The shift on the inverse limit of a covering projection

Abstract

A group endomorphismα : G → G is said to beweakly shift equivalent to the group endomorphismβ : H → H if there existsh ∈ H such thatα is shift equivalent to Ad[h] °β. Given covering projectionsa : X → X, b : Y → Y of compact, connected, locally path connected, semilocally simply connected metric spaces with fixed pointsx 0 ∈X,y 0 ∈Y respectively, the inverse limits $$\begin{array}{l} \sum\nolimits_a { = \lim } (X,a) = \{ (x_i )_{i \in Z^ + } ax_{i + 1} = x_1 ,i \in Z^ + \} , \\ \sum\nolimits_a { = \lim } (Y,b) = \{ (y_i )_{i \in Z^ + } by_{i + 1} = y_1 ,i \in Z^ + \} , \\ \end{array}$$ and the “shift” mapsσ a : Σ a → Σ a ,σ b : Σ b → Σ b defined byσ a((x i)i∈Z +)=(x i+1)i∈Z + ∈ Σ a ,σ b((y i)i∈Z +)=(y i + 1)i∈Z + ∈ Σ b are considered. It is proven that ifσ a andσ b are topologically conjugate thena # :π 1(X, x 0) →π 1(X, x 0) is weakly shift equivalent tob # :π 1(Y, y 0) →π 1(Y, y 0). Furthermore, ifa : X → X andb : Y → Y are expanding endomorphisms of compact differentiable manifolds, weak shift equivalence is a complete invariant of topological conjugacy. The use of this invariant is demonstrated by giving a complete classification of the shifts of expanding maps on the klein bottle. The reader is referred to Section 4 of this work for a detailed statement of results.