Chain recurrence classes versus homoclinic classes away from homoclinic tangencies and heterodimensional cycles ($C^1$-generic diffeomorphisms and nonsingular flows)

The weak Palis conjecture — Morse–Smale systems and systems exhibiting horseshoes are jointly dense among all systems — is known for $C^1$ diffeomorphisms (Crovisier, 2010) and for $C^1$ nonsingular flows (Xiao–Zheng, 2018), but only at the approximation level: chain recurrence classes away from homoclinic tangencies and heterodimensional cycles are Hausdorff limits of nontrivial homoclinic classes, and the known results do not establish that they are homoclinic classes themselves. This problem is the two-part question posed at the end of Xiao–Zheng's paper: (i) for a $C^1$-generic diffeomorphism away from homoclinic tangencies and heterodimensional cycles, is every nontrivial chain recurrence class a homoclinic class (not merely approximated by homoclinic classes)? Since Bonatti–Crovisier already showed that a generic chain class containing a periodic orbit is a homoclinic class, part (i) reduces to whether aperiodic chain classes can persist away from the two bifurcations. (ii) For $C^1$-generic vector fields, is every nontrivial nonsingular chain recurrence class away from homoclinic tangencies and heterodimensional cycles a homoclinic class? This is known in dimension 3 and for Lyapunov stable classes, and open for general nonsingular classes from dimension 4 up; the singular-flow extension of the weak Palis conjecture in dimension at least 4 is a related open program for which the central-model strategy alone is obstructed.

Background

A horseshoe is a compact invariant set on which a diffeomorphism or flow is semiconjugate to a shift, produced whenever the stable and unstable manifolds of a hyperbolic periodic orbit meet transversely (a transverse homoclinic point); it carries positive topological entropy and is the basic mechanism of deterministic chaos. At the opposite extreme, a Morse–Smale system has a chain recurrent set consisting of finitely many hyperbolic critical elements (periodic orbits, and for flows also singularities, i.e. zeros of the vector field) whose stable and unstable manifolds intersect transversely; on closed surfaces such systems are open and dense among $C^1$ vector fields. Palis (2000) conjectured that these two pictures exhaust typical dynamics in a weak sense: Morse–Smale systems and systems exhibiting horseshoes should be jointly dense among all systems (the weak Palis conjecture). For diffeomorphisms this was proved by Crovisier (2010) and for flows without singularities by Xiao–Zheng (2018).

The finer structure behind this conjecture is expressed through chain recurrence classes. Given $\varepsilon>0$, an $\varepsilon$-pseudo-orbit is a finite or infinite sequence of orbit jumps with errors smaller than $\varepsilon$; a point is chain recurrent if it can be chained back to itself, and the chain recurrent set decomposes into maximal chain-transitive pieces called chain recurrence classes, which for generic systems are the elementary blocks of the global dynamics. A homoclinic tangency is a nontransverse intersection of the stable and unstable manifolds of a periodic orbit. A heterodimensional cycle consists of two hyperbolic periodic orbits of different indices (dimensions of the stable bundle) whose stable and unstable manifolds cross in both directions. These are the two canonical mechanisms destroying uniform hyperbolicity; a system is 'away from' them when it admits a $C^1$-neighborhood none of whose elements exhibits either bifurcation (equivalently, it is not $C^1$-approximable by systems exhibiting them), an open condition. This is the standard reading of the hypothesis, fixed by Wen's characterization of diffeomorphisms that cannot be $C^1$-approximated by ones with homoclinic tangencies.

