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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2511.05678 |
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| _version_ | 1866917069144981504 |
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| author | Simić, Slobodan N. |
| author_facet | Simić, Slobodan N. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_05678 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Livšic equation on differential forms over Anosov flows and applications Simić, Slobodan N. Dynamical Systems 37C05, 58A10 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. |
| title | The Livšic equation on differential forms over Anosov flows and applications |
| topic | Dynamical Systems 37C05, 58A10 |
| url | https://arxiv.org/abs/2511.05678 |