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Krzysztof Barański

Publications and source records attributed to Krzysztof Barański.

At least 19 recordsLinked to original sources
arXiv2025

On the dimension of the boundaries of attracting basins of entire maps

Let $f\colon \mathbb{C} \to \mathbb{C}$ be a transcendental entire map from the Eremenko-Lyubich class $\mathcal{B}$, and let $ζ$ be an attracting periodic point of period $p$. We prove that the boundaries of components of the attracting basin of (the orbit of) $ζ$ have hyperbolic (and, consequently, Hausdorff) dimension larger than $1$, provided $f^p$ has an infinite degree on an immediate component $U$ of the basin, and the singular set of $f^p|_U$ is compactly contained in $U$. The same holds for the boundaries of components of the basin of a parabolic $p$-periodic point $ζ$, under the additional assumption $ζ\notin \overline{\text{Sing}(f^p)}$. We also prove that if an immediate component of an attracting basin of an arbitrary transcendental entire map is bounded, then the boundaries of components of the basin have hyperbolic dimension larger than $1$. This enables us to show that the boundary of a component of an attracting basin of a transcendental entire function is never a smooth or rectifiable curve. The results provide a partial answer to a question from Hayman's list of problems in function theory.

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arXiv2024

Prediction of dynamical systems from time-delayed measurements with self-intersections

In the context of predicting the behaviour of chaotic systems, Schroer, Sauer, Ott and Yorke conjectured in 1998 that if a dynamical system defined by a smooth diffeomorphism $T$ of a Riemannian manifold $X$ admits an attractor with a natural measure $μ$ of information dimension smaller than $k$, then $k$ time-delayed measurements of a one-dimensional observable $h$ are generically sufficient for $μ$-almost sure prediction of future measurements of $h$. In a previous paper we established this conjecture in the setup of injective Lipschitz transformations $T$ of a compact set $X$ in Euclidean space with an ergodic $T$-invariant Borel probability measure $μ$. In this paper we prove the conjecture for all (also non-invertible) Lipschitz systems on compact sets with an arbitrary Borel probability measure, and establish an upper bound for the decay rate of the measure of the set of points where the prediction is subpar. This partially confirms a second conjecture by Schroer, Sauer, Ott and Yorke related to empirical prediction algorithms as well as algorithms estimating the dimension and number of required delayed measurements (the so-called embedding dimension) of an observed system. We also prove general time-delay prediction theorems for locally Lipschitz or Hölder systems on Borel sets in Euclidean space.

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arXiv2024

Predicting dynamical systems with too few time-delay measurements: error estimates

We study the problem of reconstructing and predicting the future of a dynamical system by the use of time-delay measurements of typical observables. Considering the case of too few measurements, we prove that for Lipschitz systems on compact sets in Euclidean spaces, equipped with an invariant Borel probability measure $μ$ of Hausdorff dimension $d$, one needs at least $d$ measurements of a typical (prevalent) Lipschitz observable for $μ$-almost sure reconstruction and prediction. Consequently, the Hausdorff dimension of $μ$ is the precise threshold for the minimal delay (embedding) dimension for such systems in a probabilistic setting. Furthermore, we establish a lower bound postulated in the Schroer--Sauer--Ott--Yorke prediction error conjecture from 1998, after necessary modifications (whereas the upper estimates were obtained in our previous work). To this aim, we prove a general theorem on the dimensions of conditional measures of $μ$ with respect to time-delay coordinate maps.

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arXiv2023

On the dimension of stationary measures for random piecewise affine interval homeomorphisms

We study stationary measures for iterated function systems (considered as random dynamical systems) consisting of two piecewise affine interval homeomorphisms, called Alsedà--Misiurewicz (AM) systems. We prove that for an open set of parameters, the unique non-atomic stationary measure for an AM-system has Hausdorff dimension strictly smaller than $1$. In particular, we obtain singularity of these measures, answering partially a question of Alsedà and Misiurewicz from 2014.

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arXiv2023

Regularity of almost-surely injective projections in Euclidean spaces

In a previous work we proved that if a finite Borel measure $μ$ in a Euclidean space has Hausdorff dimension smaller than a positive integer $k$, then the orthogonal projection onto almost every $k$-dimensional linear subspace is injective on a set of full $μ$-measure. In this paper we study the regularity of the inverses of these projections and prove that if $μ$ has a compact support $X$ such that (respectively) the Hausdorff, upper box-counting or Assouad dimension of $X$ is smaller than $k$, then the inverse is (respectively) continuous, pointwise $α$-Hölder for some $α\in (0,1)$ or pointwise $α$-Hölder for every $α\in (0,1)$. The results generalize to the case of typical linear perturbations of Lipschitz maps and strengthen previously known ones in the lossless analog compression literature. We provide examples showing the sharpness of the statements. Additionally, we construct a non-trivial measure on the plane which admits almost-surely injective projections in every direction, and show that no homogeneous self-similar measure has this property.

