Example: an odometer and dyadic labels

Reader card. This is the cleanest finite-stage model of the clopen-flow construction. It is useful after Chapter 1 and before the technical curvature comparison.

The odometer is the cleanest laboratory for the finite clopen-flow part of the proof.

The finite-quotient transport calculation below is explicit. Technical Engine A's PV/Barlak comparison makes it an input to the proved absorption theorem.

Height dynamics

Let

\[ \Sigma=\mathbb Z_2 \]

be the two-adic integers and let \(T_h(x)=x+1\). This is a minimal Cantor homeomorphism. For \(m\geq0\), the cylinder

\[ C_m=2^m\mathbb Z_2 \]

has measure

\[ \mu(C_m)=2^{-m}. \]

Choose commuting transverse translations \(T_i(x)=x+a_i\), with \(a_i\in\mathbb Z_2\). Every locally constant function factors through a finite quotient \(\mathbb Z/2^N\mathbb Z\). On that finite cyclic orbit, the difference

\[ (\alpha_i-1)f \]

has sum zero, and therefore is a discrete derivative \((\alpha_h-1)h_i\). The transfer function \(h_i\) is obtained by taking partial sums around the cycle.

flowchart LR
    C0["one clopen tower of height 2^N"]
    D["difference (alpha_i - 1)f"]
    Z["sum around tower = 0"]
    H["partial sums give h_i"]
    W["finite PE transport W_i"]
    C0 --> D --> Z --> H --> W

This is the finite-quotient version of the clopen-flow construction in Chapter 1.

Mixed dyadic-magnetic traces

Take \(f=1_{C_m}\). For any elementary transverse Rieffel module \(E_T\), the absorption theorem gives

\[ \tau_\mu([\mathcal E_{T,f}]) =2^{-m}\tau_{\Theta_G}([E_T]). \]

For a basic transverse two-torus class this reads

\[ \tau_\mu([\mathcal E_{f}])=2^{-m}\Theta_{12} \]

up to orientation. Thus a single system produces an entire dyadic family of mixed labels.

The abstract theorem does not require the full \(\mathbb Z^p\)-action to be faithful. If one wants a free geometric model, the odometer can instead be used as the transversal of a limit-periodic Delone suspension or placed in a suitable skew product. The height-one proof itself is unchanged.