, Volume 151, Issue 1, pp 29-63

Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps

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Abstract.

We show equivalence of several standard conditions for non-uniform hyperbolicity of complex rational functions, including the Topological Collet-Eckmann condition (TCE), Uniform Hyperbolicity on Periodic orbits, Exponential Shrinking of components of pre-images of small discs, backward Collet-Eckmann condition at one point, positivity of the infimum of Lyapunov exponents of finite invariant measures on the Julia set. The condition TCE is stated in purely topological terms, so we conclude that all these conditions are invariant under topological conjugacy.¶For rational maps with one critical point in Julia set all the conditions above are equivalent to the usual Collet-Eckmann and backward Collet-Eckmann conditions. Thus the latter ones are invariant by topological conjugacy in the unicritical setting. We also prove that neither part of this stronger statement is valid in the multicritical case.

Oblatum 2-IV-2002 & 2-V-2002¶Published online: 6 August 2002
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ID="*"All authors are supported by the European Science Foundation program PRODYN. The first author is also supported by the Foundation for Polish Sciences and Polish KBN grant 2P03A 00917. The second author is grateful to IMPAN and KTH for hospitality and is also supported by a Polish-French governmental agreement, Fundacion Andes and a “Beca Presidente de la Republica,” Chile. The third author is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation.