Extremal distributions of partially hyperbolic systems: the Lipschitz threshold

Abstract: We prove a sharp phase transition in the regularity of the extremal distribution $E^s \oplus E^u$ for $C^\infty$ volume-preserving partially hyperbolic diffeomorphisms on closed $3$-manifolds: if $E^s \oplus E^u$ is Lipschitz, then it is automatically $C^\infty$. This extends the rigidity phenomenon established by Foulon--Hasselblatt for conservative Anosov flows in dimension $3$ to the partially hyperbolic setting. This gain in regularity has several applications to rigidity problems. In particular, we study the relationship between the $\ell$-integrability condition introduced by Eskin--Potrie--Zhang and joint integrability in the conservative setting, yielding rigidity results for $u$-Gibbs measures. We also obtain a $C^\infty$ classification of $3$-dimensional conservative partially hyperbolic diffeomorphisms with Lipschitz distributions, thereby answering a question of Carrasco--Hertz--Pujals in the conservative setting under minimal regularity assumptions.

• The paper establishes that if the extremal distribution E^s ⊕ E^u is Lipschitz, it is necessarily C∞, marking a sharp phase transition in regularity.
• It employs advanced cocycle rigidity and bootstrapping techniques to lift Lipschitz regularity to C∞, addressing both dynamical and cohomological aspects.
• The results lead to a full C∞ classification for conservative partially hyperbolic diffeomorphisms on closed 3-manifolds, resolving several longstanding rigidity conjectures.

Extremal Distributions in Partially Hyperbolic Systems and the Lipschitz Regularity Threshold

Introduction

This work addresses the regularity properties of the extremal distribution $E^s \oplus E^u$ in $C^\infty$ volume-preserving partially hyperbolic diffeomorphisms on closed $3$-manifolds. It establishes a rigidity phenomenon at the Lipschitz regularity level: if $E^s \oplus E^u$ is Lipschitz, then it must be $C^\infty$. This result significantly extends prior rigidity theorems known for conservative Anosov flows and unifies several lines of inquiry in the rigidity and classification of partially hyperbolic dynamics.

Statement of Main Results

Sharp Regularity Threshold

The central theorem demonstrates a sharp phase transition in the regularity of $E^s \oplus E^u$. Specifically, for $C^\infty$ volume-preserving partially hyperbolic diffeomorphisms $f$ of closed $3$-manifolds, the following dichotomy holds:

• If $E^s \oplus E^u$ is Lipschitz, then it is in fact $C^\infty$0.
• The Lipschitz property of $C^\infty$1 cannot occur unless the system displays the highest possible smooth rigidity in the context of partially hyperbolic diffeomorphisms.

This extends the Foulon–Hasselblatt rigidity results for Anosov flows to the broader context of partially hyperbolic diffeomorphisms.

Dynamical and Cohomological Consequences

Under the Lipschitz regularity of $C^\infty$2, one has:

• Dynamical coherence holds automatically.
• The center cocycle $C^\infty$3 is cohomologous to a constant. If the constant differs from $C^\infty$4, $C^\infty$5 must be an Anosov diffeomorphism for which $C^\infty$6 is integrable.
• There is a dichotomy: either $C^\infty$7 is integrable, or $C^\infty$8 is accessible and preserves a smooth contact form, and is $C^\infty$9-conjugate (up to finite cover and iterate) to either an isometric extension of an Anosov diffeomorphism or the time-one map of an Anosov contact flow.

Classification Consequences

When $3$0 is Lipschitz and the center foliation is absolutely continuous, a full $3$1 classification is obtained: $3$2 is, up to finite data, $3$3-conjugate to an Anosov automorphism of $3$4, an isometric extension over $3$5, or the time-one map of an Anosov flow, generalizing and resolving the question raised by Carrasco–Hertz–Pujals in the conservative category.

Methods and Key Arguments

Foulon–Hasselblatt Cocycle Construction

The rigidity mechanism is encoded in an invariant cocycle generalizing the Foulon–Hasselblatt longitudinal KAM cocycle to the discrete-time and partially hyperbolic context. This cocycle, constructed using adapted charts and polynomial normal forms, produces an obstruction to enhanced regularity. If it vanishes (i.e., is a twisted coboundary), Lipschitz regularity escalates to $3$6.

