Crovisier and Yang (2017) proved that every C1C^1 vector field on a compact three-dimensional manifold can be C1C^1-approximated either by a singular-hyperbolic vector field or by one exhibiting a homoclinic tangency associated to a hyperbolic periodic orbit, answering a conjecture of Palis. In the same paper they raise a stronger conjecture: the tangency alternative is unnecessary, i.e. every C1C^1 vector field on a compact 3-manifold can be approximated in the C1C^1 topology by singular-hyperbolic ones. The problem is to prove or refute this conjecture. The authors identify a key step (ruling out, for C1C^1-generic vector fields, non-trivial chain-recurrence classes containing a singularity with a complex eigenvalue), and a natural refutation would be a C1C^1 Newhouse domain for flows (a non-empty C1C^1-open set in which homoclinic tangencies are dense), of which no example is known in any regularity below C2C^2. Even the non-singular case of the conjecture is open and contains a flow analogue of Smale's conjecture on density of hyperbolicity.

Background

For a C1C^1 vector field XX on a compact boundaryless manifold MM with flow φ\varphi, the classical hyperbolic theory (Axiom A systems) describes the case where the non-wandering set splits into finitely many pieces on which the derivative contracts/expands uniformly in all directions, and Smale's spectral decomposition then organizes the dynamics into finitely many basic pieces. Two mechanisms destroy this picture. First, a homoclinic tangency — a non-transverse intersection of the stable and unstable manifolds of a hyperbolic periodic orbit — is a non-hyperbolic bifurcation; Newhouse (1970, 1979) showed that in CrC^r regularity with r≥2r\geq 2 there exist open sets (Newhouse domains) of surface diffeomorphisms in which homoclinic tangencies are persistent, so hyperbolicity is not CrC^r-dense; Newhouse domains also exist for three-dimensional flows in CrC^r, r≥2r\geq 2. The construction uses the thickness of Cantor sets, a tool that fails at C1C^1, and no example of a C1C^1 Newhouse domain is known for diffeomorphisms or flows. Second, equilibria (singularities) of a flow: hyperbolic fixed points have stable manifolds of dimension given by their index, and an invariant set containing singularities of different indices cannot carry a uniformly hyperbolic structure for the tangent flow. This is the mechanism of the Lorenz attractor: a robustly non-hyperbolic attractor containing an equilibrium. Morales, Pacifico and Pujals (2004) introduced singular hyperbolicity to capture such Lorenz-like dynamics: a compact invariant set Λ\Lambda is singular hyperbolic if (for XX or −X-X) the tangent bundle over Λ\Lambda splits into continuous invariant sub-bundles TM∣Λ=Es⊕EcuTM|_\Lambda = E^s \oplus E^{cu} such that EsE^s is uniformly contracted, EcuE^{cu} is sectionally expanding (every two-plane in EcuE^{cu} is uniformly area-expanding for the future flow), and the splitting is dominated (EsE^s is uniformly weaker than EcuE^{cu}). Away from singularities this reduces to ordinary hyperbolicity. A vector field is called singular hyperbolic if its chain-recurrent set (the set of points chained to themselves: yy is ε\varepsilon-chained to zz if finitely many orbit segments of size <ε<\varepsilon join them) is a finite union of singular-hyperbolic sets; this is an open condition in the C1C^1 topology and implies good topological and ergodic behavior (finiteness of attractors, spectral decomposition à la Smale). Palis (2000) proposed a global program: every dynamical system should be approximable either by hyperbolic-type systems with finitely many attractors or by systems exhibiting a homoclinic bifurcation (tangency or heterodimensional cycle). For surface diffeomorphisms, Pujals and Sambarino (2000) proved the dichotomy 'hyperbolicity or homoclinic tangency'; Arroyo and Rodriguez Hertz (2003) obtained the analogous statement for three-dimensional flows without singularities (with a singular-cycle alternative); Crovisier and Yang (2015 announcement; 2017 complete proof) settled the full three-dimensional case: any C1C^1 vector field on a compact 3-manifold is C1C^1-approximated either by singular-hyperbolic vector fields or by vector fields exhibiting a homoclinic tangency of a regular hyperbolic periodic orbit. The proof introduces a compactification of the rescaled sectional Poincaré flow (the linear Poincaré flow is the derivative cocycle restricted to a normal bundle transverse to the flow direction, which degenerates near singularities and must be rescaled) and a Mañé–Pujals–Sambarino-type contraction theorem for local fibered flows. The dichotomy, however, leaves the decisive question unanswered: since Newhouse domains are not known to exist in C1C^1, the tangency alternative may be an artifact of the method rather than a genuine obstruction, and one may hope that singular-hyperbolicity alone is C1C^1-dense in dimension three. The conjecture is closely tied to the structure of chain-recurrence classes (the equivalence classes of the chain-relation; a class is non-trivial if it contains more than one orbit) for C1C^1-generic vector fields: Crovisier and Yang note that the conjecture would follow if every chain-recurrence class of a C1C^1-generic field admitted a dominated splitting (a decomposition TM∣Λ=E⊕FTM|_\Lambda = E \oplus F invariant by the derivative cocycle with EE uniformly weaker than FF), and that an important sub-case is to exclude non-trivial classes containing a singularity with a complex (non-real) eigenvalue. The conjecture also asserts that, for typical 3-dimensional vector fields, non-trivial singular behavior occurs only inside Lorenz-like attractors and repellers. Even for vector fields without singularities the conjecture is open; it asks for the density of hyperbolicity and has a counterpart for surface diffeomorphisms sometimes called Smale's conjecture.

Problem Statement

Let MM be a compact boundaryless three-dimensional manifold and X1(M)\mathcal{X}^1(M) the space of C1C^1 vector fields on MM with the C1C^1 topology. Call X∈X1(M)X \in \mathcal{X}^1(M) singular hyperbolic if its chain-recurrent set is a finite union of singular-hyperbolic sets (an open condition in X1(M)\mathcal{X}^1(M)). Crovisier and Yang proved that any XX can be C1C^1-approximated by singular-hyperbolic fields or by fields exhibiting a homoclinic tangency associated to a regular hyperbolic periodic orbit. The problem is their conjecture: prove or refute that, for every compact 3-manifold MM, any vector field can be approximated in X1(M)\mathcal{X}^1(M) by singular-hyperbolic ones — i.e. that the tangency alternative in the Palis dichotomy can be entirely removed in the C1C^1 topology in dimension three. A refutation consists of a compact 3-manifold MM and a non-empty C1C^1-open set U⊂X1(M)\mathcal{U}\subset\mathcal{X}^1(M) containing no singular-hyperbolic vector field (the natural candidate being a Newhouse domain for flows: an open set in which homoclinic tangencies occur on a dense subset). Two intermediate milestones are explicitly identified by the authors: (i) show that for C1C^1-generic vector fields every chain-recurrence class admits a dominated splitting, from which the conjecture follows; and (ii) as an important sub-case, rule out for C1C^1-generic vector fields the existence of non-trivial chain-recurrence classes containing a singularity with a complex (non-real) eigenvalue. The conjecture would imply that, for typical 3-dimensional vector fields, non-trivial singular behavior occurs only inside Lorenz-like attractors and repellers; note that even its restriction to non-singular vector fields, which asks for the C1C^1-density of hyperbolicity (a flow counterpart of Smale's conjecture for surface diffeomorphisms), is open.

References

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