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Anosov flows in dimension 3 from gluing building blocks with quasi-transverse boundary
Authors
Neige Paulet
Publication date
12 March 2024
Publisher
View
on
arXiv
Abstract
We prove a new result allowing to construct Anosov flows in dimension 3 by gluing building blocks. By a building block, we mean a compact 3-manifold with boundary
P
P
P
, equipped with a
C
1
C^1
C
1
vector field
X
X
X
, such that the maximal invariant set
β©
t
β
R
X
t
(
P
)
\cap_{t \in \mathbb{R}} X^t (P)
β©
t
β
R
β
X
t
(
P
)
is a saddle hyperbolic set, and the boundary
β
P
\partial P
β
P
is quasi-transverse to
X
X
X
, i.e. transverse except for a finite number of periodic orbits contained in
β
P
\partial P
β
P
. Our gluing theorem is a generalization of a recent result of F. B\'eguin, C. Bonatti, and B. Yu who only considered the case where the block does not contain attractors nor repellers, and the boundary
β
P
\partial P
β
P
is transverse to
X
X
X
. The quasi-transverse setting is much more natural. Indeed, our result can be seen as a counterpart of a theorem by Barbot and Fenley which roughly states that every 3-dimensional Anosov flow admits a canonical decomposition into building blocks (with quasi-transverse boundary). We will also show a number of applications of our theorem.Comment: 160 pages, 98 figure
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Last time updated on 28/09/2024