arXiv subjects

Nicolae Mihalache

Publications and source records attributed to Nicolae Mihalache.

14 recordsLinked to original sources

arXiv2026

Precise asymptotics at the tip of the Mandelbrot set

For the quadratic family $f_c(z)=z^2+c$, the only parameters in the Mandelbrot set $\cal M$ for which the Julia set $\cal J_c$ has Hausdorff dimension $1$ are $c=0$ and $c=-2$. Near $c=0$, Ruelle's theory gives a real-analytic expansion of the dimension. The tip $c=-2$ of $\cal M$, however, is a non-hyperbolic parameter and the dimension function $c\mapsto \mathrm{dim_H}(\cal J_c)$ is highly discontinuous there. We prove the sharp first-order asymptotic for the lower envelope of the Hausdorff dimension at the tip: If $c\in \cal M $ then $\mathrm{dim_H}(\cal J_c)$ lies asymptotically above $1+ Ω\sqrt{|c+2|}$ with the Jaksztas constant $Ω=\sqrt{\frac{2}{3}}\frac{1}{π\log 2}$. This is a surprisingly precise contribution to the Yoccoz problem about unfolding attractors. The proof develops a thermodynamic formalism for degenerating families of box mappings. At each scale, for parameters $c\to -2$, the induced dynamics exhibit a uniform property of exponential tails, generating improved control of their pressure functions.

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arXiv2025

Factorization of the quadratic Misiurewicz-Thurston polynomials

This note provides the complete factorization of the Misiurewicz-Thurston polynomial $q_{\ell,n}=p_{\ell+n}(z) - p_\ell(z)$ over $\mathbb{C}$, which plays a central role in the study of the Mandelbrot set, where \[ p_0(z) = 0, \qquad p_{n+1}(z) = p_n(z)^2 + z. \] The roots can be classified into two categories. First, there are hyperbolic points $\operatorname{hyp}(k)$ for any divisor $k$ of $n$, which are parameters whose critical orbits are of exact period $k$. Those are roots of $q_{\ell,n}$ with multiplicity $\left\lfloor \frac{\ell -1}{k} \right\rfloor + 2$. Next are the points $\operatorname{mis}(j,k)$ for $2\leq j\leq \ell$ whose critical orbits are pre-periodic of exact period $k$ with an exact pre-period $j$. Those are simple roots of $q_{\ell,n}$.

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arXiv2024

How to split a tera-polynomial

This article presents a new algorithm to compute all the roots of two families of polynomials that are of interest for the Mandelbrot set $\mathcal{M}$ : the roots of those polynomials are respectively the parameters $c\in\mathcal{M}$ associated with periodic critical dynamics for $f_c(z)=z^2+c$ (hyperbolic centers) or with pre-periodic dynamics (Misiurewicz-Thurston parameters). The algorithm is based on the computation of discrete level lines that provide excellent starting points for the Newton method. In practice, we observe that these polynomials can be split in linear time of the degree. This article is paired with a code library [Mandel] that implements this algorithm. Using this library and about 723 000 core-hours on the HPC center Roméo (Reims), we have successfully found all hyperbolic centers of period $\leq 41$ and all Misiurewicz-Thurston parameters whose period and pre-period sum to $\leq 35$. Concretely, this task involves splitting a tera-polynomial, i.e. a polynomial of degree $\sim10^{12}$, which is orders of magnitude ahead of the previous state of the art. It also involves dealing with the certifiability of our numerical results, which is an issue that we address in detail, both mathematically and along the production chain. The certified database is available to the scientific community. For the smaller periods that can be represented using only hardware arithmetic (floating points FP80), the implementation of our algorithm can split the corresponding polynomials of degree $\sim10^{9}$ in less than one day-core. We complement these benchmarks with a statistical analysis of the separation of the roots, which confirms that no other polynomial in these families can be split without using higher precision arithmetic.

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arXiv2023

A short ODE proof of the Fundamental Theorem of Algebra

We propose a short proof of the Fundamental Theorem of Algebra based on the ODE that describes the Newton flow and the fact that the value $|P(z)|$ is a Lyapunov function. It clarifies an idea that goes back to Cauchy.

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arXiv2022

On the local connectivity of attractors of Markov IFS

We prove an extension of M. Hata's theorem [4] for planar Markov Iterated Function Systems satisfying a strong version of the Open Set Condition. More precisely, if the attractor of such a system is connected, then it is locally connected. We construct counterexamples to show that all additional hypothesis are necessary.

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arXiv2022

Hausdorff dimension of Julia sets in the logistic family

A closed interval and circle are the only smooth Julia sets in polynomial dynamics. D. Ruelle proved that the Hausdorff dimension of unicritical Julia sets close to the circle depends analytically on the parameter. Near the tip of the Mandelbrot set M, the Hausdorff dimension is generally discontinuous. Answering a question of J-C. Yoccoz in the conformal setting, we observe that the Hausdorff dimension of quadratic Julia sets depends continuously on $c$ and find explicit bounds at the tip of M for most real parameters in the the sense of 1-dimensional Lebesgue measure.

