Fix a positive integer \(N\) and consider the half-open interval \(X_N:=[0,\frac{1}{N})\). Define the map \(T_N:X_N\rightarrow X_N\) as \(T_N(x)=x-\frac{1}{N}+\frac{1}{k}+\frac{1}{k+1}\), where \(k\) is the integer part of \(\frac{1}{1/N -x}\). If we consider \(X_N\) as to be divided in subintervals of the form \([\frac{1}{N} - \frac{1}{k},\frac{1}{N}-\frac{1}{k+1})\), the map \(T_N\) can be viewed as the dynamical system that reverses the order of such subintervals. In particular, if \(N=1\) we obtain an infinite interval exchange transformation. Let \(\Lambda_N\) denote the set of point which are aperiodic (not periodic) under \(T_N\). Continuing with the notation, let \(\mathcal{N}\) be the set of sequences in \(\{0,1\}^{\mathbb N}\) which end in an infinite sequence of ones.
The main results in the paper are:
1) For each integer \(N\) and \(T_N\) there is a continuous bijection \(h_N\) from \(\{0,1\}^{\mathbb N}\setminus \mathcal{N}\) to the aperiodic set \(\Lambda_N\subset X_N\) such that \(T_N\circ h_N(\alpha)=h_N\circ f(\alpha)\) for all \(\alpha\in\{0,1\}^{\mathbb N}\), where \(f:\{0,1\}^{\mathbb N}\rightarrow\{0,1\}^{\mathbb{N}}\) is the \(2\)-adic odometer or addition-by-one map. As an interesting open question, the authors propose to guess the set of periods of \(T_N\) as well as to find, for each period \(p\), the Lebesgue measure of its corresponding set of periodic points of period \(p\).
2) For each \(N\) there is a Cantor set in \([0,\frac{1}{N}]\) of Hausdorff dimension zero such that \(x\) is a periodic point of \(T_N\) if and only if there exists \(\varepsilon>0\) such that \((x,x+\varepsilon)\) does not contain points of the Cantor set, therefore any point lying outside the Cantor set is periodic. Moreover, the set \(\bar{\Lambda}_N\) arises as a Cantor set.