Hyperbolicity or heterodimensional cycles for general $C^1$ partially hyperbolic systems with one-dimensional center: removing the $C^2$ and nonzero-Lyapunov-exponent hypotheses

Hayashi (2007) proved a dichotomy for $C^2$ diffeomorphisms whose ergodic measures all have nonzero Lyapunov exponents, on compact invariant sets with a partially hyperbolic splitting and a one-dimensional center bundle: either the closure of the supports of the hyperbolic ergodic measures intersected with the set is a uniformly hyperbolic set, or arbitrarily $C^1$-small perturbations create a heterodimensional cycle localized near that region. He presented this as partial progress toward Palis's $C^1$ conjecture, explicitly limited by two hypotheses: $C^2$ differentiability and the absence of zero Lyapunov exponents. The problem is to remove both restrictions, i.e. to establish (or refute) the Palis-type alternative for general $C^1$ diffeomorphisms with partially hyperbolic invariant sets with one-dimensional center, including systems admitting ergodic measures with zero center Lyapunov exponents. In the later literature (Hayashi's own follow-ups, the Crovisier–Pujals generic program, Abdenur–Bonatti–Crovisier, and the 2020s work on nonhyperbolic measures and cycles), only generic, conservative, or approximate variants are known; the deterministic $C^1$ statement without the nonzero-exponent assumption remains open.

Background

A diffeomorphism $f$ of a compact manifold $M$ is uniformly hyperbolic (satisfies Axiom A) if its nonwandering set splits into finitely many transitive pieces on which the tangent bundle decomposes into uniformly contracting and expanding invariant directions; such systems are structurally stable and well understood, but they are not dense in the space $\mathrm{Diff}^r(M)$ of $C^r$ diffeomorphisms. Palis conjectured (in his 2005 global perspective on non-conservative dynamics) that hyperbolic diffeomorphisms, together with diffeomorphisms exhibiting a homoclinic tangency (a nontransverse intersection of the stable and unstable manifolds of a hyperbolic periodic point) or a heterodimensional cycle (a pair of hyperbolic periodic points of different stable dimensions — different indices — whose stable and unstable manifolds intersect cyclically, possible only in dimension at least three), are dense in $\mathrm{Diff}^r(M)$ for every $r\ge 1$; in the $C^1$ topology this is the Palis conjecture toward which the source paper works. The conjecture is known for surface diffeomorphisms (Pujals–Sambarino: $C^1$ density of hyperbolicity or homoclinic tangencies) and for maps of the interval/circle, and is open in dimension three and higher.

A compact $f$-invariant set $\Lambda$ is partially hyperbolic if the tangent bundle over $\Lambda$ admits a dominated splitting $TM|\Lambda = E^s\oplus E^c\oplus E^u$ (a splitting $A\oplus B$ is dominated if $A$ is exponentially stronger than $B$ under iteration, uniformly over the set), with $E^s$ uniformly contracting, $E^u$ uniformly expanding, and a center bundle $E^c$ that is intermediate; here $\dim E^c=1$ and $E^s,E^u\neq\{0\}$, which forces $\dim M\ge 3$. Nonuniform hyperbolicity is expressed through Lyapunov exponents — the exponential growth rates $\chi(\mu)=\lim_{n\to\infty}\frac{1}{n}\log\|Df^n v\|$ along Oseledets directions for an ergodic invariant probability measure $\mu$; $\mu$ is hyperbolic if none of its exponents vanish. The exponent along the one-dimensional center, $\chi^c(\mu)$, is the only exponent not controlled by partial hyperbolicity, and it is the mechanism by which nonhyperbolicity appears: hyperbolic ergodic measures with $\chi^c<0$ and $\chi^c>0$ can coexist on the same invariant set, giving periodic points of adjacent stable indices whose connection under small $C^1$ perturbations yields a heterodimensional cycle. The $C^1$ perturbation tools are Hayashi's connecting lemma and Mañé's ergodic closing lemma, which converts long recurrent orbit segments into periodic orbits.

