Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The geometric index and attractors of homeomorphisms of $\mathbb{R}^3$   In this paper we focus on compacta $K \subseteq \mathbb{R}^3$ which possess a
neighbourhood basis that consists of nested solid tori $T_i$. We call these
sets toroidal. In \cite{hecyo1} we defined the genus of a toroidal set as a
generalization of the classical notion of genus from knot theory. Here we
introduce the self-geometric index of a toroidal set $K$, which captures how
each torus $T_{i+1}$ winds inside the previous $T_i$. We use this index in
conjunction with the genus to approach the problem of whether a toroidal set
can be realized as an attractor for a flow or a homeomorphism of
$\mathbb{R}^3$. We obtain a complete characterization of those that can be
realized as attractors for flows and exhibit uncountable families of toroidal
sets that cannot be realized as attractors for homeomorphisms.

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The categories of this paper are: eess
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The categories of this paper are: nucl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: adap-org
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The categories of this paper are: plasm-ph
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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: quant-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-th
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex