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Ilia Binder

Publications and source records attributed to Ilia Binder.

At least 19 recordsLinked to original sources
arXiv2025

Computable thermodynamic formalism

We investigate the theory of thermodynamic formalism from the perspective of computable analysis, with a special focus on the computability of equilibrium states. Specifically, we develop two complementary general approaches to verify the computability of equilibrium states for nonuniformly expanding computable dynamical systems. The first approach applies to dynamical systems whose topological pressure functions admit effective approximations and whose measure-theoretic entropy functions are upper semicontinuous. As a concrete application, we establish the computability of the equilibrium states for Misiurewicz-Thurston rational maps with H\"older continuous potentials. The second approach exploits prescribed Jacobians of equilibrium states through a local analysis and applies to settings where the measure-theoretic entropy functions may lack upper semicontinuity.

math.DS↗
arXiv2025

Analysis of Boundary Behaviour of Quasidisks and Jordan Repellers

We investigate the fine properties of harmonic measure and boundary rotation. By focusing on quasidisks and, in particular, on connected Jordan Repellers arising from conformal expanding dynamical systems, we explore those using the deep interplay between geometric function theory and dynamical systems as a unified framework. While some of the results presented here are regarded as 'folklore' among experts, they lack rigorous proofs in the existing literature. We fill this gap by providing a comprehensive, referable treatment using a novel approach that also expands existing results.

math.CV↗
arXiv2025

Green\'s Mapping and Julia Sets

In March 1999, the first named author (Binder) posed the problem of showing that a ``good direction'' $\psi\in [0,2]$ exists, for any Green's mapping $T:H\rightarrow\tilde \Omega$, i.e., \begin{equation}\label{binder} \int\limits_0\limits^{1} |T''(re^{i\pi\psi})|dr <\infty, \quad\text{ for at least one } \quad \psi\in [0,2]. \end{equation} Presently this problem is open even in the special case where $\partial \Omega $ is a uniformly perfect subset of the real line. In this paper we obtain a positive solution when $\Omega = \overline{C} \setminus E_0$ where $E_0 \subset R $ is the Julia set of an expanding quadratic polynomial.

math.CV↗
arXiv2024

Orthodiagonal Maps, Tilings of Rectangles, and their Convergence to Conformal Maps

A classic result of Brooks, Smith, Stone and Tutte associates to any finite planar network with distinguished source and sink vertices, a tiling of a rectangle by smaller subrectangles whose aspect ratios are given by the conductances of corresponding edges in the network. This tiling can be viewed as a discrete analogue of the uniformizing conformal map that maps a simply connected domain with four distinguished prime ends to a rectangle, so that the four prime ends are mapped to the four corners of the rectangle. \\ \\ We make this intuition precise by showing that if $\Omega$ is a simply connected domain with four distinguished prime ends $A,B,C,D$ in counterclockwise order and $(\Omega_{n})_{n\geq{1}}$ is a sequence of orthodiagonal maps with distinguished boundary vertices $A_{n}, B_{n}, C_{n}, D_{n}$ in counterclockwise order, that are finer and finer approximations of $\Omega$ with its distinguished boundary points $A,B,C,D$, then the corresponding ``rectangle tiling maps" converge uniformly on compacts to the aforementioned conformal map on $\Omega$.

math.CV↗
arXiv2024

Power rate of convergence of discrete curves: framework and applications

We provide a general framework of estimates for convergence rates of random discrete model curves approaching Schramm Loewner Evolution (SLE) curves in the lattice size scaling limit. We show that a power-law convergence rate of an interface to an SLE curve can be derived from a power-law convergence rate for an appropriate martingale observable provided the discrete curve satisfies a specific bound on crossing events, the Kempannien-Smirnov condition, along with an estimate on the growth of the derivative of the SLE curve. We apply our framework to show that the exploration process for critical site percolation on hexagonal lattice converges to the SLE$_6$ curve with a power-law convergence rate.

math.PR↗
arXiv2024

Decoupling and Multipoint moments for the Inverse of the Gaussian multiplicative chaos

In this article we study the decoupling structure and multipoint moment of the inverse of the Gaussian multiplicative chaos. It is also the second part of preliminary work for extending the work in "Random conformal weldings" (by K. Astala, P. Jones, A. Kupiainen, E. Saksman) to the existence of Lehto welding for the inverse. In particular, we prove that the dilatation of the inverse homeomorphism on the positive real line is in $L^{1}([0,1]\times[0,2])$.

math.PR↗
arXiv2023

Inverse of the Gaussian multiplicative chaos: Lehto welding of Independent Quantum disks

In this article, we use the framework of "Random conformal weldings" (by K. Astala, P. Jones, A. Kupiainen, E. Saksman) to prove the existence of Lehto-welding for the inverse for $γ<0.1818$ and independent copies for $γ_{2}\leqγ_{1}<0.1818$. In particular, we obtain the existence of some conformaly invariant loop $Γ$ that glues two disks with boundary length given by independent copies of GMC on the unit circle. It is still unclear how to show that those loops $Γ$ are in fact SLE-loops in a parallel proof to "Integrability of SLE via conformal welding of random surfaces" (by Morris Ang, Nina Holden, and Xin Sun).

