Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Classifying expanding attractors on figure eight knot complement space
  and non-transitive Anosov flows on Franks-Williams manifold   The path closure of figure eight knot complement space, $N_0$, supports a
natural DA (derived from Anosov) expanding attractor. Using this attractor,
Franks-Williams constructed the first example of non-transitive Anosov flow on
the manifold $M_0$ obtained by gluing two copies of $N_0$ through identity map
along their boundaries, named by Franks-Williams manifold. In this paper, our
main goal is to classify expanding attractors on $N_0$ and non-transitive
Anosov flows on $M_0$. We prove that, up to orbit-equivalence, the DA expanding
attractor is the unique expanding attractor supported by $N_0$, and the
non-transitive Anosov flow constructed by Franks and Williams is the unique
non-transitive Anosov flow admitted by $M_0$. Moreover, more general cases are
also discussed. In particular, we completely classify non-transitive Anosov
flows on a family of infinitely many toroidal $3$-manifolds with two hyperbolic
pieces, obtained by gluing two copies of $N_0$ through any gluing
homeomorphism.

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The categories of this paper are: patt-sol
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The categories of this paper are: plasm-ph
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The categories of this paper are: econ
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The categories of this paper are: hep-ex
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The categories of this paper are: cs
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The categories of this paper are: mtrl-th
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The categories of this paper are: supr-con
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The categories of this paper are: atom-ph
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The categories of this paper are: q-alg
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The categories of this paper are: stat
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: astro-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn