Domination and entropy rigidity on non-n-Fuchsian Hitchin fibers in Higgs-bundle moduli

Establish the domination and entropy-rigidity results for regular fibers of the Hitchin map under the non-abelian Hodge correspondence, beyond the fibers associated with n-Fuchsian representations.

Background

The paper studies bending fibers of Hitchin representations in the representation variety and proves domination of Hilbert and translation length spectra, together with entropy rigidity in the Hilbert setting and, for translation entropy, in the n-Fuchsian case.

The authors relate these representation-theoretic bending fibers to regular fibers of the Hitchin map in Higgs-bundle moduli via the non-abelian Hodge correspondence. They explicitly state that the analogous domination and entropy-rigidity results for those fibers remain conjectural, with known results restricted to n-Fuchsian fibers.

References

In particular, the “bending fiber” in Theorem 1.4 can be thought of as an analogue of a regular fiber of the Hitchin map, and the corresponding domination and entropy rigidity results there are still conjectural, known only for the fibers of n-Fuchsian representations (see [DL22]).

— Entropy and domination for quasi-Hitchin representations  (2608.27939 - Barman et al., 28 Aug 2026) in Final paragraph of the Introduction, p. 4