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arXiv2022

On the Shroer-Sauer-Ott-Yorke predictability conjecture for time-delay embeddings

Shroer, Sauer, Ott and Yorke conjectured in 1998 that the Takens delay embedding theorem can be improved in a probabilistic context. More precisely, their conjecture states that if $μ$ is a natural measure for a smooth diffeomorphism of a Riemannian manifold and $k$ is greater than the information dimension of $μ$, then $k$ time-delayed measurements of a one-dimensional observable $h$ are generically sufficient for a predictable reconstruction of $μ$-almost every initial point of the original system. This reduces by half the number of required measurements, compared to the standard (deterministic) setup. We prove the conjecture for ergodic measures and show that it holds for a generic smooth diffeomorphism, if the information dimension is replaced by the Hausdorff one. To this aim, we prove a general version of predictable embedding theorem for injective Lipschitz maps on compact sets and arbitrary Borel probability measures. We also construct an example of a $C^\infty$-smooth diffeomorphism with a natural measure, for which the conjecture does not hold in its original formulation.

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arXiv2020

A probabilistic Takens theorem

Let $X \subset \mathbb{R}^N$ be a Borel set, $μ$ a Borel probability measure on $X$ and $T:X \to X$ a Lipschitz and injective map. Fix $k \in \mathbb{N}$ greater than the (Hausdorff) dimension of $X$ and assume that the set of $p$-periodic points has dimension smaller than $p$ for $p=1, \ldots, k-1$. We prove that for a typical polynomial perturbation $\tilde{h}$ of a given Lipschitz map $h : X \to \mathbb{R}$, the $k$-delay coordinate map $x \mapsto (\tilde{h}(x), \tilde{h}(Tx), \ldots, \tilde{h}(T^{k-1}x))$ is injective on a set of full measure $μ$. This is a probabilistic version of the Takens delay embedding theorem as proven by Sauer, Yorke and Casdagli. We also provide a non-dynamical probabilistic embedding theorem of similar type, which strengthens a previous result by Alberti, Bölcskei, De Lellis, Koliander and Riegler. In both cases, the key improvements compared to the non-probabilistic counterparts are the reduction of the number of required measurements from $2\dim X$ to $\dim X$ and using Hausdorff dimension instead of the box-counting one. We present examples showing how the use of the Hausdorff dimension improves the previously obtained results.

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arXiv2019

Singular stationary measures for random piecewise affine interval homeomorphisms

We show that the stationary measure for some random systems of two piecewise affine homeomorphisms of the interval is singular, verifying partially a conjecture by Alsedà and Misiurewicz and contributing to a question of Navas on the absolute continuity of stationary measures, considered in the setup of semigroups of piecewise affine circle homeomorphisms. We focus on the case of resonant boundary derivatives.

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arXiv2017

Fatou components and singularities of meromorphic functions

We prove several results concerning the relative position of points in the postsingular set $P(f)$ of a meromorphic map $f$ and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljević-Brandt and Rempe-Gillen. For wandering domains we show that if the iterates $U_n$ of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values $p_n$ such that ${\rm dist}(p_n,\partial U_n)\to 0$ as $n\to \infty$. We also prove that if $U_n \cap P(f)=\emptyset$ and the postsingular set of $f$ lies at a positive distance from the Julia set (in $\mathbb C$) then any sequence of iterates of wandering domains must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values.

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arXiv2016

Conformal measures for meromorphic maps

In this paper we study the relation between the existence of a conformal measure on the Julia set $J(f)$ of a transcendental meromorphic map $f$ and the existence of zero of the topological pressure function $t \mapsto P(f, t)$ for the map $f$. In particular, we show that if $f$ is hyperbolic and there exists a $t$-conformal measure which is not totally supported on the set of escaping points, then $P(f, t) = 0$. On the other hand, for a wide class of maps $f$, including arbitrary maps with at most finitely many poles and finite set of singular values and hyperbolic maps with at most finitely many poles and bounded set of singular values, if $P(f, t) = 0$, we construct a $t$-conformal measure on $J(f)$. This partially answers a question of R.D. Mauldin.

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arXiv2015

Escaping points in the boundaries of Baker domains

We study the dynamical behaviour of points in the boundaries of simply connected invariant Baker domains $U$ of meromorphic maps $f$ with a finite degree on $U$. We prove that if $f|_U$ is of hyperbolic or simply parabolic type, then almost every point in the boundary of $U$ with respect to harmonic measure escapes to infinity under iteration. On the contrary, if $f|_U$ is of doubly parabolic type, then almost every point in the boundary of $U$ with respect to harmonic measure has dense forward trajectory in the boundary of $U$, in particular the set of escaping points in the boundary of $U$ has harmonic measure zero. We also present some extensions of the results to the case when $f$ has infinite degree on $U$, including classical Fatou example.

