A star flow is a C1C^1 vector field all of whose periodic orbits remain hyperbolic under C1C^1 perturbations. In dimensions 3 and 4, generic star flows are classified by singular hyperbolicity, and da Luz constructed a robust open set of 5-dimensional star flows in which hyperbolic singularities with different stable indices share one chain recurrence class, ruling out singular hyperbolicity and refuting Palis's strong density conjecture in dimensions at least 5. The framework created in response — multisingular hyperbolicity, equivalently singular domination of the reparametrized linear Poincaré flow — implies the star property, and conversely holds for a C1C^1-open and dense subset of star flows, for all star flows whose periodic orbits robustly share one index (hence all 3-dimensional star flows), and for star flows with robust chain recurrence classes. What remains open is the full converse in the mixed-index regime: whether every star flow in every dimension, including dimensions at least 4 where singularities of different indices may coexist in one class, has all chain recurrence classes multisingular hyperbolic.

Background

For a C1C^1 flow generated by a vector field XX on a closed manifold, a compact invariant set is hyperbolic if the tangent bundle over it splits into invariant subbundles that are uniformly contracted and expanded by the derivative of the flow. Two refinements matter here. A point is chain recurrent if it can be connected to itself by pseudo-orbits with arbitrarily small jumps, and the equivalence classes of this relation, the chain recurrence classes, are the indecomposable pieces of the dynamics. A vector field is a star flow if every periodic orbit of XX and of every C1C^1-near vector field is hyperbolic (the singularities of a star flow are then hyperbolic as well); this robust hyperbolicity of the closed orbits is the flow analogue of the class of diffeomorphisms studied in Mañé's characterization of hyperbolicity. In the absence of singularities, star flows satisfy Axiom A with no cycles — the non-wandering set is a finite union of hyperbolic basic sets with no cycles among them — so the essential difficulty of the star class is the coexistence of singularities (zeros of the vector field) with regular orbits inside one indecomposable piece.

Morales, Pacifico and Pujals introduced singular hyperbolicity — a dominated splitting of the tangent bundle over the invariant set into a uniformly contracted bundle and a bundle that expands area (volume expansion in dimension 2 subspaces) containing the flow direction — and proved that robust transitive singular sets of 3-dimensional flows are singular hyperbolic; generic star flows on 3-manifolds have all chain recurrence classes hyperbolic or singular hyperbolic, and Gan, Shi and Wen extended this classification to generic star flows in dimension 4. Singular hyperbolicity, however, forces all singularities accumulated by regular orbits inside one class to have the same stable index (the number of contracting eigenvalues of the derivative at the singularity). Palis conjectured that every vector field can be approximated by a singular hyperbolic one or by one exhibiting a homoclinic tangency (a non-transverse intersection between the stable and unstable manifolds of a periodic orbit) or a heterodimensional cycle (a cycle between periodic orbits of different stable index); Crovisier and Yang proved this strong form in dimension 3, and Bonatti, Li and Yang had already built a robustly chain transitive attractor with singularities of different indices, hinting at the limit of the notion. da Luz then constructed an open set of star flows on a 5-manifold (product of a singular hyperbolic flow with a surface factor, modified by plugs and an isotopy) in which two hyperbolic singularities with different stable indices belong robustly to the same chain recurrence class. The center direction of this example has dimension at least 3, so no partial hyperbolicity of the tangent bundle can hold: this refutes Palis's strong density conjecture and the star-flow classification conjecture in all dimensions at least 5.