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arXiv2022

An invitation to rough dynamics: zipper maps

In the field of dynamical systems, it is not rare to meet irregular functions, which are typically H{ö}lder but not Lipschitz (e.g. the Weierstrass functions). Our goal is to scratch the surface of the following question: what happens if we consider irregular maps and iterate them? We introduce the family of "zipper maps", which are irregular in the above sense, and study some of their dynamical properties. For a large set of parameters, the corresponding zipper map admits horseshoe of all orders; as an immediate consequence, every order on $k\ell$ points can be realized by $k$ orbits of length $\ell$ of the map.These maps have infinite topological entropy, and we refine this statement by showing that they have positive metric mean dimension with respect to the Euclidean metric, as well as by introducing other notions of higher complexity.Finally, we prove that every interval map (thus including zipper maps) have vanishing absolute metric mean dimension, proving a small case of the conjecture that the absolute metric mean dimension coincides with the topological mean dimension.

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arXiv2022

Fast Evaluation of Real and Complex Polynomials

We propose an algorithm for quickly evaluating polynomials. It pre-conditions a complex polynomial $P$ of degree $d$ in time $O(d\log d)$, with a low multiplicative constant independent of the precision. Subsequent evaluations of $P$ computed with a fixed precision of $p$ bits are performed in average arithmetic complexity $O\big(\sqrt{d(p+\log d)}\big)$ and memory $O(dp)$. The average complexity is computed with respect to points $z \in \mathbb{C}$, weighted by the spherical area of $\overline{\mathbb{C}}$. The worst case does not exceed the complexity of H{ö}rner's scheme. In particular, our algorithm performs asymptotically as $O(\sqrt{d\log d})$ per evaluation. For many classes of polynomials, in particular those with random coefficients in a bounded region of $\mathbb{C}$, or for sparse polynomials, our algorithm performs much better than this upper bound, without any modification or parameterization.The article contains a detailed analysis of the complexity and a full error analysis, which guarantees that the algorithm performs as well as H\''orner's scheme, only faster. Our algorithm is implemented in a companion library, written in standard C and released as an open-source project [MV22].Our claims regarding complexity and accuracy are confirmed in practice by a set of comprehensive benchmarks.

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arXiv2017

Collet, Eckmann and the bifurcation measure

The moduli space $\mathcal{M}_d$ of degree $d\geq2$ rational maps can naturally be endowed with a measure $μ_\mathrm{bif}$ detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure $μ_\mathrm{bif}$ has positive Lebesgue measure. To do so, we establish a general sufficient condition for the conjugacy class of a rational map to belong to the support of $μ_\mathrm{bif}$ and we exhibit a large set of Collet-Eckmann rational maps which satisfy this condition. As a consequence, we get a set of Collet-Eckmann rational maps of positive Lebesgue measure which are approximated by hyperbolic rational maps.

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arXiv2016

Diabolical entropy

Milnor and Thurston's famous paper proved monotonicity of the topological entropy for the real quadratic family. Guckenheimer showed that it is Hölder continuous. We obtain a precise formula for the Hölder exponent at almost every quadratic parameter. Furthermore, the entropy of most parameters is proven to be in a set of Hausdorff dimension smaller than one, while most values of the entropy arise from a set of parameters of dimension smaller than one.

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arXiv2012

Metric properties of mean wiggly continua

We study lower and upper bounds of the Hausdorff dimension for sets which are wiggly at scales of positive density. The main technical ingredient is a construction, for every continuum K, of a Borel probabilistic measure μwith the property that on every ball B(x,r), with x in K, the measure is bounded by a universal constant multiple of r\exp(-g(x,r)), where g(x,r) > 0 is an explicit function. The continuum K is mean wiggly at exactly those points x in K where g(x, r) has a logarithmic growth to infinity as r goes to 0. The theory of mean wiggly continua leads, via the product formula for dimensions, to new estimates of the Hausdorff dimension for Cantor sets. We prove also that asymptotically flat sets are of Hausdorff dimension 1 and that asymptotically non-porous continua are of the maximal dimension. Another application of the theory is geometric Bowen's dichotomy for Topological Collet-Eckmann maps in rational dynamics. In particular, mean wiggly continua are dynamically natural as they occur as Julia sets of quadratic polynomials for parameters from a generic set on the boundary of the Mandelbrot set.

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arXiv2011

Symmetric Itinerary Sets

We consider a one parameter family of dynamical systems W :[0, 1] -> [0, 1] constructed from a pair of monotone increasing diffeomorphisms Wsub(i), such that Wsub(i)(inverse): [0, 1] -> [0, 1], (i = 0, 1). We characterise the set of symbolic itineraries of W using an attractor of an iterated closed relation,in the terminology of McGehee, and prove that there is a member of the family for which is symmetrical.

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arXiv2009

Julia and John revisited

We show that Fatou components of a semi-hyperbolic rational map are John domains and that the converse does not hold. This generalizes a famous result of Carleson, Jones and Yoccoz. We show that a connected Julia set is locally connected for a large class of non-uniformly hyperbolic rational maps. This class is more general than semi-hyperbolicity and includes Collet-Eckmann and Topological Collet-Eckmann maps and maps verifying a summability condition (as considered by Graczyk and Smirnov).

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arXiv2008

Two counterexamples in rational and interval dynamics

In rational dynamics, we prove the existence of a polynomial that satisfies the Topological Collet-Eckmann condition, but which has a recurrent critical orbit that is not Collet-Eckmann. This shows that the converse of the main theorem in [11] does not hold. In interval dynamics, we show that the Collet-Eckmann property for recurrent critical orbits is not a topological invariant for real polynomials with negative Schwarzian derivative. This contradicts a conjecture of Swiatek [22].

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