A Priori Bounds for Some Infinitely Renormalizable Quadratics, III. Molecules

Tan, Lei; Yin, Ya‐Xia

doi:10.1201/b10617-7

“…We should emphasise that complex bounds do not hold in general for complex maps; there are infinitely renormalizable complex quadratic maps for which complex bounds are known to fail, see for example [Le2]. However, they do hold for unicritical maps with certain combinatorics, see [Ka,KL3,KL4]. For an early work on non-renormalizable mappings with specific combinatorics, where the shape of puzzle pieces was well controlled we draw the readers attention to [Sm3].…”

Section: Previous Results On Complex Boundsmentioning

confidence: 99%

Complex Bounds for Real Maps

Clark

¹

,

Strien

²

,

Trejo

³

2017

Commun. Math. Phys.

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Abstract:In this paper we prove complex bounds, also referred to as a priori bounds for C 3 , and, in particular, for analytic maps of the interval. Any C 3 mapping of the interval has an asymptotically holomorphic extension to a neighbourhood of the interval. We associate to such a map, a complex box mapping, which provides a kind of Markov structure for the dynamics. Moreover, we prove universal geometric bounds on the shape of the domains and on the moduli between components of the range and domain. Such bounds show that the first return maps to these domains are well controlled, and consequently such bounds form one of the corner stones in many recent results in one-dimensional dynamics, for example: renormalization theory, rigidity, density of hyperbolicity, and local connectivity of Julia sets.

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“…In [Kah06], it is proved for infinitely primitively renormalizable maps of bounded type. In [KL08], it is proved for parameters satisfying a decorations condition and in [KL09b], under a molecule condition.…”

Section: Cheraghimentioning

confidence: 99%

“…In [Kah06], it is proved for infinitely primitively renormalizable maps of bounded type. In [KL08], it is proved for parameters satisfying a decorations condition and in [KL09b], under a molecule condition. Here we prove that the a priori bounds property, under the SL condition, implies the combinatorial rigidity conjecture for infinitely renormalizable maps.…”

Section: Cheraghimentioning

confidence: 99%

Combinatorial rigidity for some infinitely renormalizable unicritical polynomials

Cheraghi

¹

2010

Conform. Geom. Dyn.

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“…Examples of unicritical polynomials having a priori bounds include at most finitely renormalizable polynomials without neutral cycles [4,6,7], all real polynomials [10,11], and some infinitely renormalizable complex polynomials [5,12]. Recall that for an integer s ≥ 1, a unicritical polynomial f of degree d ≥ 2 is renormalizable of period s, if there are Jordan disks U and V such that U is contained in V , and such that the following properties hold:…”

Section: Definitionmentioning

confidence: 99%