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We prove that cubic polynomial maps with a fixed Siegel disk and a critical orbit eventually landing inside that Siegel disk lie in the support of the bifurcation measure. This answers a question of Dujardin in positive. Our result implies the existence of holomorphic disks in the support of the bifurcation measure, and also implies that the set of rigid parameters is not closed in the moduli space of cubic polynomials.

In this paper we build a geometric model for the renormalisation of irrationally indifferent fixed points. The geometric model incorporates the fine arithmetic properties of the rotation number at the fixed point. Using this model for the renormalisation, we build a topological model for the dynamics of a holomorphic map near an irrationally indifferent fixed point. Then, we explain the topology of the maximal invariant set for the model, and also explain the dynamics of the map on the maximal invariant set., Comment: 73 pages, this article is the first part of the article arXiv:1706.02678, which has been extended to a complete model for the renormalisation. The second part of arXiv:1706.02678 will appear under that title with an updated version. Latest version: Minor modifications following referee comments

We prove that for a dense set of irrational numbers $\alpha$, the analytic centraliser of the map $e^{2\pi i \alpha} z+ z^2$ near $0$ is trivial. We also prove that some analytic circle diffeomorphisms in the Arnold family, with irrational rotation numbers, have trivial centralisers. These provide the first examples of such maps with trivial centralisers., Comment: 13 Pages, No figures

In this paper we study the geometry of the attractors of holomorphic maps with an irrationally indifferent fixed point. We prove that for an open set of such holomorphic systems, the local attractor at the fixed point has Hausdorff dimension two, provided the asymptotic rotation at the fixed point is of sufficiently high type and does not belong to Herman numbers. As an immediate corollary, the Hausdorff dimension of the Julia set of any such rational map with a Cremer fixed point is equal to two. Moreover, we show that for a class of asymptotic rotation numbers, the attractor satisfies Karpi\'{n}ska's dimension paradox. That is, the the set of end points of the attractor has dimension two, but without those end points, the dimension drops to one., Comment: 42 pages, 4 figures

We construct transcendental automorphims of $\mathbb{C}^{2}$ having an unbounded and regular Siegel domain., Comment: 10 pages, no figures

We introduce a topological object, called hairy Cantor set, which in many ways enjoys the universal features of objects like Jordan curve, Cantor set, Cantor bouquet, hairy Jordan curve, etc. We give an axiomatic characterisation of hairy Cantor sets, and prove that any two such objects in the plane are ambiently homeomorphic. Hairy Cantor sets appear in the study of the dynamics of holomorphic maps with infinitely many renormalisation structures. They are employed to link the fundamental concepts of polynomial-like renormalisation by Douady-Hubbard with the arithmetic conditions obtained by Herman-Yoccoz in the study of the dynamics of analytic circle diffeomorphisms., Comment: 40 pages, 4 figures

We study the post-critical set of a class of holomorphic systems with an irrationally indifferent fixed point. We prove a trichotomy for the topology of the post-critical set based on the arithmetic of the rotation number at the fixed point. The only options are Jordan curves, a one-sided hairy Jordan curves, and Cantor bouquet. This explains the degeneration of the closed invariant curves inside the Siegel disks, as one varies the rotation number., Comment: 58 pages, 9 figures. The original preprint under this tile is now divided into two papers. The first part of the paper on the model now appears as arXiv:2112.14557

We prove the uniform hyperbolicity of the near-parabolic renormalization operators acting on an infinite-dimensional space of holomorphic transformations. This implies the universality of the scaling laws, conjectured by physicists in the 70's, for a combinatorial class of bifurcations. Through near-parabolic renormalizations the polynomial-like renormalizations of satellite type are successfully studied here for the first time, and new techniques are introduced to analyze the fine-scale dynamical features of maps with such infinite renormalization structures. In particular, we confirm the rigidity conjecture under a quadratic growth condition on the combinatorics. The class of maps addressed in the paper includes infinitely-renormalizable maps with degenerating geometries at small scales (lack of a priori bounds)., Comment: 71 pages, comments welcome

We describe the statistical properties of the dynamics of the quadratic polynomials P_a(z):=e^{2\pi a i} z+z^2 on the complex plane, with a of high return times. In particular, we show that these maps are uniquely ergodic on their measure theoretic attractors, and the unique invariant probability is a physical measure describing the statistical behavior of typical orbits in the Julia set. This confirms a conjecture of Perez-Marco on the unique ergodicity of hedgehog dynamics, in this class of maps., Comment: 58 pages; pre-publication version

Marmi Moussa and Yoccoz conjectured that some error function Upsilon, related to the approximation of the size of Siegel disk by some arithmetic function of the rotation number theta, is a Holder continuous function of theta with exponent 1/2. Using the renormalization invariant class of Inou and Shishikura, we prove this conjecture for the restriction of Upsilon to a class of high type numbers., Comment: 71 pages, pre publication version

We describe the topological behavior of typical orbits of complex quadratic polynomials P_alpha(z)=e^{2\pi i alpha} z+z^2, with alpha of high return type. Here we prove that for such Brjuno values of alpha the closure of the critical orbit, which is the measure theoretic attractor of the map, has zero area. Then combining with Part I of this work, we show that the limit set of the orbit of a typical point in the Julia set is equal to the closure of the critical orbit., Comment: 38 pages, 5 figures; fixed the issues with processing the figures

We investigate the quantitative and analytic aspects of the near-parabolic renormalization scheme introduced by Inou and Shishikura in 2006. These provide techniques to study the dynamics of some holomorphic maps of the form $f(z) = e^{2\pi i \alpha} z + \mathcal{O}(z^2)$, including the quadratic polynomials $e^{2\pi i \alpha} z+z^2$, for some irrational values of $\alpha$. The main results of the paper concern fine-scale features of the measure-theoretic attractors of these maps, and their dependence on the data. As a bi-product, we establish an optimal upper bound on the size of the maximal linearization domain in terms of the Siegel-Brjuno-Yoccoz series of $\alpha$., Comment: 80 pages; Corrections have been made according to referee reports. A new statement is added to the paper

We prove combinatorial rigidity of infinitely renormalizable unicritical polynomials, P_c :z \mapsto z^d+c, with complex c, under the a priori bounds and a certain "combinatorial condition". This implies the local connectivity of the connectedness loci (the Mandelbrot set when d = 2) at the corresponding parameters., Comment: 37 Pages; 7 figures; pre-publication version