APOTP FOR THE INVERSE LIMIT SPACES

The connection between APOTP （asymptotic pseudo orbit tracing property） for a continuous map on a compact metric space and that for the shift map on the inverse limit space is investigated. As an application, it is showed that the shift map on Henderson pseudoarc has APOTP.

Inverse limits of hereditarily almost expandable class

In this paper, the hereditarily almost expandability of inverse limits is investigated, with two results obtained. Let X be the inverse limit space of an inverse system （Xα, π^αβ, ∧}. （1） Suppose X is hereditarily κ-metacompact, if each Xα is hereditarily pointwise collectionwise normal （almost θ-expandable, almost discrete θ-expandable）, then so is X; （2） Suppose X is hereditarily κ-σ-metacompact, if each Xα is hereditarily almost σ-expandable （almost discrete σ-expandable = σ-pointwise collectionwise normal）, then so is X.

B(LF,ω～2 )-refinability of inverse limits

Let X be the limit of an inverse system {Xα, παβ, ∧} and and let λ be the cardinal number of A. Assume that each projection πα ： X → Xα is an open and onto map and X is A-paracompact. We prove that if each Xα is B（LF, ω^2）-refinable （hereditarily B（LF, ω^2）- refinable）, then X is B（LF, ω^2）-refinable （hereditarily B（LF,ω ^2）-refinable）. Furthermore, we show that B（LF, ω^2）-refinable spaces can be preserved inversely undcr closed maps.

Inverse Limit of Spaces of Upper Semicontinuous Functions

Given a compact Hausdorff space X, U(X) denotes the compact Hausdorff space of all the upper semicontinuous functions from X to the unit interval with the dual lim inf topology. Then U is an endofunctor on compact Hausdorff space. It is proved in this note that this functor preserves inverse limits.

An Arc as the Inverse Limit of a Single Bonding Map of Type N on [0,1]

Let I=[0,1],c_1,c_2 ∈(0,1)with c_1<c_2 and f:I→I be a continuous map satisfying:f|_[0,c_1] and f|_[c_2,1] are both strictly increasing and f|_[c_1,c_2]is strictly decreasing.Let A ={z ∈[0,c_1]|f(x)=x}, a=maxA,a_1=max(A\{a}),and B={x∈[c_2,1]|f(x)=x},b=minB,b_1=min(B\{b}).Then the in- verse limit(I,f)is an arc if and only if one of the following three conditions holds: (1)If c_1<f(c_1)≤c_2(resp.c_1≤f(c_2)<c_2),then f has a single fixed point,a period two orbit, but no points of period greater than two or f has more than one fixed point but no points of other periods,furthermore,if A≠φ and B≠φ,then f(c2)>a(resp.f(c_1)<b). (2)If f(c_1)≤c_1(resp.f(c_2)≥c_2),then f has more than one fixed point,furthermore,if B≠φ and A\{a}≠φ,f(c_2)≥a or if a_1<f(c_2)<a,f^2(c_2)>f(c_2),(resp.f has more than one fixed point,furthermore,if A≠φ and B\{b}≠φ,f(c_1)≤b or if b<f(c_2)<b_1,f^2(c_1)<f(c_1)). (3)If f(c_1)>c_2 and f(c_2)<c_1,then f has a single fixed point,a single period two orbit lying in I\(u,v)but no points of period greater than two,where u,v ∈[c_1,c_2] such that f(u)=c_2 and f(v)=c_1.

On Some Questions of Barge and Toledo

Barge asked if each homeomorphism of a hereditarily decomposable chainable continuum has zero topological entropy and Toledo asked if a homeomorphism of a chainable continuum can always be induced by square commuting diagram on inverse systems of finite graphs. We show in this note that if Toledos question has a positive answer then Barges question also has a positive answer.

Inverse Limits in Representations of a Restricted Lie Algebra

Let （g, [p]） be a restricted Lie algebra over an algebraically closed field of characteristic p 〉 O. Then the inverse limits of ＂higher＂ reduced enveloping algebras {uxs （g） I s ∈ N} with X running over g＊ make representations of g split into different ＂blocks＂. In this paper, we study such an infinite- dimensional algebra Ax （g） ：=lim Uxs （g） for a given X C g＊. A module category equivalence is built between subcategories of U（g）-rnod and Ax（g）-mod. In the case of reductive Lie algebras, （quasi） generalized baby Verma modules and their properties are described. Furthermore, the dimensions of projective covers of simple modules with characters of standard Levi form in the generalized x-reduced module category are precisely determined, and a higher reciprocity in the case of regular nilpotent is obtained, generalizing the ordinary reciprocity.

Hereditarily covering properties of inverse sequence limits

Let {Xi,πki,ω} be an inverse sequence and X -- lim{Xi,πki,ω）. If each Xi is hereditarily （resp. metaLindelSf, σ-metaLindelSf, σ-orthocompact, weakly suborthocompact, δθ-refinable, weakly θ-refinable, weakly δθ-refinable）, then so is X.

Representations of Free Groups on Rational Line Q

A.M.W. Glass and S.H.McCleary have given the 2 transitive representation of the countable free l group F η(1<η≤ω 0 ).In this paper we shall give the highly ordered transitive representation of countable free groups on the rational line Q, which generalizes their results. As applications,we obtain the highly ordered transitive representation for the direct product of countable free groups,and the inverse limit of countable free groups must be an action on the set Q.

Entropy,Periodicity,and Graphs with Zero Euler Characteristic

Let G be a graph which contains exactly one simple closed curve.We prove that a continuous map f:G→G has zero topological entropy if and only if there exist at most k■[(Edg(G)+End(G)+ 3)/2]different odd numbers n_1,...,n_k such that Per(f)is contained in ∪_i^k=1 ∪_j~∞=0 n_i2~j,where Edg(G) is the number of edges of G and End(G)is the number of end points of G.