One-Dimensional Dynamical Systems Of Amenable Group Actions

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.