Below follows abstracts for the talks given at the conference. The abstracts can also be downloaded as a pdf-file.
We shall prove that for every complex quadratic polynomial \(f\) with Cremer's fixed point \(z_0\) (or periodic orbit) for every \(\delta \gt 0\), there is at most one periodic orbit of minimal period \(n\) for all \(n\) large enough, entirely in the disc (ball) \(B(z_0, \exp -\delta n)\) (at most \(2np\) for a Cremer orbit of period \(p\)). Next, it is proved that the number of periodic orbits of period \(n\) in a bunch \(P_n\), that is for all \(x,y\in P_n\), \(|f^j(x)- f^j(y)|\le \exp -\delta n\) for all \(j=0,...,n-1\), does not exceed \(\exp \delta n\). We conclude that the geometric pressure defined with the use of periodic points coincides with the one defined with the use of preimages of an arbitrary typical point. I. Binder, K. Makarov and S. Smirnov (Duke Math. J. 2003) proved this for all polynomials but assuming all periodic orbits to be hyperbolic, and asked about general situations. We prove here a positive answer for all quadratic polynomials.
Joint work with Marta Kosek.
We consider a sequence \((p_n)_{n=1}^\infty\) of polynomials with uniformly bounded zeros and \(\deg p_1\geq 1\), \(\deg p_n\geq 2\) for \(n\geq 2\), satisfying certain additional asymptotic conditions (guided sequences). We show that the non-autonomous filled Julia set \(\mathcal{K}[(p_{n})_{n=1}^\infty]\) generated by such a sequence \((p_{n})_{n=1}^\infty\), defined as $$ \left\{z\in\mathbb{C}\!:\! \left((p_n\circ\dots\circ p_1)(z)\right)_{n=1}^\infty \text{ is bounded}\right\}, $$ is nonempty, compact and regular with respect to the Green function. Using Klimek's metric we also prove that the function sequence $$ \left(\frac{1}{\deg p_n\cdot...\cdot \deg p_1}\log^+|p_n\circ...\circ p_1|\right)_{n=1}^\infty $$ is uniformly convergent in \(\mathbb{C}\). We discuss in more detail our toy example, generated by \(t_n=\frac{1}{2^{n-1}}T_n,\ n=1,2,...\), where \(T_n\) is the classical Chebyshev polynomial of degree \(n\).
The possibilities for limit functions on a Fatou component for the iteration of a single polynomial or rational function are well understood and quite restricted. In non-autonomous iteration, where one considers compositions of arbitrary polynomials with suitably bounded degrees and coefficients, with so much freedom in choosing the particular polynomial sequence, one expects to observe a far greater range of behaviour. We show this is indeed the case and that it is possible to obtain the \emph{whole} of the classical Schlicht family of normalized univalent functions on the unit disc as limit functions on a \emph{single} Fatou component for a \emph{single} bounded sequence of quadratic polynomials. The main ideas behind this are quasiconformal surgery and the feature of dynamics on Siegel discs where suitable high iterates of a single polynomial with a Siegel disc \(U\) approximate the identity closely on compact subsets of \(U\). This allows us both to approximate many functions from the Schlicht family on a Fatou component and to correct the small but inevitable errors arising from these approximations. Do almost nothing and you can do almost anything!
We present some results concerning closed discs contained in the filled Julia sets associated with polynomials \(f_c:\mathbb{C} \ni z \longmapsto z^{2}+c \in \mathbb{C}\) mostly for \(c\in \left[-2,\frac{1}{4} \right]\) but also some other values. We investigate also a few non-autonomous Julia sets. As an application we mention that the pluricomplex Green function associated with the non-autonomous Julia set of the sequence \((f_{c_n})_{n=1}^{\infty}\) is H{\"o}lder continuous, provided \((c_n)_{n=1}^{\infty} \subset \overline{D}\left(0,\frac{1}{4} \right)\).
Joint work with Erin Milña-Díaz.
Given a compact set \(K \subset \mathbb{C}\), the Chebyshev polynomial of degree \(n\) is the unique monic polynomial that minimizes the supremum norm on \(K\). When \(K\) is infinite, such a polynomial exists and is uniquely defined for each degree. Although explicit formulas are generally unavailable, Chebyshev polynomials can be analyzed through families of near-minimal polynomials. One such family is that of the Faber polynomials, which arise naturally from the conformal map associated with the Green function of \(K\). In this talk, I will present recent results that establish connections between Chebyshev and Faber polynomials along equipotential curves.
Joint work with Chelo Ferreira, José López and Ester Pérez Sinusía.