Bootstrapping Regularity

The argument proceeds by:

1. Proving that Lipschitz bundles can be promoted to $3$7, then to $3$8 regularity along the accessibility class (using cohomological equations and the strong bunching condition).
2. Utilizing the precise structure of ergodic accessibility classes and the classification of $3$9-manifold partially hyperbolic diffeomorphisms.
3. Applying cocycle rigidity techniques (Livshits Theorem and Journé’s regularity lemma) to yield global smoothness.
4. Employing contact geometric arguments: the dichotomy between integrability (foliated by $E^s \oplus E^u$0-tori) and the existence of a contact structure with center direction as Reeb vector field; this last case can be ruled out unless $E^s \oplus E^u$1 is a special model case.

Measure Rigidity Applications

The paper interprets regularity threshold results in the context of $E^s \oplus E^u$2-Gibbs states and their supports.

• $E^s \oplus E^u$3-integrability for $E^s \oplus E^u$4 (surfaces with $E^s \oplus E^u$5 submanifolds tangent to $E^s \oplus E^u$6) forces Lipschitz regularity, and thus smoothness.
• A complete equivalence: $E^s \oplus E^u$7 is $E^s \oplus E^u$8-integrable for some $E^s \oplus E^u$9 if and only if $C^\infty$0 is integrable.
• This governs the geometric and ergodic properties of $C^\infty$1-Gibbs states: if $C^\infty$2 is not integrable, any $C^\infty$3-Gibbs state with center exponent $C^\infty$4 and positive volume support is the normalized volume on an open accessibility class and satisfies the QNI condition.

Discussion and Implications

Phase Transition and Rigidity

This work demonstrates a phase transition in regularity at the Lipschitz threshold, confirming that ‘extremal’ distributions in partially hyperbolic systems are only Lipschitz in highly rigid, algebraically structured models. This answers foundational rigidity questions and provides optimal regularity thresholds.

The results also demonstrate that the pathological or ‘exotic’ examples constructed in previous works cannot realize Lipschitz (let alone smooth) $C^\infty$5 distributions unless they are smoothly reducible to known algebraic models.

Structural and Ergodic Implications

These rigidity results have several structural implications:

• The dichotomy between accessibility and joint integrability supports a refined measure-theoretic classification of partially hyperbolic systems.
• The equivalence of $C^\infty$6-integrability and integrability of $C^\infty$7 sharpens the connection between smooth dynamical features and the statistical properties of invariant measures.
• The vanishing of the center Lyapunov exponent on the closure of each open accessibility class is shown to be a universal feature under the Lipschitz assumption.

Extensions and Limitations

While the regularity threshold is established as optimal (examples exist that are $C^\infty$8 for all $C^\infty$9 but not Lipschitz), in situations with additional hyperbolicity or bunching, lower regularity already forces smoothness.

Examples are provided throughout the text illustrating the indispensability of both the Lipschitz regularity assumption for $E^s \oplus E^u$0 and absolute continuity for the center foliation in the classification results.

Future Directions

Potential directions for further research include:

• Generalization to higher dimensions where the topology of the manifold and the partially hyperbolic splitting is more intricate.
• Exploration of other rigidity thresholds for intermediate bundles and for systems with weaker bunching conditions.
• Application of these structural results to spectral and statistical properties of the dynamics (e.g., decay of correlations, classification of physical measures).
• Further elucidation of the role played by accessibility and its interplay with topological obstructions (e.g., non-solvable fundamental group).

Conclusion

This work establishes a sharp and optimal rigidity threshold for the regularity of the extremal distribution in conservative partially hyperbolic diffeomorphisms of $E^s \oplus E^u$1-manifolds. The Lipschitz property of $E^s \oplus E^u$2 is demonstrated to be highly constraining, ensuring full $E^s \oplus E^u$3 regularity and robustly restricting the global dynamics to known algebraic or geometric models. Correspondingly, it provides comprehensive answers to long-standing conjectures in the smooth classification theory of partially hyperbolic systems, with significant implications for ergodic and geometric rigidity phenomena.