unstable self-homeomorphisms of a compact space,Vanderbilt University Thesis,1954.
[12]Burslem L.and Wilkinson A.,Global rigidity of solvable group actions on S~1,Geom.Topol,8(2004),877-924(electronic).
[13]Cairns G.,Davis G.,Elton D.,Kolganova A.and Perversi P.,Chaotic group actions,Enseign.Math.,41(1995),123-133.
[14]Cairns G.and Kolganova A.,Chaotic actions of free groups,Nonlinearity,9(1996),1015-1021.
[15]Cairns G.and Nielsen A.,On the dynamics of the linear action of SL(n,Z),Bull.Austral.Math.Soc.,71(2005),no.3,359-365.
[16]Dani S.G.and Raghavan S.,Orbits of Euclidean frames under discrete linear groups,Israel J.Math.,36(1980),300-320.
[17]Devaney R.L.,An Introduction to Chaotic Dynamical Systems,Addison-Wesley Publishing Company,Redwood City,1989.
[18]Efremova L.S.and Makhrova E.N.,The dynamics of monotone maps of dendrites,Sb.Math.,192(2001),807-821.
[19]Efremova L.S.and Makhrova E.N.,On the center of continuous maps of dendrites,J.Diff.Equa.Appl.,9(2003),381-392.
[20]Farb B.and Franks J.,Group actions on one-manifolds.Ⅱ.Extensions of Holder's Theorem,Trans.Amer.Math.Soc.,355(2003),4385-4396.
[21]Farb B.and Franks J.,Groups of homeomorphism of one-manifolds.Ⅲ.Nilpotent subgroups,Ergod.Th.Dynam.Sys.,23(2003),1467-1484.
[22]Furstenberg H.,Disjointness in ergodic theory,minimal sets,and a problem in Diophantine approximation,Math.Systems Theory,1(1967),1-49.
[23]Ghys E.,Actions de reseaux sur le cercle,Invent.Math.,137(1999),199-231.
[24]Ghys E.,Groups acting on the circle,L'Enseign.Math.,47(2001),329-407.
[25]Ghys E.,Langevin R.and Walczak P.,Entropie geometrique des feuilletages,Acta Math.,160(1988),105-142.
[26]Glasner E.,Ergodic Theory via Joinings,Mathematical Surveys and Monographs,101,American Mathematical Society,Providence,RI,2003.
[27]Gromov M.,Groups of polynomial growth and expanding maps,Publ.Math.Inst.Hautes Etudes Sci.,53(1981),53-78.
[28]Harpe P.,Free groups in linear groups,Enseign.Math,29(1983),129-144.
[29]Harpe P.,Topics in geometric group theory,Chigago Lectures in Math.,University of Chicago Press,Chicago,IL,2000.
[30]Hasselblatt B.and Katok A.,Handbook of Dynamical Systems,Volume 1A,Elsevier,Amsterdam,2002.
[31] Hurder S., Dynamics of expansive. group actions on the circle, Available at http: //www.math.uic.edu/hurder/.
[32] Jakobsen J. F. and Utz W. R., The nonexistence of expansive homeomorphisms of a closed 2-cell, Pacific J. Math., 10(1960), 1319-1321.
[33] Kato H., The nonexistence of expansive homeomorphisms of dendroids, Fund. Math., 136(1990), 37-43.
[34] Kato H., The nonexistence of expansive homeomorphisms of 1-dimensional compact ANRs, Proc. Amer. Math. Soc, 108(1990), 267-269.
[35] Kato H., The nonexistence of expansive homeomorphisms of Peano continua in the plane, Topology Appl., 34(1990), 161-165.
[36] Kato H., A note on periodic points and recurrent points of maps of dendrites, Bull. Austral. Math. Soc, 51(1995), 459-461.
[37] Kato H., The nonexistence of expansive homeomorphisms of chainable continua, Fund. Math., 149(1996), 119-126.
[38] Kato H., Topological entropy of monotone maps and confluent maps on regular curves, Topology Proc, 28(2004), no. 2, 587-593.
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