Hayashi (2007) proved (Theorem A) that if $f$ is $C^2$ and every ergodic measure of $f$ has no zero Lyapunov exponents, then for such a set $\Lambda$, with $S(f)$ denoting the closure of the supports of the hyperbolic ergodic measures of $f$, the following dichotomy holds: $S(f)\cap\Lambda$ is a hyperbolic set, or, for every $C^1$ neighborhood $\mathcal{U}$ of $f$ and every neighborhood $U$ of $S(f)\cap\Lambda$, some $g\in\mathcal{U}$ has a heterodimensional cycle contained in $U$. The proof modifies a theorem of Mañé from his work on the $C^1$ stability conjecture — which produces periodic orbits with prescribed averaged contraction/expansion along an invariant subbundle, given a family of hyperbolic ergodic measures of common index over a set with dominated splitting and contracting $E$ — by adding the property that a substring of the periodic orbit is arbitrarily Hausdorff-close to the support of one of the measures (Theorem B), obtained however under a stronger hypothesis than Mañé's original statement.

Later work clarified the landscape but left the source's limitations untouched. Bonatti–Díaz–Pujals (2003) proved a $C^1$-generic dichotomy (weak forms of hyperbolicity or infinitely many sinks or sources); Bonatti–Díaz (2008) showed that $C^1$-generically, chain recurrence classes containing saddles of two distinct indices yield robust heterodimensional cycles by perturbation; Crovisier (2010) proved any diffeomorphism is $C^1$-approximated by Morse–Smale ones (finitely many hyperbolic periodic points whose stable and unstable manifolds meet transversally, so there is no other recurrence) or ones with a transverse homoclinic intersection (a weak Palis-type statement); Crovisier (2011) showed $C^1$-generic diffeomorphisms far from homoclinic bifurcations are essentially partially hyperbolic with center dimension at most two; Crovisier–Pujals (2015) proved any diffeomorphism is $C^1$-approximated by one with a homoclinic bifurcation or by an essentially hyperbolic one; Crovisier–Sambarino–Yang (2015) related partial hyperbolicity and tangencies; Abdenur–Bonatti–Crovisier (2011, Theorem 3.8) proved that for $C^1$-generic diffeomorphisms every ergodic measure is simultaneously approximated by periodic measures in the weak, Hausdorff, and Lyapunov-exponent senses. Hayashi himself extended the ergodic closing lemma (2010) and proved a $C^2$-generic trichotomy (2014): Axiom A without cycles, or Kupka–Smale diffeomorphisms (all periodic points hyperbolic and all intersections of their invariant manifolds transverse, a genericity condition) admitting zero Lyapunov exponents, or the $C^1$ creation of homoclinic bifurcations. All of these are generic, conservative, or approximation statements; none gives the deterministic $C^1$ dichotomy for partially hyperbolic sets with one-dimensional center, and none handles zero center exponents in that deterministic setting.

Problem Statement

Let $M$ be a compact boundaryless smooth manifold with $\dim M\ge 3$, let $f\in\mathrm{Diff}^1(M)$ be a $C^1$ diffeomorphism (not assumed $C^2$), and let $\Lambda\subset M$ be a compact $f$-invariant set admitting a partially hyperbolic splitting $TM|\Lambda=E^s\oplus E^c\oplus E^u$ with $\dim E^c=1$, $E^s$ uniformly contracting, $E^u$ uniformly expanding, and $E^s,E^u\neq\{0\}$. Let $S(f)$ be the closure of the supports of the hyperbolic ergodic $f$-invariant probability measures (ergodic measures none of whose Lyapunov exponents vanish). Prove, or refute by an explicit example, the following extension of Hayashi's Theorem A, in which the two hypotheses of the source — $C^2$ differentiability of $f$ and nonvanishing of all Lyapunov exponents of all ergodic measures of $f$ — are removed: at least one of the following alternatives holds: either $S(f)\cap\Lambda$ is a hyperbolic set (it admits a uniform hyperbolic splitting with no center direction), or for every $C^1$ neighborhood $\mathcal{U}$ of $f$ and every neighborhood $U$ of $S(f)\cap\Lambda$ there exists $g\in\mathcal{U}$ exhibiting a heterodimensional cycle contained in $U$ (two hyperbolic periodic points of different stable indices whose stable and unstable manifolds intersect cyclically, with full orbits in $U$). In particular, settle the case in which some ergodic measure of $f$ has zero center Lyapunov exponent: determine whether the presence of such zero-exponent measures still forces the hyperbolicity-or-cycle alternative above, or whether — as in Hayashi's $C^2$-generic trichotomy — the zero-exponent regime constitutes a genuine third alternative that must be part of the statement. Hayashi's Theorem A is presented in the source as a partial result toward Palis's $C^1$ density conjecture, and the present problem continues that direction in the one-dimensional-center partially hyperbolic setting; a resolution should make explicit which alternative holds for systems that are merely $C^1$ and may possess zero center exponents.

References

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