On Open Maps between Dendrites
نویسندگان
چکیده
In this paper we use a result by J. Krasinkiewicz to present a description of the topological behavior of an open map defined between dendrites. It is shown that, for every such map f : X → Y, there exist n subcontinua X1, X2,. . ., Xn of X such that X = X1∪X2∪· · ·∪Xn, each set Xi∩Xj consists of at most one element which is a critical point of f , and each map f|Xi : Xi → Y is open, onto and can be lifted, in a natural way, to a product space Zi × Y for some compact and zero-dimensional space Zi. We also study the ω-limit sets ω(x) of a self-homeomorphism f : X → X defined on a dendrite X. It is shown that ω(x) is either a periodic orbit or a Cantor set (and if this is the case, then f|ω(x) is an adding machine).
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