Renormalizing an infinite rational IET

Abstract

We study an interval exchange transformation of [0, 1] formed by cutting the interval at the points 1 n and reversing the order of the intervals. We find that the transformation is periodic away from a Cantor set of Hausdorff dimension zero. On the Cantor set, the dynamics are nearly conjugate to the 2-adic odometer.

Rotated odometers

“…If 𝑒 2𝜋𝑖∕2 𝑚 is an eigenvalue of the Koopman operator of the minimal subsystem, then by Lemma 8.2, 2 𝑚 must divide ĥ(𝑛) 𝑖 for 0 ⩽ 𝑖 ⩽ 3, and 𝑛 large enough. Inverting the first block of the matrix in (21), we find…”

“…16, 5.20, and 5.21 show that 𝐼 𝑝𝑒𝑟 may be an empty set, or a finite or infinite union of half-open intervals. An infinite IET that contains an infinite collection of intervals of periodic points was also considered in [21], see Example 3.5.…”

“…We note that the property of having an infinite collection of intervals of periodic points is also exhibited by infinite IETs which are not rotated odometers. For instance, in [21] half-open subintervals of 𝐼 are rearranged in the manner similar to the von Neumann-Kakutani map, but the lengths of the subintervals are different from the lengths in (1). Namely, 𝑏 ∶ 𝐼 → 𝐼 is given by…”

“…Example We note that the property of having an infinite collection of intervals of periodic points is also exhibited by infinite IETs which are not rotated odometers. For instance, in [21] half‐open subintervals of $$I$$ are rearranged in the manner similar to the von Neumann–Kakutani map, but the lengths of the subintervals are different from the lengths in (). Namely, $$b:I \rightarrow I$$ is given by $$$$\begin{equation*} b(x) = x - 1 + k^{-1} + (k+1)^{-1} \quad \textrm { for } 1 - k^{-1} \leqslant x < 1- (k+1)^{-1}, \, k \in \mathbb {N}.$$…”

“…Examples 5.16, 5.20, and 5.21 show that $$I_{per}$$ may be an empty set, or a finite or infinite union of half‐open intervals. An infinite IET that contains an infinite collection of intervals of periodic points was also considered in [21], see Example 3.5. Remark The existence of a unique minimal aperiodic set imposes strong restrictions on the behavior of the system.…”