On a bi-lateral Adding Machine and its characterization
Pouya Mehdipour, Rebeca M. S. dos Santos
TL;DR
The paper extends the classical odometer to a bilateral adding machine on a zip space, defining the $\alpha$-adic space $\Delta_{\alpha}$ and the $(Z,S)$-full zip space $\Sigma_{Z,S}$ with transition $\tau$, and proves the bilateral adding machine map $f_\alpha$ is a homeomorphism that is not expansive. It contrasts zip shifts (S-expansive local homeomorphisms) with the non-zip bilateral odometer and provides a conjugacy framework: a compact metric space $(X,d)$ with a continuous map $f$ is topologically conjugate to $f_\alpha$ iff there exists a two-sided partition-refinement structure (via cylinder-like coverings) and a surjective map $\pi:X\to\Delta_\alpha$ with $\pi\circ f=f_\alpha\circ\pi$. The work thus yields a concrete, two-sided symbolic model for minimal systems with regular recurrence and clarifies when such systems arise as bilateral adding machines. These results sharpen the understanding of bilateral odometers within symbolic dynamics and separate them from zip-shift dynamics, with implications for modeling complex two-sided dynamical behaviors.
Abstract
In this paper, we introduce a bilateral adding machine based on a zip space with two sets of alphabets. We demonstrate that these adding machines are homeomorphisms and provide necessary and sufficient conditions for their characterization.