The $C^2$ Palis conjecture: eliminating zero Lyapunov exponents from the generic trichotomy for $C^2$ diffeomorphisms

Hayashi (Trans. Amer. Math. Soc., 2014) proved that a $C^2$-generic diffeomorphism of a compact manifold satisfies one of three alternatives: uniform hyperbolicity (Axiom A with no cycles); an ergodic measure with a zero Lyapunov exponent; or the creation of a homoclinic tangency or a heterodimensional cycle by a $C^1$-small perturbation. This trichotomy isolates exactly what still separates the state of the art from the $C^2$ case of the Palis density conjecture, which predicts that densely in $\mathrm{Diff}^r(M)$, $r \ge 1$, diffeomorphisms are either hyperbolic or exhibit homoclinic bifurcations. The problem asks whether the zero-Lyapunov-exponent alternative can be removed by $C^2$-small perturbations on a residual set and whether $C^1$-creation can be upgraded to $C^2$-creation, so that the Palis dichotomy holds $C^2$-generically.

Background

Let $M$ be a compact smooth manifold and let $\mathrm{Diff}^r(M)$ denote the space of $C^r$ diffeomorphisms of $M$ with the $C^r$ topology. A diffeomorphism is uniformly hyperbolic (satisfies Axiom A with the no-cycles condition) if its nonwandering set decomposes into finitely many disjoint invariant pieces on each of which the tangent bundle splits into $Df$-invariant subbundles that are uniformly contracted and expanded; since the work of Smale and Palis these systems are regarded as the elementary building blocks of dynamics. The two classical mechanisms by which hyperbolicity fails are: a homoclinic tangency, where the stable and unstable manifolds of the same periodic orbit meet non-transversally, and a heterodimensional cycle, where the stable and unstable manifolds of periodic orbits of different indices (different numbers of expanding directions) intersect cyclically, producing a mismatch of stable dimensions. The Palis density conjecture (Palis, 2005) asserts that for every $r \ge 1$, densely in $\mathrm{Diff}^r(M)$, diffeomorphisms are either hyperbolic or exhibit homoclinic tangencies or heterodimensional cycles: these two bifurcations should account for all of non-hyperbolicity on a dense set.

For $r=1$ a substantial but still partial theory exists. The $C^1$ connecting lemma (Hayashi, 1997), which creates true orbits from pseudo-orbits by $C^1$-small perturbations, and the generic recurrence structure of $C^1$-generic diffeomorphisms (Bonatti and Crovisier, 2004) together yield the weak form of the conjecture: a $C^1$-generic diffeomorphism whose chain recurrence classes carry no homoclinic tangency and no heterodimensional cycle is essentially hyperbolic (Crovisier, 2011). All of these arguments perturb only in the $C^1$ topology, and the toolbox genuinely fails at higher regularity: Gutierrez (1987) constructed a counterexample to the $C^2$ closing lemma, and the one $C^2$ connecting tool available, Mañé's $C^2$ connecting lemma (Mañé, 1982; see Hayashi, 2010), connects orbits only under measure-theoretic hypotheses on invariant measures and cannot drive genericity arguments. A Lyapunov exponent of $f$ for an $f$-invariant ergodic probability measure $\mu$ is an exponential rate $\chi = \lim_{n\to\infty} \frac{1}{n}\log\|Df^n v\|$ of contraction or expansion along orbits, guaranteed to exist for $\mu$-almost every point by Oseledets' theorem; an exponent equal to zero for some ergodic measure obstructs the uniform contraction/expansion splitting that hyperbolicity requires. A diffeomorphism is Kupka-Smale if all its periodic points are hyperbolic and all intersections of their invariant manifolds are transversal, a property that holds on a residual subset of $\mathrm{Diff}^r(M)$.

Hayashi (2014) proved the $C^2$ generic trichotomy: for a residual set of $C^2$ diffeomorphisms, either (a) $f$ is Axiom A with no cycles; or (b) $f$ is Kupka-Smale and some ergodic measure has a zero Lyapunov exponent; or (c) $f$ is Kupka-Smale and a $C^1$-small perturbation of $f$ creates a homoclinic tangency or a heterodimensional cycle. The theorem falls short of the $C^2$ Palis conjecture in two precise ways: the zero-exponent alternative (b) may be an irreducible third obstruction, and the bifurcation creation in (c) is by $C^1$ rather than $C^2$ perturbations (a $C^1$-small perturbation of a $C^2$ diffeomorphism need not be $C^2$-close). Whether either gap can be closed is open; Hayashi identifies the absence of a general $C^2$ connecting lemma and of $C^2$ general density theorems (density of periodic points in limit sets) as the obstacle to running the $C^1$ genericity scheme inside $\mathrm{Diff}^2(M)$.

Problem Statement

Let $M$ be a compact smooth manifold and let $\mathrm{Diff}^2(M)$ be the space of $C^2$ diffeomorphisms of $M$ with the $C^2$ topology. The problem is the $C^2$ case of the Palis density conjecture, as isolated by Hayashi's $C^2$ generic trichotomy:

(1) Is there a residual subset $\mathcal{R}$ of $\mathrm{Diff}^2(M)$ such that every $f \in \mathcal{R}$ satisfies the dichotomy: either $f$ is uniformly hyperbolic (Axiom A with the no-cycles condition), or in every $C^2$-neighborhood of $f$ there is a diffeomorphism exhibiting a homoclinic tangency or a heterodimensional cycle?

In the formulation inherited from the trichotomy, (1) asks jointly that the alternative 'some ergodic measure has a zero Lyapunov exponent' be removed $C^2$-generically and that the $C^1$-creation of a homoclinic tangency or a heterodimensional cycle be upgraded to creation by $C^2$-small perturbations.

A solution is a proof or refutation of (1). The problem retains its full generality: it concerns all compact manifolds and the whole space $\mathrm{Diff}^2(M)$, with no restriction on dimension, class of examples, or answer form.

References

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