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arXiv2015

Connectivity of Julia sets of Newton maps: A unified approach

In this paper we give a unified proof of the fact that the Julia set of Newton's method applied to a holomorphic function of the complex plane (a polynomial of degree large than $1$ or an entire transcendental function) is connected. The result was recently completed by the authors' previous work, as a consequence of a more general theorem whose proof spreads among many papers, which consider separately a number of particular cases for rational and transcendental maps, and use a variety of techniques. In this note we present a unified, direct and reasonably self-contained proof which works for all situations alike.

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arXiv2014

On the dimension of the graph of the classical Weierstrass function

This paper examines dimension of the graph of the famous Weierstrass non-differentiable function \[ W_{λ, b} (x) = \sum_{n=0}^{\infty}λ^n\cos(2πb^n x) \] for an integer $b \ge 2$ and $1/b < λ< 1$. We prove that for every $b$ there exists (explicitly given) $λ_b \in (1/b, 1)$ such that the Hausdorff dimension of the graph of $W_{λ, b}$ is equal to $D = 2+\frac{\logλ}{\log b}$ for every $λ\in(λ_b,1)$. We also show that the dimension is equal to $D$ for almost every $λ$ on some larger interval. This partially solves a well-known thirty-year-old conjecture. Furthermore, we prove that the Hausdorff dimension of the graph of the function \[ f (x) = \sum_{n=0}^{\infty}λ^nϕ(b^n x) \] for an integer $b \ge 2$ and $1/b < λ< 1$ is equal to $D$ for a typical $\mathbb Z$-periodic $C^3$ function $ϕ$.

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arXiv2014

Accesses to infinity from Fatou components

We study the boundary behaviour of a meromorphic map $f: \mathbb C \to \widehat{\mathbb C}$ on its invariant simply connected Fatou component $U$. To this aim, we develop the theory of accesses to boundary points of $U$ and their relation to the dynamics of $f$. In particular, we establish a correspondence between invariant accesses from $U$ to infinity or weakly repelling points of $f$ and boundary fixed points of the associated inner function on the unit disc. We apply our results to describe the accesses to infinity from invariant Fatou components of the Newton maps.

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arXiv2014

Absorbing sets and Baker domains for holomorphic maps

We consider holomorphic maps $f: U \to U$ for a hyperbolic domain $U$ in the complex plane, such that the iterates of $f$ converge to a boundary point $ζ$ of $U$. By a previous result of the authors, for such maps there exist nice absorbing domains $W \subset U$. In this paper we show that $W$ can be chosen to be simply connected, if $f$ has parabolic I type in the sense of the Baker--Pommerenke--Cowen classification of its lift by a universal covering (and $ζ$ is not an isolated boundary point of $U$). Moreover, we provide counterexamples for other types of the map $f$ and give an exact characterization of parabolic I type in terms of the dynamical behaviour of $f$.

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arXiv2012

Brushing the hairs of transcendental entire functions

Let f be a hyperbolic transcendental entire function of finite order in the Eremenko-Lyubich class (or a finite composition of such maps), and suppose that f has a unique Fatou component. We show that the Julia set of $f$ is a Cantor bouquet; i.e. is ambiently homeomorphic to a straight brush in the sense of Aarts and Oversteegen. In particular, we show that any two such Julia sets are ambiently homeomorphic. We also show that if $f\in\B$ has finite order (or is a finite composition of such maps), but is not necessarily hyperbolic, then the Julia set of f contains a Cantor bouquet. As part of our proof, we describe, for an arbitrary function $f\in\B$, a natural compactification of the dynamical plane by adding a "circle of addresses" at infinity.

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arXiv2011

Bowen's formula for meromorphic functions

Let $f$ be an arbitrary transcendental entire or meromorphic function in the class $\mathcal S$ (i.e. with finitely many singularities). We show that the topological pressure $P(f,t)$ for $t > 0$ can be defined as the common value of the pressures $P(f,t, z)$ for all $z \in \mathbb C$ up to a set of Hausdorff dimension zero. Moreover, we prove that $P(f,t)$ equals the supremum of the pressures of $f|_X$ over all invariant hyperbolic subsets $X$ of the Julia set, and we prove Bowen's formula for $f$, i.e. we show that the Hausdorff dimension of the radial Julia set of $f$ is equal to the infimum of the set of $t$, for which $P(f,t)$ is non-positive. Similar results hold for (non-exceptional) transcendental entire or meromorphic functions $f$ in the class $\mathcal B$ (i.e. with bounded set of singularities), for which the closure of the post-singular set does not contain the Julia set.

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