Smooth Orbit Equivalence of Multidimensional Borel Flows

The paper proves that every free Borel R^d-flow (d ≥ 2) is smoothly equivalent to a special flow (Theorem 20) and then uses Miller–Rosendal’s one-dimensional classification to conclude that all non-tame free Borel R^d-flows are smoothly equivalent (Theorem 21). The construction hinges on building coherent Borel families of disk-shaped regions with compatible diffeomorphisms across levels (Lemma 10 and subsequent lemmas), defining a straightened action via local charts, extracting an invariant one-dimensional spine, and identifying a special product representation; the identity is shown to be a smooth equivalence and the Borelness of the cocycle is carefully justified via countably many shapes (items 5(v) and 10(v)). These steps are clearly reflected in the text around Lemmas 10–19, Theorem 20, and Theorem 21 . The candidate solution mirrors this strategy: (A) builds coherent disk regions and compatible charts; (B) defines a straightened action and identifies an invariant spine Σ giving a special flow Σ×R^{d−1}; (C) invokes Miller–Rosendal for the base R-flows and lifts equivalences. Minor differences are cosmetic (e.g., the model suggests a rational-grid refinement to ensure countably many shapes, while the paper ensures this via explicit Borel partitions and a countable catalog in Theorem 5(v) and Lemma 10(v) ). Both arguments are correct and essentially the same in substance.