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The Livšic equation on differential forms over Anosov flows and applications

Slobodan N. Simić

TL;DR

The paper develops a Livšic-type theory for the Lie derivative $L_X$ of differential forms along an Anosov flow on a closed manifold, focusing on the solvability of $L_X\xi=\eta$ across form degrees. It proves a sharp existence and uniqueness result for intermediate degrees $2\le k \le n-2$ in the asymmetric codimension-one setting, and links the $L^2$-closure of the image of $L_X$ on $(n-1)$-forms to a geometric integrability condition: the sum of the strong bundles is uniquely integrable, which implies a topological conjugacy to a suspension of an Anosov diffeomorphism. The work uses elementary, non-mmicrolocal methods together with the Gol'dshtein–Troyanov complex to study regularization, cohomology, and duality, connecting dynamical rigidity to differential-form cohomology. The results provide a criterion tying dynamical geometric structure to the solvability of cohomological equations and to global cross-section-type behavior of the flow. The findings have implications for understanding when asymmetric Anosov flows admit suspension representations and illuminate the relationship between flow geometry and analytic properties of $L_X$ on differential forms.

Abstract

The goal of this paper is to explore the relationship between the geometric properties of an Anosov flow on a closed manifold $M$ and the analytic properties of its infinitesimal generator $X$ as a linear operator on the space of smooth differential forms of all degrees. In particular, we study the solvability of the Livšic equation $L_X ξ= η$ on the space of differential forms and show, for instance, that if the Anosov flow is \emph{asymmetric}, then the equation has a unique solution in the continuous category in degrees $2 \leq k \leq n-2$, where $n = \dim M$. Intuitively, an Anosov flow is asymmetric if in negative time it shrinks the volume of any $(n-2)$-dimensional parallelepiped exponentially fast when at least one side of it is in the strong unstable direction. As an application, we show that for volume-preserving asymmetric Anosov flows, the following result holds: the $L^2$-closure of the image of $L_X$ restricted to differential forms of degree $(n-1)$ contains the space of $L^2$-exact $(n-1)$-forms if and only if the sum of the strong bundles of the flow is uniquely integrable, in which case the flow is therefore topologically conjugate to a suspension of an Anosov diffeomorphism.

The Livšic equation on differential forms over Anosov flows and applications

TL;DR

The paper develops a Livšic-type theory for the Lie derivative of differential forms along an Anosov flow on a closed manifold, focusing on the solvability of across form degrees. It proves a sharp existence and uniqueness result for intermediate degrees in the asymmetric codimension-one setting, and links the -closure of the image of on -forms to a geometric integrability condition: the sum of the strong bundles is uniquely integrable, which implies a topological conjugacy to a suspension of an Anosov diffeomorphism. The work uses elementary, non-mmicrolocal methods together with the Gol'dshtein–Troyanov complex to study regularization, cohomology, and duality, connecting dynamical rigidity to differential-form cohomology. The results provide a criterion tying dynamical geometric structure to the solvability of cohomological equations and to global cross-section-type behavior of the flow. The findings have implications for understanding when asymmetric Anosov flows admit suspension representations and illuminate the relationship between flow geometry and analytic properties of on differential forms.

Abstract

The goal of this paper is to explore the relationship between the geometric properties of an Anosov flow on a closed manifold and the analytic properties of its infinitesimal generator as a linear operator on the space of smooth differential forms of all degrees. In particular, we study the solvability of the Livšic equation on the space of differential forms and show, for instance, that if the Anosov flow is \emph{asymmetric}, then the equation has a unique solution in the continuous category in degrees , where . Intuitively, an Anosov flow is asymmetric if in negative time it shrinks the volume of any -dimensional parallelepiped exponentially fast when at least one side of it is in the strong unstable direction. As an application, we show that for volume-preserving asymmetric Anosov flows, the following result holds: the -closure of the image of restricted to differential forms of degree contains the space of -exact -forms if and only if the sum of the strong bundles of the flow is uniquely integrable, in which case the flow is therefore topologically conjugate to a suspension of an Anosov diffeomorphism.

Paper Structure

This paper contains 12 sections, 12 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Anosov flows.
  4. Notation, standing assumptions, and facts.
  5. The case of codimension one Anosov flows.
  6. The $L^2$-structure on the space of differential forms.
  7. The Gol'dshtein-Troyanov complex
  8. A Livšic theorem on the space of differential forms
  9. Invariant forms.
  10. The action of $L_X$ in all degrees.
  11. Proof of the Main Theorem
  12. Acknowledgement.

Key Result

Proposition 2.1

Let $\Phi = \{ f_t \}$ be a volume preserving codimension one Anosov flow on a closed manifold $M$ of dimension $n \geq 4$. Assume, without loss, that $E^{uu}$ is 1-dimensional and orientable, and let $Y$ be a non-vanishing section of $E^{uu}$. Assume as before that with respect to a fixed Riemannia for all $t \geq 0$ and $v \in E^{ss}$, where $c, \nu > 0$ are as in $(\spadesuit)$. Then for every

Theorems & Definitions (23)

  • Definition 1.1
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Proposition 2.5
  • proof
  • Theorem 2.6: The regularization and homotopy operators
  • Lemma 2.7
  • ...and 13 more