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QUASISYMMETRIC RIGIDITY in ONE-DIMENSIONAL DYNAMICS May 24, 2018 QUASISYMMETRIC RIGIDITY IN ONE-DIMENSIONAL DYNAMICS May 24, 2018 TREVOR CLARK AND SEBASTIAN VAN STRIEN Abstract. In the late 1980's Sullivan initiated a programme to prove quasisymmetric rigidity in one-dimensional dynamics: interval or circle maps that are topologically conju- gate are quasisymmetrically conjugate (provided some obvious necessary assumptions are satisfied). The aim of this paper is to conclude this programme in a natural class of C3 mappings. Examples of such rigidity were established previously, but not, for example, for real polynomials with non-real critical points. Our results are also new for analytic mappings. The main new ingredients of the proof in the real analytic case are (i) the existence of infinitely many (complex) domains associated to its complex analytic extension so that these domains and their ranges are compatible, (ii) a methodology for showing that combinatorially equivalent complex box mappings are qc conjugate, (iii) a methodology for constructing qc conjugacies in the presence of parabolic periodic points. For a C3 mapping, the dilatation of a high iterate of any complex extension of the real map will in general be unbounded. To deal with this, we introduce dynamically defined qcnbg partitions, where the appropriate mapping has bounded quasiconformal dilatation, except on sets with `bounded geometry'. To obtain such a partition we prove that we have very good geometric control for infinitely many dynamically defined domains. Some of these results are new even for real polynomials, and in fact an important sequence of domains turn out to be quasidiscs. This technology also gives a new method for dealing with the infinitely renormalizable case. We will briefly also discuss why quasisymmetric rigidity is such a useful property in one-dimensional dynamics. Contents 1. Introduction and statements of main results 2 1.1. Previous results 4 1.2. Applications of quasisymmetric ridigity 8 1.3. Necessity of assumptions 11 arXiv:1805.09284v1 [math.DS] 23 May 2018 1.4. Statement of the main theorems 11 2. Ingredients of the paper and a sketch of the proof 17 2.1. Old and new ingredients of this paper 17 2.2. The QC Criterion and qcnbg mappings. 22 2.3. Sketch of the proof and organization of the paper 23 2.4. A guide to the paper 27 Date: May 24, 2018. 1991 Mathematics Subject Classification. Primary 37E05; Secondary 30D05. The authors were supported by ERC AdG RGDD No 339523. Trevor Clark was supported in part by NSF EMSW21-RTG 1045119. The authors thank Lasse Rempe-Gillen, Daniel Smania and Davoud Cheraghi for several helpful comments, and would like to express their gratitude to Weixiao Shen for his many insightful remarks. 1 QUASISYMMETRIC RIGIDITY IN ONE-DIMENSIONAL DYNAMICS May 24, 2018 2 3. Preliminary definitions, remarks and some tools 28 3.1. Basic definitions and terminology 28 3.2. Quasiconformal and quasisymmetric maps 30 3.3. Asymptotically holomorphic mappings 31 3.4. Poincar´edisks 33 4. Real dynamics 36 4.1. A real partition 39 4.2. Renormalizable maps 41 4.3. A starting partition 42 4.4. A decomposition of the set of critical points 43 4.5. Real bounds, old and new 44 5. Compatible complex box mappings and complex bounds 50 5.1. Compatible complex box mappings and bounds at persistently recurrent points 51 5.2. Compatible complex box mappings and bounds at reluctantly recurrent points 55 5.3. Complex bounds for central cascades 57 5.4. Chains without long central cascades 66 5.5. The good nest and complex bounds 73 6. Dynamics away from critical points: Touching box mappings 76 6.1. Touching box mappings and petals at (possibly parabolic) periodic points 77 6.2. Angle control in the presence of parabolic points 81 6.3. Global touching box mappings 84 7. Combinatorial equivalence and external conjugacies 87 7.1. Existence of external conjugacies 88 7.2. Rigidity of non-renormalizable analytic complex box mappings 89 7.3. Restricting the puzzle to control dilatation 90 7.4. Statement of the QCnBG Partition of a Central Cascade 92 7.5. A Real Spreading Principle 94 7.6. Central cascades, the proof of a QCnBG Partition of a Central Cascade 102 8. Quasisymmetric rigidity 121 8.1. The immediate basin of attraction of a periodic attractor 122 8.2. The infinitely renormalizable, analytic case 124 8.3. The infinitely renormalizable, C3 case 126 8.4. The persistently recurrent, finitely renormalizable case 136 8.5. The reluctantly recurrent case 137 8.6. Conclusion 138 9. Table of notation and terminology 139 References 139 1. Introduction and statements of main results In this paper, we conclude a programme which was initiated by Sullivan in the late 1980's, see for example [Su1], [Su2] and [McM], namely that one has quasisymmetric rigidity in real one-dimensional dynamics. Around the same time, Herman asked a similar question for critical circle maps [He2]. Indeed we will prove that topologically conjugate smooth QUASISYMMETRIC RIGIDITY IN ONE-DIMENSIONAL DYNAMICS May 24, 2018 3 mappings satisfying certain hypotheses are quasisymmetrically conjugate, see Theorem A in Section 1.4. For real-analytic maps this result implies the following. Theorem 1.1 (QS-rigidity of real-analytic maps). Let M be either [0; 1] or S1 and suppose that f; f~: M ! M are real-analytic, have at least one critical point or periodic point, and are topologically conjugate. Moreover, assume that the topological conjugacy is a bijection between • the sets of parabolic periodic points, • the sets of critical points, and that the orders of corresponding critical points are the same. Then f and f~ are qua- sisymmetrically conjugate. In our approach to extending this result to C3 mappings f; f~: M ! M, where M is either [0; 1] or S1; we will make use of complex tools. For these to be applicable, it is essential to assume that at each critical point c of f, f(x) = ±(φ(x))` + f(c), where φ is a local C3 diffeomorphism, φ(c) = 0, and ` ≥ 2 is an integer, and likewise at each critical point of f~. Under these hypotheses, we have the following. Theorem 1.2 (QS-rigidy of C3 maps with no parabolic cycles). Suppose that f and f~ are C3 topologically conjugate mappings, as above, with no parabolic cycles. Assume that f and f~ each has at least one critical point or periodic point, that the conjugacy is a bijection between the sets of critical points of f and f~, and that corresponding critical points have the same orders. Then f and f~ are quasisymmetrically conjugate. Remark 1.1. The assumption in Theorem 1.2 that neither f nor f~ has parabolic cycles can be removed under some weak additional smoothness and genericity assumptions, see Theorem A. Remark 1.2. If f; f~: M ! M have periodic points, then we do not require that they have critical points. In particular, our theorems hold for interval maps and for covering maps of the circle of degree 6= 1. They also hold for maps f; f~: [0; 1] ! R; which are topologically conjugate on the non-escaping parts of their dynamics. For example, if f : R ! R satisfies jf(x)j=jxj ! 1 as jxj ! 1, then there exists an interval [a; b] so that f(fa; bg) ⊂ fa; bg and n limn!1 jf (x)j ! 1 for each x2 = [a; b]. Our theorems imply qs-rigidity of the non-escaping part of the dynamics. Remark 1.3. Note that the parabolic multiplicity of corresponding parabolic points, that is, the order of contact at the parabolic points, does not need to be the same for the two maps. Remark 1.4. For both analytic and smooth mappings, we obtain the quasisymmetric conju- gacy as restriction of a quasiconformal mapping of the plane. In the analytic case, the qc mapping is a conjugacy on a necklace neighbourhood of the real line; however, for smooth mappings this is not the case. Remark 1.5. Quasisymmetric mappings are H¨older, but they are not necessarily Lipschitz or even absolutely continuous. Indeed one can give examples of topologically conjugate mappings as in Theorem 1.1 for which no conjugacy satisfies these properties, see for exam- ple [BJ]. Moreover, it was proved in [MdM] that if two mappings with neither solenoidal nor wild attractors are conjugate by an absolutely continuous mapping, then they are C2 conjugate, so that any conjugacy which fails to preserve multipliers is singular with respect QUASISYMMETRIC RIGIDITY IN ONE-DIMENSIONAL DYNAMICS May 24, 2018 4 to Lebesgue measure. The condition that the mappings have neither solenoidal nor wild at- tractors is necessary. In [MoSm] it was proved that two Feigenbaum mappings with critical points of the same even order are always conjugate by an absolutely continuous mapping, and that provided that the degree is sufficiently high, the same is true for two Fibonacci mappings. The above theorems are essentially optimal: in Subsection 1.3 we will show that the conditions in these theorems are all necessary in general. In particular, if f is a map of the circle, then rigidity in full generality, that is, of all maps, requires that f has at least one critical point. 1.1. Previous results. There has been substantial work on quasisymmetric rigidity over the past three decades. 1.1.1. A summary of past results. Let us briefly summarize the previous results on qs-rigidity: • For polynomials, rigidity was, in general, only known for real polynomials with all critical points real. • For analytic mappings, rigidity was only known in some special cases, and often the results were semi-local. • For smooth mappings, rigidity was only known in a few very special cases.
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