Flexibility of measurable and topological nilfactors in dynamical systems

We construct examples of minimal and uniquely ergodic systems realizing all possible behaviors in the interplay of measurable and topological nilfactors. To build such examples, we adapt an idea that stems from Furstenberg’s construction of a minimal but not uniquely ergodic system on $\mathbb{T}^2$.

💡 Research Summary

The paper investigates the relationship between measurable nilfactors (denoted Zₖ) and topological nilfactors (denoted Xₖ) in dynamical systems, showing that essentially any prescribed pattern of coincidence or divergence between them can be realized within a single strictly ergodic system. The authors build on Furstenberg’s classic construction of a minimal but not uniquely ergodic transformation on the two‑torus, extending the idea of a “measurable but not continuous coboundary” to higher‑dimensional toral extensions.

The main result, Theorem 1.1, states that for any integers 0 ≤ j ≤ ℓ ≤ k with k ≥ 1 there exists a minimal, uniquely ergodic (strictly ergodic) system such that: 1. Zₖ is a non‑trivial extension of Zₖ₋₁; 2. For all i ≤ j the measurable and topological i‑step nilfactors coincide (Zᵢ ≅ Xᵢ); 3. For j + 1 ≤ i ≤ k the measurable nilfactor Zᵢ is not topologically isomorphic to Xᵢ (although they are measurably isomorphic); 4. For all i ≥ ℓ the topological nilfactors stabilize, i.e. Xᵢ ≅ Xᵢ₊₁.

Moreover, the theorem asserts that any minimal, ergodic, non‑weak‑mixing system can be modeled by a system satisfying some choice of parameters (j, ℓ, k), confirming the universality of the construction.

The construction proceeds as follows. Choose an irrational rotation α on the circle and a measurable coboundary f : 𝕋 → 𝕋 that has no continuous solution to the cohomological equation ρ = F∘Tα − F. Using Fourier series with carefully chosen coefficients (based on Diophantine approximations of α), the authors produce such an f. They then define a skew‑product transformation on the k‑torus: \