Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A connecting lemma for rational maps satisfying a no growth condition   We introduce and study a non uniform hyperbolicity condition for complex
rational maps, that does not involve a growth condition. We call this condition
Backward Contraction. We show this condition is weaker than the Collet-Eckmann
condition, and than the summability condition with exponent~1.
  Our main result is a connecting lemma for Backward Contracting rational maps,
roughly saying that we can perturb a rational map to connect each critical
orbit in the Julia set with an orbit that does not accumulate on critical
points. The proof of this result is based on Thurston's algorithm and some
rigidity properties of quasi-conformal maps. We also prove that the Lebesgue
measure of the Julia set of a Backward Contracting rational map is zero, when
it is not the whole Riemann sphere. The basic tool of this article are sets
having a Markov property for backward iterates, that are holomorphic analogues
of nice intervals in real one-dimensional dynamics.

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The categories of this paper are: astro-ph
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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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: stat
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: nucl-th
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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: gr-qc
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The categories of this paper are: patt-sol
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The categories of this paper are: adap-org