The homoclinic class $H(P)$ of a hyperbolic periodic orbit $P$ is the closure of the set of transverse intersection points of the stable and unstable manifolds of $P$; it is transitive and carries the horseshoe-type dynamics associated with $P$. For $C^1$-generic diffeomorphisms, Bonatti–Crovisier showed that homoclinic classes are themselves chain recurrence classes; consequently every chain class of a generic diffeomorphism that contains a periodic orbit is a homoclinic class, and a chain class that is not a homoclinic class contains no periodic orbit at all (an aperiodic class). Crovisier's central-model theorem further shows that aperiodic chain-transitive sets that are partially hyperbolic with one-dimensional center bundle (i.e. carry a dominated splitting with one uniformly contracting and one uniformly expanding direction around a one-dimensional central direction) are Hausdorff limits of nontrivial relative homoclinic classes — but these approximating classes may lie outside the given class, and an aperiodic class can never equal a homoclinic class since it contains no periodic orbit. For flows, the analogous statements are harder: perturbations of vector fields are constrained along orbit directions, so the pseudo-orbit connecting lemmas behind the diffeomorphism results do not transfer, and partial hyperbolicity for nonsingular flows is formulated through the linear Poincaré flow (the derivative flow projected to the normal bundle of the flow direction). Gan–Yang resolved the three-dimensional flow case, and Xiao–Zheng reduced the nonsingular case in any dimension to a dichotomy: each nontrivial nonsingular chain class of a generic vector field away from both bifurcations either is a homoclinic class or contains a minimal set partially hyperbolic with one-dimensional center for the linear Poincaré flow, and in the latter case the class is approximated by nontrivial homoclinic classes. Whether the second alternative can actually occur away from the bifurcations — i.e. whether the approximation can be upgraded to equality — is the open question.

Problem Statement

This is the Question posed at the end of the introduction of Xiao–Zheng (Discrete Contin. Dyn. Syst. 38 (2018), 1809–1832, DOI 10.3934/dcds.2018074), in two parts. Throughout, a system is 'away from' homoclinic tangencies and heterodimensional cycles if it admits a $C^1$-neighborhood none of whose elements exhibits either bifurcation — an open condition, equivalent to not being $C^1$-approximable by systems exhibiting them; this is the standard reading (Wen, Gan–Yang, Crovisier–Pujals) under which the known partial results are stated.

(i) Diffeomorphisms. Let $M$ be a compact boundaryless manifold and let $f$ belong to a residual (dense $G_\delta$) subset of the space $\mathrm{Diff}^1(M)$ of $C^1$ diffeomorphisms with the $C^1$ topology. Assume $f$ is away from homoclinic tangencies and heterodimensional cycles in the sense just defined. Is every nontrivial chain recurrence class of $f$ — every class not reduced to a periodic orbit — a homoclinic class $H(P,f)$ of some hyperbolic periodic orbit $P$, and not merely a Hausdorff limit of homoclinic classes? By Bonatti–Crovisier, every chain class of a generic diffeomorphism that contains a periodic orbit is already a homoclinic class, so part (i) is equivalent to: can such a generic diffeomorphism possess an aperiodic chain recurrence class (a chain class containing no periodic orbit)? A positive answer requires proving that aperiodic chain classes force homoclinic tangencies or heterodimensional cycles nearby; a negative answer requires constructing, generically inside an open set free of both bifurcations, a nontrivial chain class that is not a homoclinic class.

(ii) Nonsingular flows. Let $X$ belong to a residual subset of the space of $C^1$ vector fields on a compact boundaryless manifold. Is every nontrivial chain recurrence class of $X$ that contains no singularity of $X$ (a nonsingular class), under the assumption that $X$ is away from homoclinic tangencies and heterodimensional cycles in the same sense, a homoclinic class of a hyperbolic periodic orbit? This is known in dimension 3 (where the away-from-tangencies hypothesis suffices, by Gan–Yang) and for Lyapunov stable classes in any dimension (by Gan–Yang–Zheng); the question is open for general nonsingular classes in manifolds of dimension at least 4, where the dichotomy of Xiao–Zheng leaves open whether the aperiodic alternative (a minimal set partially hyperbolic with one-dimensional center for the linear Poincaré flow) can actually occur away from the two bifurcations.

References

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