Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence and uniqueness of global Koopman eigenfunctions for stable
  fixed points and periodic orbits   We consider $C^1$ dynamical systems having an attracting hyperbolic fixed
point or periodic orbit and prove existence and uniqueness results for $C^k$
(actually $C^{k,\alpha}_{\text{loc}}$) linearizing semiconjugacies -- of which
Koopman eigenfunctions are a special case -- defined on the entire basin of
attraction. Our main results both generalize and sharpen Sternberg's $C^k$
linearization theorem for hyperbolic sinks, and in particular our corollaries
include uniqueness statements for Sternberg linearizations and Floquet normal
forms. Using our main results we also prove new existence and uniqueness
statements for $C^k$ Koopman eigenfunctions, including a complete
classification of $C^\infty$ eigenfunctions assuming a $C^\infty$ dynamical
system with semisimple and nonresonant linearization. We give an intrinsic
definition of "principal Koopman eigenfunctions" which generalizes the
definition of Mohr and Mezi\'{c} for linear systems, and which includes the
notions of "isostables" and "isostable coordinates" appearing in work by
Ermentrout, Mauroy, Mezi\'{c}, Moehlis, Wilson, and others. Our main results
yield existence and uniqueness theorems for the principal eigenfunctions and
isostable coordinates and also show, e.g., that the (a priori non-unique)
"pullback algebra" defined in \cite{mohr2016koopman} is unique under certain
conditions. We also discuss the limit used to define the "faster" isostable
coordinates in \cite{wilson2018greater,monga2019phase} in light of our main
results.

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The categories of this paper are: nlin
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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: patt-sol
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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
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The categories of this paper are: supr-con
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The categories of this paper are: stat
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: physics
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The categories of this paper are: cs
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The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph