Transversality in the setting of hyperbolic and parabolic maps
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GENADI LEVIN, WEIXIAO SHEN AND SEBASTIAN VAN STRIEN Dedicated to Lawrence Zalcman Abstract. In this paper we consider families of holomorphic maps defined on subsets of the complex plane, and show that the technique developed in [24] to treat unfolding of critical relations can also be used to deal with cases where the critical orbit converges to a hyperbolic attracting or a parabolic periodic orbit. As before this result applies to rather general families of maps, such as polynomial-like mappings, provided some lifting property holds. Our Main Theorem states that either the multiplier of a hyperbolic attracting periodic orbit depends univalently on the parameter and bifurcations at parabolic periodic points are generic, or one has persistency of periodic orbits with a fixed multiplier.
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Introduction
When studying families of maps defined on an open subset of the complex plane, it is useful to have certain transversality properties. For example, do multipliers of attracting periodic points depend univalently on the parameter and do parabolic periodic points undergo generic bifurcations? Building on a method developed in [24] we establish such transversality results in a very general setting. The conclusion of our Main Theorem states that one has either such transversality or persistency of periodic points with the same multiplier holds. The key assumption in our Main Theorem is a so-called lifting property defined in §2.2. It turns out that this assumption is applicable in rather general settings, including families of maps with an infinite number of singular values, such as polynomial-like mappings and also maps with essential singularities. Although the Main Theorem applies to complex maps, let us first mention applications to certain families of real maps. For example, consider the periodic doubling cascade associated to the family fλ = λx(1 − x),
x ∈ [0, 1], λ ∈ [0, 4]. 247
´ JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 141 (2020) DOI 10.1007/s11854-020-0130-7
248
G. LEVIN, W. SHEN AND S. VAN STRIEN
It is well-known that the multiplier κ(λ) of attracting periodic orbit decreases in λ diffeomorphically in each interval for which κ(λ) ∈ [−1, 1) and that one has generic bifurcations when κ(λ) = ±1. An application of our result is that the same conclusion holds for families of the form fλ (x) = λf (x) and similarly for gc (z) = g(z) + c where f and g are rather general interval maps. An important feature of our method is that we obtain information about the sign of the derivative of κ′ , namely that κ′ > 0; see §8. The main aim of this paper is to obtain results which apply to maps which are defined only locally, e.g., having essential singularities. It turns out that many more classical results can be recovered also. Before stating our results more formally, let us discuss previous results and the approach that is used in this paper. The study of the dependence of a multiplier on parameters goes back to Douady–Hubbard [10, 11], who obtained the celebrated result (using Sullivan’s quasiconformal surgery) that the
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