Odomutants and flexibility results for quantitative orbit equivalence

TL;DR

The paper introduces odomutants—explicit distortions of odometers—to study quantitative orbit equivalence with controllable cocycle integrability. It leverages Kerr–Li entropy results for Shannon orbit equivalence while proving optimality results: entropy preservation is almost sharp, Belinskaya-type rigidity fails under sublinear integrability, and strong orbit equivalence refinements extend Boyle–Handelman phenomena to the measured and topological Cantor settings. The odomutant construction yields systems with the same point spectrum as their odometers but with richer entropy and Bernoulli properties, enabling precise counterexamples and constructive proofs. The work also connects measure-theoretic and topological dynamics via Cantor minimal homeomorphisms and Bratteli–Vershik representations, highlighting fundamental limits of entropy and looser Bernoulli behavior under quantitative orbit equivalences.

Abstract

We introduce new systems that we call odomutants, built by distorting the orbits of an odometer. We use these transformations for flexibility results in quantitative orbit equivalence. It follows from the work of Kerr and Li that if the cocycles of an orbit equivalence are $\log$-integrable, the entropy is preserved. Although entropy is also an invariant of even Kakutani equivalence, we prove that this relation and $L^{\frac{1}{2}}$ orbit equivalence are not the same, using a non-loosely Bernoulli system of Feldman which is an odomutant. We also show that Kerr and Li's result on preservation of entropy is optimal, namely we find odomutants of all positive entropies orbit equivalent to an odometer, with almost $\log$-integrable cocycles. We actually build a strong orbit equivalence between uniquely ergodic Cantor minimal homeomorphisms, so our result is a refinement of a famous theorem of Boyle and Handelman. We finally prove that Belinskaya's theorem is optimal for all the odometers, namely for every odometer, we find a odomutant which is almost-integrably orbit equivalent to it but not flip-conjugate. This yields an extension of a theorem by Carderi, Joseph, Le Maître and Tessera.

Figures (9)

• Figure 1: Example of odometer with $q_0=3$, $q_1=2$, $q_2=3$ (so $h_1=3$, $h_2=6$, $h_3=18$).
• Figure 2: Example of the first two steps in the construction of an odometer (on the left) and an associated odomutant (on the right). For a permutation $\sigma$ of the set $\{0,\ldots,k-1\}$, the notation $\sigma=(i_0\ldots i_{k-1})$ means that $\sigma$ is defined by $\sigma(j)=i_j$ for every $j\in\{0,\ldots,k-1\}$. The area coloured in purple (resp. orange) is the subset on which $S$ and $T$ are not yet defined at the end of the first step (resp. second step), it is equal to $\{N^{+}=1\}$ (resp. $\{N^{+}=2\}$) for the odometer, $\{N^{+}\circ\psi=1\}$ (resp. $\{N^{+}\circ\psi=2\}$) for the odomutant.
• Figure 3: At the top, the second step of a less restrictive cutting-and-stacking construction that we want to describe with an odomutant. At the bottom, the way we encode it with such a system. Here, the dynamics of the yellow tower appears twice in each new towers, so we divide it in two subtowers. Note that the partition $\tilde{\mathcal{P}}(2)$ is exactly the partition which gives the colour (yellow or blue) and the level in the $h_0$-tower to each points of the space, so that we cannot distinguish between points of the two yellow subtowers which are at the same level, contrary to the partition $\mathcal{P}(2)$. For the third step of the construction, the value of $q_2$ will depend on the number ($c_{2,0}$ and $c_{2,1}$) of copies for the dynamics of the two current towers in the next ones.
• Figure 4: Example of ordered Bratteli diagram $B$. The image of $(e_0,e_1,e_2,\ldots)$ by $T_B$ is $(f_0,f_1,e_2,\ldots)$.
• Figure 5: An ordered Bratteli diagram describing the odometer on $\prod_{n\geq 0}{\{0,\ldots,q_n-1\}}$. The image of $(e_0,e_1,e_2,e_3,\ldots)$ by $T_B$ is $(f_0,f_1,f_2,e_3,\ldots)$.

Theorems & Definitions (107)

• Theorem : kerrEntropyVirtualAbelianness2024
• Lemma
• Theorem
• Theorem 1: See Theorem \ref{['thAbis']}
• Remark 1.1
• Theorem 2
• Theorem 3: See Theorem \ref{['thBbis']}
• Theorem : boyleEntropyOrbitEquivalence1994
• Theorem : carderiBelinskayaTheoremOptimal2023
• Theorem 4: See Theorem \ref{['thCbis']}