## Publications

7 results found

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

Cheraghi D, 2016, Typical orbits of quadratic polynomials with a neutral fixed point: non-Brjuno type

We investigate the quantitative and analytic aspects of the near-parabolicrenormalization scheme introduced by Inou and Shishikura in 2006. These providetechniques 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 mainresults of the paper concern fine-scale features of the measure-theoreticattractors of these maps, and their dependence on the data. As a bi-product, weestablish an optimal upper bound on the size of the maximal linearizationdomain in terms of the Siegel-Brjuno-Yoccoz series of $\alpha$.

Cheraghi D, ChÃ©ritat 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

Avila A, Cheraghi D, 2014, Statistical properties of quadratic polynomials with a neutral fixed point

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 highreturn times. In particular, we show that these maps are uniquely ergodic ontheir measure theoretic attractors, and the unique invariant probability is aphysical measure describing the statistical behavior of typical orbits in theJulia set. This confirms a conjecture of Perez-Marco on the unique ergodicityof hedgehog dynamics, in this class of maps.

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

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Cheraghi D, 2010, Combinatorial rigidity for some infinitely renormalizable unicritical polynomials, *Conformal Geometry and Dynamics of the American Mathematical Society*, Vol: 14, Pages: 219-219

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