The replacement notion is multisingular hyperbolicity (Bonatti–da Luz), defined not on the tangent flow but on the linear Poincaré flow (the derivative cocycle restricted to the normal bundle of the flow direction over the set of regular orbits), reparametrized by cocycles associated with the singularities and considered over an extended invariant set that includes the singular directions. Crovisier, da Luz, Yang and Zhang gave an equivalent, more checkable formulation through singular domination — a dominated splitting structure for the linear Poincaré flow that is robust under perturbation — avoiding blow-ups and renormalization cocycles. The known implications are: multisingular hyperbolicity implies the star property (Bonatti–da Luz, Lemma 13); conversely, a C1C^1-open and dense subset of star flows has its chain recurrent set contained in finitely many filtrating regions (regions isolating pieces of the dynamics by a finite filtration of the manifold) on which the flow is multisingular hyperbolic (Bonatti–da Luz, Theorem 3), star flows whose periodic orbits are all robustly of the same index are multisingular hyperbolic — hence all 3-dimensional star flows are (Bohorquez, da Luz and Jaque) — and star flows with robust chain recurrence classes in any dimension are. da Luz's mixed-index example itself is multisingular hyperbolic. The gap that survives is precisely the genericity hypothesis: no result as of 2026 establishes, and no example refutes, multisingular hyperbolicity for every star flow in dimensions at least 4 when periodic orbits of different indices share a chain recurrence class that is not robust.

Problem Statement

Let MM be a closed manifold and XX a C1C^1 star vector field on MM (every periodic orbit of every C1C^1-near vector field is hyperbolic). Is every chain recurrence class of XX — every equivalence class of the relation that connects a point to itself by pseudo-orbits with arbitrarily small jumps — multisingular hyperbolic, in the sense of Bonatti–da Luz — equivalently, in the singular-domination formulation of Crovisier–da Luz–Yang–Zhang for the reparametrized linear Poincaré flow? The content of the question lies in dimensions at least 4 and outside the settled cases: it is known for a C1C^1-open and dense subset of star flows in all dimensions, for star flows all of whose periodic orbits are robustly of the same stable index (hence for all star flows on three-manifolds), and for star flows with robust chain recurrence classes. The remaining regime is a star flow, in dimension at least 4, in which periodic orbits or singularities of different stable indices coexist in a single chain recurrence class that is not robust. An affirmative answer, with proof, establishes the equivalence of the star property and multisingular hyperbolicity in all dimensions and removes the genericity hypothesis from Bonatti–da Luz's Theorem 3; a negative answer establishes that some star flow has a chain recurrence class that provably fails multisingular hyperbolicity (for example by failing singular domination of the linear Poincaré flow).

References

  1. Adriana da Luz, Star flows with singularities of different indices, arXiv:1806.09011 (v2, 2020), DOI 10.48550/arXiv.1806.09011, https://arxiv.org/abs/1806.09011
  2. Christian Bonatti, Adriana da Luz, Star flows and multisingular hyperbolicity, Journal of the European Mathematical Society 23 (2021), no. 8, 2649–2705, DOI 10.4171/jems/1064, https://doi.org/10.4171/jems/1064 (arXiv:1705.05799)
  3. Sylvain Crovisier, Adriana da Luz, Dawei Yang, Jinhua Zhang, On the notions of singular domination and (multi-)singular hyperbolicity, Science China Mathematics 63 (2020), no. 9, 1721–1744, DOI 10.1007/s11425-019-1764-x, https://doi.org/10.1007/s11425-019-1764-x (arXiv:2003.07099)
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  6. Yi Shi, Shaobo Gan, Lan Wen, On the singular-hyperbolicity of star flows, Journal of Modern Dynamics 8 (2014), 191–219, DOI 10.3934/jmd.2014.8.191, https://doi.org/10.3934/jmd.2014.8.191 (arXiv:1310.8118)
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  10. Christian Bonatti, Ming Li, Dawei Yang, A robustly chain transitive attractor with singularities of different indices, Journal of the Institute of Mathematics of Jussieu 12 (2013), 449–501, DOI 10.1017/S1474748012000710, https://doi.org/10.1017/S1474748012000710
  11. María José Pacifico, Fan Yang, Jiagang Yang, Rusong Zheng, Komuro Expansivity and Periodic Orbit Growth for Multi-Singular Hyperbolic Flows, arXiv:2608.02186 (2026), DOI 10.48550/arXiv.2608.02186, https://arxiv.org/abs/2608.02186
  12. Ming Li, Fan Yang, Jiagang Yang, Rusong Zheng, An example derived from Lorenz attractor, Ergodic Theory and Dynamical Systems 46 (2026), 1189–1218, DOI 10.1017/etds.2026.10275, https://doi.org/10.1017/etds.2026.10275 (arXiv:2302.02098)