arXiv subjects
Publications and source records attributed to Jan Dudák.
Extending a recent result from [Bull. Belg. Math. Soc. Simon Stevin 33 (2026), 138-144], we prove that the space of continuous functions $C(X)$ on any countable compact space $X$ admits an isometric copy in $C([0,1])$ consisting, except for the zero function, entirely of Besicovitch functions, i.e., functions that have no one-sided derivative (finite or infinite) at any point.
This article consists of two papers: $\textit{Typical dynamics of Newton's method}$ by Steele and $\textit{Erratum to "Typical dynamics of Newton's method"}$ by Dudák and Steele. Let $C^1(M)$ be the space of continuously differentiable real-valued functions defined on $[-M,M]$. We show that for the typical element $f$ in $C^1(M)$, there exists a set $S \subseteq [-M,M]$, both residual and of full measure in $[-M,M]$, such that for any $x \in S$, the trajectory generated by Newton's method using $f$ and $x$ either diverges, converges to a root of $f$, or generates a Cantor set as its attractor. Whenever the Cantor set is the attractor, the dynamics on the attractor are described by a single type of adding machine, so that the dynamics on all of these attracting Cantor sets are topologically equivalent.
We deal with topological spaces homeomorphic to their respective squares. Primarily, we investigate the existence of large families of such spaces in some subclasses of compact metrizable spaces. As our main result we show that there is a family of size continuum of pairwise non-homeomorphic compact metrizable zero-dimensional spaces homeomorphic to their respective squares. This answers a question of W. J. Charatonik. We also discuss the situation in the classes of continua, Peano continua and absolute retracts.
It is well known due to Hahn and Mazurkiewicz that every Peano continuum is a continuous image of the unit interval. We prove that an assignment, which takes as an input a Peano continuum and produces as an output a continuous mapping whose range is the Peano continuum, can be realized in a Borel measurable way. Similarly, we find a Borel measurable assignment which takes any nonempty compact metric space and assigns a continuous mapping from the Cantor set onto that space. To this end we use the Burgess selection theorem. Finally, a Borel measurable way of assigning an arc joining two selected points in a Peano continuum is found.
We are dealing with the complexity of the homeomorphism equivalence relation on some classes of metrizable compacta from the viewpoint of invariant descriptive set theory. We prove that the homeomorphism equivalence relation of absolute retracts in the plane is Borel bireducible with the isomorphism equivalence relation of countable graphs. In order to stress the sharpness of this result, we prove that neither the homeomorphism relation of locally connected continua in the plane nor the homeomorphism relation of absolute retracts in $\mathbb R^3$ is Borel reducible to the isomorphism relation of countable graphs. We also improve the recent results of Chang and Gao by constructing a Borel reduction from both the homeomorphism equivalence relation of compact subsets of $\mathbb R^n$ and the ambient homeomorphism equivalence relation of compact subsets of $[0,1]^n$ to the homeomorphism equivalence relation of $n$-dimensional continua in $\mathbb R^{n+1}$.