Related papers: A non-Borel special alpha-limit set in the square
Special $\alpha$-limit sets ($s\alpha$-limit sets) combine together all accumulation points of all backward orbit branches of a point $x$ under a noninvertible map. The most important question about them is whether or not they are closed.…
We prove in this paper that there exists some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [Automates et Th\'eorie Descriptive, Ph. D. Thesis, Universit\'e Paris 7, March…
Asking which sets are fixed-parameter tractable for a given parameterization constitutes much of the current research in parameterized complexity theory. This approach faces some of the core difficulties in complexity theory. By focussing…
In the paper we study what sets can be obtained as $\alpha$-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those $\alpha$-limit sets are $\omega$-limit sets and for all but finitely many…
We demonstrate that the set $L^\infty(X, [-1,1])$ of all measurable functions over a Borel measure space $(X, \mathcal B, \mu )$ with values in the unit interval is typically non-polyhedric when interpreted as a subset of a dual space. Our…
We show that there is a bijection between the renormalizations and proper completely invariant closed sets of expanding Lorenz map, which enable us to distinguish periodic and non-periodic renormalizations. Based on the properties of…
We study the descriptive complexity of sets of points defined by placing restrictions on statistical behaviour of their orbits in dynamical systems on Polish spaces. A particular examples of such sets are the set of generic points of a…
We characterize analytic curves that contain non-trivial self-affine sets. We also prove that compact algebraic surfaces cannot contain non-trivial self-affine sets.
We study non-recurrence sets for weakly mixing dynamical systems by using linear dynamical systems. These are systems consisting of a bounded linear operator acting on a separable complex Banach space X, which becomes a probability space…
We study the Borel complexity of sets of normal numbers in several numeration systems. Taking a dynamical point of view, we offer a unified treatment for continued fraction expansions and base $r$ expansions, and their various…
We show that there is a set which is not a set of multiple recurrence despite being a set of recurrence for nil-Bohr sets. This answers Huang, Shao, and Ye's \enquote{higher-order} version of Katznelson's Question on Bohr recurrence and…
We construct certain non-degenerate maps and sets, mainly in the complex-analytic category. For example, we show that for every countable subset S in an irreducible complex space X there exists a holomorphic map from the unit disk to X such…
We construct some example of a closed nondegenerate nonflexible polyhedron $P$ in Euclidean 3-space that is the limit of a sequence of nondegenerate flexible polyhedra each of which is combinatorially equivalent to $P$. This implies that…
Metrizable spaces are studied in which every closed set is an $\alpha$-limit set for some continuous map and some point. It is shown that this property is enjoyed by every space containing sufficiently many arcs (formalized in the notion of…
Understanding when linear immersions of nonlinear dynamical systems exist is important since such immersions allow us to leverage the rich tools of linear system theory to analyze nonlinear dynamics. Recently, Liu et al. (2023) showed that…
Let $X$ be a regular curve and let $f: X\to X$ be a monotone map. In this paper, nonwandering set of $f$ and the structure of special $\alpha$-limit sets for $f$ are investigated. We show that AP$(f)= \textrm{R}(f) =\Omega(f)$, where…
Inspired by the work of Pujals and Sambarino on dominated splitting, we present billiards with a modified reflection law which constitute simple examples of dynamical systems with limit sets with dominated splitting and where the dynamics…
We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with…
If $X$ is an analytic metric space satisfying a very mild doubling condition, then for any finite Borel measure $\mu$ on $X$ there is a set $N\subseteq X$ such that $\mu(N)>0$, an ultrametric space $Z$ and a Lipschitz bijection $\phi:N\to…
We investigate $\mathcal F$-Borel topological spaces. We focus on finding out how a~complexity of a~space depends on where the~space is embedded. Of a~particular interest is the~problem of determining whether a~complexity of given space $X$…