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.