arXiv subjects

Krzysztof Lech

Publications and source records attributed to Krzysztof Lech.

3 recordsLinked to original sources

arXiv2023

Equilibrium measures on Julia sets of random quadratic polynomials

We consider sequences of compositions of quadratic polynomials $f_{c_n} (z) = z^2 + c_n$. For such sequences one can naturally generalize the definitions of the Julia set and basin of infinity from the autonomous case. In this setting the Julia set depends on a sequence $ω= (c_0, c_1, ...)$. We study the equilibrium (harmonic) measure on such Julia sets. In particular, we calculate the Hausdorff dimension of the equilibrium measure and study its dependence on the ''scale of randomness''.

math.DS↗

arXiv2020

Julia sets of random exponential maps

For a sequence $(λ_n)$ of positive real numbers we consider the exponential functions $f_{λ_n} (z) = λ_n e^z$ and the compositions $F_n = f_{λ_n} \circ f_{λ_{n-1}} \circ ... \circ f_{λ_1}$. For such a non-autonomous family we can define the Fatou and Julia sets analogously to the usual case of autonomous iteration. The aim of this document is to study how the Julia set depends on the sequence $(λ_n)$. Among other results, we prove the Julia set for a random sequence $\{λ_n \}$, chosen uniformly from a neighbourhood of $\frac{1}{e}$, is the whole plane with probability $1$. We also prove the Julia set for $\frac{1}{e} + \frac{1}{n^p}$ is the whole plane for $p < \frac{1}{2}$, and give an example of a sequence $\{λ_n \} $ for which the iterates of $0$ converge to infinity starting from any index, but the Fatou set is non-empty.

math.DS↗

arXiv2020

Total disconnectedness of Julia sets of random quadratic polynomials

For a sequence of complex parameters $\{c_n\}$ we consider the compositions of functions $f_{c_n} (z) = z^2 + c_n$, which is the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disk. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just disks, in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

math.DS↗