Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extremal ergodic measures and the finiteness property of matrix
  semigroups   Let $\bS=\{S_1,...,S_K\}$ be a finite set of complex $d\times d$ matrices and
$\varSigma_{K}^+$ the compact space of all one-sided infinite sequences
$i_{\bcdot}\colon\mathbb{N}\rightarrow\{1,...,K\}$. An ergodic probability
$\mu_*$ of the Markov shift
$\theta\colon\varSigma_{K}^+\rightarrow\varSigma_{K}^+;\ i_{\bcdot}\mapsto
i_{\bcdot+1}$, is called "extremal" for $\bS$, if
${\rho}(\bS)=\lim_{n\to\infty}\sqrt[n]{\norm{S_{i_1}...S_{i_n}}}$ holds for
$\mu_*$-a.e. $i_{\bcdot}\in\varSigma_{K}^+$, where $\rho(\bS)$ denotes the
generalized/joint spectral radius of $\bS$. Using extremal norm and Kingman
subadditive ergodic theorem, it is shown that $\bS$ has the spectral finiteness
property (i.e. $\rho(\bS)=\sqrt[n]{\rho(S_{i_1}...S_{i_n})}$ for some
finite-length word $(i_1,...,i_n)$) if and only if for some extremal measure
$\mu_*$ of $\bS$, it has at least one periodic density point
$i_{\bcdot}\in\varSigma_{K}^+$.

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The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
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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: alg-geom
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The categories of this paper are: math-ph