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On non-uniform hyperbolicity assumptions in one-dimensional dynamics. (English) Zbl 1219.37032

In this paper, the authors study the relations between the widely adopted non-uniform hyperbolicity conditions and a new different type of non-uniform hyperbolicity condition which is usually called backward contraction condition. For complex polynomials of degree at least two, which are at most finitely renormalizable and have only hyperbolic periodic points, they give an essentially equivalent formulation of the backward contracting property in terms of expansion along the orbits of critical values. The conclusion also holds for all \(C^3\) interval maps with non-flat critical points.

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37E05 Dynamical systems involving maps of the interval
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