C1-robust strong pluripotency for two-dimensional horseshoes

Establish whether there exists a two-dimensional diffeomorphism that is C1-robustly strongly pluripotent for a horseshoe.

Background

The paper proves C1-robust strong pluripotency for a dense majority-condition subset of a wild blender-horseshoe in manifolds of dimension at least three. It contrasts this result with the two-dimensional Colli–Vargas horseshoe model, for which corresponding robust strong pluripotency is not established.

The authors explain that the two-dimensional argument requires continuous variation of the stable and unstable thicknesses of the horseshoe in a neighborhood of the model. Moreira’s result shows that thickness continuity fails in the C1 category, and the paper therefore leaves the two-dimensional existence question unresolved.

References

Is there a two-dimensional diffeomorphism which is $C1$-robustly (strongly) pluripotent for a horseshoe? In contrast to the higher-dimensional case, in dimension two, the process of proving strong pluripotency needs the fact that the stable and unstable thicknesses of the horseshoe vary continuously in a neighborhood of the Colli--Vargas model $f_0$. However, in the $C1$-category, Moreira proved that the continuity of thickness fails. Therefore the above problem remains open.

— $C^1$-robust strong pluripotency for blender-horseshoes  (2608.22784 - Soma et al., 24 Aug 2026) in Section 1, Introduction, immediately before Section 2