Product-foliation structure of the C0 foliation L(k;infinity)

Determine whether the C0 foliation \mathcal{L}_{(k;\infty)} constructed on the bridge blocks of the wild affine blender-horseshoe is C1-diffeomorphic to a product foliation.

Background

The paper constructs a C0 foliation \mathcal{L}_{(k;\infty)} whose leaves are C1 arcs adaptable to the unstable cone field. The union of these foliations is used to build the pseudo-orbits and perturbations underlying the C1-robust strong-pluripotency theorem.

Although the authors establish local C1 closeness of nearby leaves, they explicitly state that they do not know whether the resulting foliation has the global product structure represented by a product foliation. This structural question is not needed for the main theorem and remains unresolved in the paper.

References

From the construction of $\mathcal{L}{(k;\infty)}$, we also know that, for any $\epsilon>0$ and any leaf $l$ of $\mathcal{L}{(k;\infty)}$, there exists a neighborhood $\mathcal{N}(l)$ of $l$ in $\mathbb{B}$ such that any leaf $l'$ of $\mathcal{L}{(k;\infty)}$ contained in $\mathcal{N}(l)$ is $C1$-$\epsilon_$-close to $l$. Even so, the authors are not certain whether $\mathcal{L}_{(k;\infty)}$ is $C1$-diffeomorphic to a product foliation.

— $C^1$-robust strong pluripotency for blender-horseshoes  (2608.22784 - Soma et al., 24 Aug 2026) in Section 3, immediately after Definition of \mathcal{L}_{\infty} in the discussion preceding Section 4