Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Mean dimension and a sharp embedding theorem: extensions of aperiodic
  subshifts   We show that if $(X,T)$ is an extension of an aperiodic subshift (a subsystem
of $({1,2,...,l}^{\mathbb{Z}},\mathrm{shift})$ for some $l\in\mathbb{N}$) and
has mean dimension $mdim(X,T)<\frac{D}{2}$ $(D\in \mathbb{N}$), then it embeds
equivariantly in (([0,1]^{D})^{\mathbb{Z}},\mathrm{shift})$. The result is
sharp. If $(X,T)$ is an extension of an aperiodic zero-dimensional system then
it embeds equivariantly in $(([0,1]^{D+1})^{\mathbb{Z}},\mathrm{shift})$.

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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga
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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-ex
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The categories of this paper are: physics
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The categories of this paper are: chao-dyn
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The categories of this paper are: eess
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an