In 2008 José L. López introduced a new method for computing asymptotic expansions of integral transforms \(F(x) = \int_0^{\infty} f(t) \, h(xt) \, dt\) for small \(x\gt 0\). Besides requiring mild conditions on the functions \(f\) and \(h\) and being easy to apply, the method is a very general technique that unifies a certain set of classical methods of the asymptotic analysis of integrals.
In this talk we revisit this method and consider its extension to two-disional integrals. We show that such an extension is possible to apply to integrals of the form \(F(x) = \int_0^{\infty} \int_0^{\infty} f(t) g(s) h(xts) dt ds\), with small \(x\gt 0\). We obtain a complete asymptotic expansion of this kind of integrals and show the applicability of the method by computing the asymptotic expansion of a certain family of integrals that has acquiered attention in a model of \(2D\) quantum gravity.
Joint work with José López, Pablo Palacios and Ester Pérez Sinusía.
The Fourier transform \(\int_{-\infty}^{\infty}f(t)e^{itx} dt\) of a function \(f(t)\) plays a central role in ma\-the\-ma\-tics, statistics and physics. Asymptotic expansions of this integral for small values of the variable \(x\) are known only for certain specific families of functions \(f(t)\), but not for general functions \(f(t)\).
In this talk we derive an analytic representation of the one-sided Fourier transform integral \(F(x):=\int_{0}^{\infty}f(t)e^{itx} dt\) for a very large family of functions \(f(t)\). It is given in the form of a series that has an asymptotic character for small values of the variable \(x\). This expansion is derived using the asymptotic method of Mellin convolution integrals. Some numerical experiments show the accuracy of the approximation supplied by the first few terms of the series.
Starbursts are the characteristic light intensity patterns perceived when ob- serving small, bright sources at night, such as stars. These patterns arise due to the formation of caustics (light concentration zones) on the retina, caused by wavefront aberrations of the eye. A striking and previously unexplained feature of starbursts is their typical p-fold symmetry. We present a theoretical framework that explains the number of points and symmetries observed in starbursts, based on the geometrical and algebraic properties of the wave aberration function expressed through its Zernike polynomial expansion. We analyze the number and spatial distribution of Gauss saddle cusps of the Hessian of the wave aberration, and establish their relationship with the symmetry and number of starburst points.
In complex dynamics, the celebrated Brolin equidistribution theorem asserts that for every value \(a\in\mathbb{C}\) but at most one, the averaged preimages \((f^n)^*\delta_a/d^n\) (as purely atomic probability measures in \(\mathbb{C}\)) under the iterations \(f^n\) of a complex polynomial \(f\) of degree \(d\) more than one tend to the canonical equilibrium measure of \(f\) (\(=\) the potential theoretic equilibrium measure on the Julia set of \(f\) with pole at \(\infty\)) as \(n\to\infty\). The question on whether such an equidistribution assertion is also true for the sequence of derivatives \((f^n)'\) of the iterated polynomials was studied by Gauthier--Vigny using higher dimensional techniques and settled by Okuyama (the speaker) using Nevanlinna theoretic ones, and then the same question for the sequence of (higher) \(m\)th order derivatives \((f^n)^{(m)}\) was settled by Okuyama--Vigny and Okuyama. We will talk about those equidistribution results for any order derivatives of iterated polynomials, emphasizing on the essential dynamical (and also a bit arithmetic) natures of the problems.
The study of periodic Jacobi matrices on the line is a classical subject in spectral theory, motivated by condensed matter physics and having connections to orthogonal polynomial theory, potential theory and other areas of mathematics. In this talk we will discuss ongoing work that attempts to generalize the theory to more general trees, with an emphasis on Floquet theory which describes the structure of generalized eigenfunctions. We will describe some results obtained in joint works with Jess Banks, Jorge Garza Vargas, Eyal Seelig and Barry Simon.
Building on our work with Jasmin Raissy and using a software developped by Arnaud Ch\'eritat, we investigate the connection between the discrete dynamics of germs tangent to the identity in \(C^2\), the continuous dynamics of homogeneous vector fields in \(C^2\) and the geodesic flow in Riemann surfaces equipped with a complex affine structure. For example, iteration of \(f(x,y) = (x+y^2,y+x^2)\) is strongly related the the ODE \((x',y') = (y^2,x^2)\) and to the trajectories in an equilateral triangular billiard.
Consider a Coulomb gas of charged particles confined to a set in the complex plane. I will discuss the following question: How does the asymptotic expansion of the free energy depend on the geometry of the set, as the number of particles tends to infinity? When the set is a Jordan domain, curve or arc, this problem is related to Faber polynomials and Grunsky operators associated to the set, revealing a close connection to the Loewner energy and other interesting domain functionals. Based joint works with K. Courteaut (NYU) and K. Johansson (KTH).
