Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Joint spectral radius, Sturmian measures, and the finiteness conjecture   The joint spectral radius of a pair of 2x2 real matrices $(A_0,A_1)\in
M_2(\mathbb{R})^2$ is defined to be $r(A_0,A_1)= \limsup_{n\to\infty} \max
\{\|A_{i_1}...A_{i_n}\|^{1/n}: i_j\in\{0,1\}\}$, the optimal growth rate of the
norm of products of these matrices. The Lagarias-Wang finiteness conjecture,
asserting that $r(A_0,A_1)$ is always the nth root of the spectral radius of
some length-n product $A_{i_1}...A_{i_n}$, has been refuted by Bousch &
Mairesse, with subsequent counterexamples presented by Blondel, Theys &
Vladimirov; Kozyakin; Hare, Morris, Sidorov & Theys. In this article we
introduce a new approach to generating finiteness counterexamples, and use this
to exhibit an open subset of $M_2(\mathbb{R})^2$ with the property that each
member $(A_0,A_1)$ of the subset generates uncountably many counterexamples of
the form $(A_0, tA_1)$. Our methods employ ergodic theory, in particular the
analysis of Sturmian invariant measures; this approach allows a short proof
that the relation between the parameter $t$ and the Sturmian parameter $P(t)$
is a devil's staircase.

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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
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The categories of this paper are: physics
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: patt-sol
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The categories of this paper are: astro-ph