Lyapunov stability and the ring of \(p\)-adic integers (Q2390113)

Let \(P=(p_1,p_2,p_3,\dots)\) be a sequence of prime numbers. For all \(N\in\mathbb N\), let \(P_N:=p_1\cdot p_2\cdots p_N\) (define \(P_0:=1\)). Any integer \(z\in\mathbb Z\) has a unique \(P\)-adic expansion \(z = \pm \sum_{n=0}^\infty z_n P_n\), where \(0\leq z_n\leq p_n\) for all \(n\in\mathbb N\), with only finitely many \(z_n\) being nonzero. The ring \({\mathcal Z}_P\) of \(P\)-adic integers is then defined as the set of all formal sums of this kind where infinitely many \(z_n\) are allowed to be nonzero. It can also be defined as the inverse limit of the sequence of cyclic groups \(\mathbb Z_{/P_1} \leftarrow\mathbb Z_{/P_2} \leftarrow\mathbb Z_{/P_3} \leftarrow\cdots\), where each of the arrows represents the obvious `mod \(p_n\)' epimorphism from \(\mathbb Z_{P_n}\) into \(\mathbb Z_{P_{n-1}}\). Define \(\alpha:{\mathcal Z}_P\rightarrow{\mathcal Z}_P\) by \(\alpha(z)=z+1\). The resulting dynamical system \(({\mathcal Z}_P,\alpha_P)\) is called the \(P\)-adic odometer or \(P\)-adic adding machine. These dynamical systems arise frequently in the study of solenoid systems, Toeplitz systems, substitution shifts, and Lyapunov-stable topological dynamical systems. This paper is really a short announcement of results which are supposedly proved in detail elsewhere. The paper announces two results. Theorem 3.2. Let \(X\) be a locally compact, locally connected metric space, let \(f:X\rightarrow X\) be a continuous map, and let \(A\subseteq X\) be a compact, \(f\)-invariant, \(f\)-transitive, \(f\)-Lyapunov stable subset. Let \(K\) be the set of connected components of \(A\); then \(f\) induces a map \(\tilde{f}:K\rightarrow K\). If \(|K|\) is infinite, then the system \((K,\tilde{f})\) is conjugate to a \(P\)-adic adding machine for some (nonunique) choice of \(P\). We define the multiplicity function \(m_P\) as follows: for any prime \(p\), let \(m_p(P)\) be the number of times \(p\) appears in \(P\) (from \(0\) to \(\infty\)). Theorem 3.3. Let \(P\) and \(P'\) be two prime sequences. The odometers \(({\mathcal Z}_P,\alpha_P)\) and \(({\mathcal Z}_{P'},\alpha_{P'})\) are conjugate if and only if \(P\) and \(P'\) have the same multiplicity function. In particular, all the different odometer representations of the system \((K,\tilde{f})\) in Theorem 3.2 have the same multiplicity function. Theorem 3.1 is really just a restatement of a result previously established by the author (Buescu) along with Ian Stewart in 1995. Theorem 3.2 is a well-known result in the odometer/Toeplitz literature. The novelty of the present approach is that Theorem 3.2 is obtained as a corollary of Theorem 3.1. Reviewer's remark: This short paper is poorly edited. There are several typos and sentence structure errors. The symbol `\(K\)' is never defined. The symbols \({\mathcal Z}_P\) and \(\mathbb Z _P\) are used interchangeably without explanation, as are the symbols \({\mathcal S}_\infty\) and \(\Sigma_P\) and the symbols \(K\) and \({\mathcal C}_\infty\).