math.PR↗
arXiv2023

On computability of equilibrium states

Equilibrium states are natural dynamical analogues of Gibbs states in thermodynamic formalism. This paper investigates their computability within the framework of Computable Analysis. We show that the unique equilibrium state for a computable, open, topologically exact, distance-expanding map $T\colon X\rightarrow X$ and a computable H\"older continuous potential $\varphi\colon X\rightarrow\mathbb{R}$ is always computable. As an application, we establish the computability of equilibrium states for computable hyperbolic rational maps and their respective geometric potentials. Moreover, we develop a constructive method to exhibit the non-uniqueness of equilibrium states for some dynamical systems. We also present some computable dynamical systems whose equilibrium states are all non-computable.

math.DS↗
arXiv2023

Conformal Dimension of the Brownian Graph

Conformal dimension of a metric space $X$, denoted by $\dim_C X$, is the infimum of the Hausdorff dimension among all its quasisymmetric images. If conformal dimension of $X$ is equal to its Hausdorff dimension, $X$ is said to be minimal for conformal dimension. In this paper we show that the graph of the one dimensional Brownian motion is almost surely minimal for conformal dimension. We also give many other examples of minimal sets for conformal dimension, which we call Bedford-McMullen type sets. In particular we show that Bedford-McMullen self-affine sets with uniform fibers are minimal for conformal dimension. The main technique in the proofs is the construction of ``rich families of minimal sets of conformal dimension one''. The latter concept is quantified using Fuglede's modulus of measures.

math.MG↗
arXiv2020

Computability in Harmonic Analysis

We study the question of constructive approximation of the harmonic measure $ω_x^Ω$ of a connected bounded domain $Ω$ with respect to a point $x\inΩ$. In particular, using a new notion of computable harmonic approximation, we show that for an arbitrary such $Ω$, computability of the harmonic measure $ω^Ω_x$ for a single point $x\inΩ$ implies computability of $ω_y^Ω$ for any $y\in Ω$. This may require a different algorithm for different points $y$, which leads us to the construction of surprising natural examples of continuous functions that arise as solutions to a Dirichlet problem, whose values can be computed at any point but cannot be computed with the use of the same algorithm on all of their domain. We further study the conditions under which the harmonic measure is computable uniformly, that is by a single algorithm, and characterize them for regular domains with computable boundaries.

math.CV↗
arXiv2018

Caratheodory convergence and harmonic measure

We give several new characterizations of Caratheodory convergence of simply connected domains. We then investigate how different definitions of convergence generalize to the multiply-connected case.

math.CV↗
arXiv2017

Almost Periodicity in Time of Solutions of the Toda Lattice

We study an initial value problem for the Toda lattice with almost periodic initial data. We consider initial data for which the associated Jacobi operator is absolutely continuous and has a spectrum satisfying a Craig-type condition, and show the boundedness and almost periodicity in time and space of solutions.

math.SP↗
arXiv2015

Almost Periodicity in Time of Solutions of the KdV Equation

We study the Cauchy problem for the KdV equation $\partial_t u - 6 u \partial_x u + \partial_x^3 u = 0$ with almost periodic initial data $u(x,0)=V(x)$. We consider initial data $V$, for which the associated Schrödinger operator is absolutely continuous and has a spectrum that is not too thin in a sense we specify, and show the existence, uniqueness, and almost periodicity in time of solutions. This establishes a conjecture of Percy Deift for this class of initial data. The result is shown to apply to all small analytic quasiperiodic initial data with Diophantine frequency vector.

math.AP↗
arXiv2015

A Dimension Spectrum for SLE Boundary Collisions

We consider chordal SLE(kappa) curves for kappa > 4, where the intersection of the curve with the boundary is a random fractal of almost sure Hausdorff dimension min {2-8/kappa,1}. We study the random sets of points at which the curve collides with the real line at a specified "angle" and compute an almost sure dimension spectrum describing the metric size of these sets. We work with the forward SLE flow and a key tool in the analysis is Girsanov's theorem, which is used to study events on which moments concentrate. The two-point correlation estimates are proved using the direct method.

math.PR↗
arXiv2014

On the Sum of the Non-Negative Lyapunov Exponents for Some Cocycles Related to the Anderson Model

We provide an explicit lower bound for the the sum of the non-negative Lyapunov exponents for some cocycles related to the Anderson model. In particular, for the Anderson model on a strip of width $ W $ the lower bound is proportional to $ W^{-ε} $, for any $ ε>0 $. This bound is consistent with the fact that the lowest non-negative Lyapunov exponent is conjectured to have a lower bound proportional to $ W^{-1} $.

math-ph↗