On projections onto odometers of dynamical systems with the compact phase space

Abstract

We investigate projections to odometers (group rotations over adic groups) of topological invertible dynamical systems with discrete time and compact Hausdorff phase space. For a dynamical system $(X, f)$ with a compact phase space we consider the category of its projections onto odometers. We examine the connected partial order relation on the class of all objects of a skeleton of this category. We claim that this partially ordered class always have maximal elements and characterize them. It is claimed also, that this class have a greatest element and is isomorphic to some characteristic for the dynamical system $(X, f)$ subset of the set $Σ$ of ultranatural numbers if and only if the dynamical system $(X, f)$ is indecomposable (the space $X$ could not be decomposed into two proper disjoint closed invariant subsets).