## Publications

12 results found

Cheraghi D, Avila A, Eliad A, 2020, Analytic maps of parabolic and elliptic type with trivial centralisers, Publisher: arXiv

We prove that for a dense set of irrational numbers α, the analytic centraliser of the map e^{2πiα} z+ z2 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.

Cheraghi D, De Zotti A, Yang F, 2020, Dimension paradox of irrationally indifferent attractors, Publisher: arXiv

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 Karpińska'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.

Cheraghi D, Berteloot F, 2020, Lacunary series, resonances, and automorphisms of ℂ^2 with a round Siegel domain, Publisher: ARxIV

We construct transcendental automorphims of ℂ^2 having an unbounded and regular Siegel domain.

Cheraghi D, Pedramfar M, 2019, Hairy cantor sets, Publisher: arXiv

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.

Cheraghi D, 2019, Typical orbits of quadratic polynomials with a neutral fixed point I: non-Brjuno type, *Annales Scientifiques de l'Ecole Normale Superieure*, Vol: 52, Pages: 59-138, ISSN: 0012-9593

We study (Lebesgue) typical orbits of quadratic polynomials $P_a(z)=e^{2\pia} z+z^2: C -> C$, with $a$ of non-Brjuno and high return type. This includesquadratic polynomials with positive area Julia set of X. Buff and A. Cheratat.As a consequence, we introduce rational maps of arbitrarily large degree forwhich the Brjuno condition is optimal for their linearizability. Our techniqueuses the near-parabolic renormalization developed by H. Inou and M. Shishikura.

Avila A, Cheraghi D, 2018, Statistical properties of quadratic polynomials with a neutral fixed point, *Journal of the European Mathematical Society*, Vol: 20, Pages: 2005-2062, ISSN: 1435-9855

We describe the statistical properties of the dynamics of the quadraticpolynomials 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.

Cheraghi D, 2017, dynamical systems, Mathematics of planet earth, a primer, Editors: crisan

Cheraghi D, 2017, Topology of irrationally indifferent attractors, Publisher: Arxiv Preprint

We study the attractors of a class of holomorphic systems with an irrationally indifferent fixed point. We prove a trichotomy for the topology of the attractor based on the arithmetic of the rotation number at the fixed point. That is, the attractor is either a Jordan curve, a one-sided hairy circle, or a Cantor bouquet. This has a number of remarkable corollaries on a conjecture of M. Herman about the optimal arithmetic condition for the existence of a critical point on the boundary of the Siegel disk, and a conjecture of A. Douady on the topology of the boundary of Siegel disks. Combined with earlier results on the topic, this completes the topological description of the behaviors of typical orbits near such fixed points, when the rotation number is of high type.

Cheraghi D, Chéritat A, 2015, A proof of the Marmi–Moussa–Yoccoz conjecture for rotation numbers of high type, *Inventiones mathematicae*, Vol: 202, Pages: 677-742, ISSN: 0020-9910

Cheraghi D, Shishikura M, 2015, Satellite renormalization of quadratic polynomials, Publisher: arXiv

We prove the uniform hyperbolicity of the near-parabolic renormalizationoperators acting on an infinite-dimensional space of holomorphictransformations. This implies the universality of the scaling laws, conjecturedby physicists in the 70's, for a combinatorial class of bifurcations. Throughnear-parabolic renormalizations the polynomial-like renormalizations ofsatellite type are successfully studied here for the first time, and newtechniques are introduced to analyze the fine-scale dynamical features of mapswith such infinite renormalization structures. In particular, we confirm therigidity conjecture under a quadratic growth condition on the combinatorics.The class of maps addressed in the paper includes infinitely-renormalizablemaps with degenerating geometries at small scales (lack of a priori bounds).

Cheraghi D, 2013, Typical orbits of quadratic polynomials with a neutral fixed point: Brjuno type, *Communications in Mathematical Physics*, Vol: 322, Pages: 999-1035, ISSN: 0010-3616

We describe the topological behavior of typical orbits of complex quadratic polynomials Pα(z)=e2παiz+z2 P α ( z ) = e 2 π α i z + z 2 , with α of high return type. Here we prove that for such Brjuno values of α the closure of the critical orbit, which is the measure theoretic attractor of the map, has zero area. Then we show that the limit set of the orbit of a typical point in the Julia set of P α is equal to the closure of the critical orbit. Our method is based on the near parabolic renormalization of Inou-Shishikura, and a uniform optimal estimate on the derivative of the Fatou coordinate that we prove here.

Cheraghi D, 2010, Combinatorial rigidity for some infinitely renormalizable unicritical polynomials, *CONFORMAL GEOMETRY AND DYNAMICS, An Electronic Journal of the American Mathematical Society*, Vol: 14, Pages: 219-255, ISSN: 1088-4173

We prove combinatorial rigidity of infinitely renormalizable unicriticalpolynomials, P_c :z \mapsto z^d+c, with complex c, under the a priori boundsand a certain "combinatorial condition". This implies the local connectivity ofthe connectedness loci (the Mandelbrot set when d = 2) at the correspondingparameters.

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