Holomorphic maps that have fixed points with multiplier equal to an irrational rotation \(e^{2\pi i \alpha}\) exhibit complicated and rich long term behaviour. Much depends on the irrationally indifferent attractor, which is equivalent to \(\omega(c)\) for some specially chosen critical point. The development of the Inou-Shishikura class that is invariant under renormalisation led authors such as Cheraghi to study these attractors via a special non-holomorphic toy model for the renormalisation tower. Remarkably this model retains important topological and geometrical features, but it depends entirely on the arithmetic of the rotation number \(\alpha\). However it has some restrictions, for instance only applying to uni-critical maps.
It turns out that in the bi-critical case there is much added complexity to the geometry of orbits, which will result in modified arithmetic conditions in the renormalisation tower. In this talk I will introduce some ideas that describe a new modified bi-critical version of toy model.
In a joint work with Gabriel Vigny, we show that periodic points of period \(n\) of a complex rational map \(f\) of degree \(d\gt 1\) equidistribute towards the equilibrium measure \(\mu_f\) of the rational map with a rate of convergence of \((nd^{-n})^{1/2}\) for \(C^1\)-observables. This is a consequence of a quantitative equidistribution of Galois invariant finite subsets of preperiodic points \`a la Favre and Rivera-Letelier. Our proof relies on the Hölder regularity of the quasi-psh Green function of a rational map, an estimate of Baker concerning Hsia kernel, as well as on the product formula and its generalization by Moriwaki for finitely generated fields over \(\mathbb{Q}\).
We outline a general framework to study the local dynamics of near-parabolic maps in the complex plane using the meromorphic 1-form introduced by X. Buff. As a sample application of this setup, we prove a tameness result for the invariant curves of near-parabolic maps. In the special case of polynomials, we obtain Hausdorff continuity of the external rays of a given periodic angle when the associated multipliers approach a root of unity non-tangentially. This is joint work with C. L. Petersen.
We give some refined \(L^\infty\) error estimates for best rational approximation of Markov functions, such as \(f(z) = 1/\sqrt{z}\), as discussed by Zolotarev. In particular, we give a worst case measure, allowing to bound above the relative error on the real axis for any other Markov function. Within the class of measures satisfying the Szegö condition, it is known that, asymptotically, the worst case measure with support in \([a, b]\) is the equilibrium measure. This is no longer true in the case of general measures supported on the real line, where we explicitly give, for each degree a worst case measure, and compare with the error estimate given by Zolotarev.
It is well known that the asymptotic distribution of zeros of orthogonal—and more generally, extremal—polynomials in the complex plane can often be described using logarithmic potential theory. Over the past decades, classical extremal problems have been extended to encompass more intricate settings, including polynomials with varying weights, non-Hermitian orthogonality, and multiple orthogonality. In particular, the Gonchar–Rakhmanov–Stahl (GRS) theory provides a powerful framework for describing weak asymptotics in the case of varying non-Hermitian orthogonality under harmonic external fields.
Yet some deceptively simple scenarios remain elusive. One such case involves orthogonality with respect to a sum of two weights—an apparently natural situation in which the associated weights often have dense zeros and the resulting external field is only piecewise harmonic. In this setting, many fundamental questions remain unresolved.
In this talk, I will present a brief overview of the GRS theory, highlight some compelling open problems, and focus on one particularly intriguing question: What type of extremal problem governs the distribution of zeros when the zeros of the orthogonality weight themselves become dense?
This presentation includes results from joint work with E. Rakhmanov and with G. L. F. Silva.
In this talk, I will survey recent advances in the study of universality limits of orthogonal polynomials. I will discuss cases where the Christoffel–Darboux kernel admits a power-law rescaling limit. Such universality limits typically arise in the bulk or at the edge of the spectrum. However, we show that at accumulation points of spectral gaps, the scaling behavior can be quite different. In particular, we discuss rescaling limits where there is not a unique limiting kernel, but rather a full limit cycle. Balanced measures on real Julia sets of arbitrary expanding polynomials provide natural examples of this type of rescaling behavior.
This talk is based on joint works with Milivoje Lukić, Brian Simanek, Harald Woracek and Peter Yuditskii.
In this joint work with Carsten Petersen and Eva Uhre, we consider a sequence of polynomials \(q_k\) whose root-distributions converge weak* to a limit measure \(\mu\). In a region \(U\) where the number of roots is uniformly bounded on compact subsets, we establish an asymtotic relation between the roots of the \(m\)th derivative \(q_k^{(m)}\), the roots of \(q_k\) and the critical points of the potential associated to \(\mu\).