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Przemysław Kucharski

Publications and source records attributed to Przemysław Kucharski.

9 recordsLinked to original sources
arXiv2026

A note on the definition of good special flows of Kanigowski and Ravotti

In this short note we show that certain definition from the work of Kanigowski and Ravotti can be relaxed. More specifically, we show that the definition of good special flows, that is, flows exhibiting a form of mechanism of mixing, shearing of Birkhoff sums can be relaxed. Specifically, we show that the disintegration corresponding to almost partitions, which are uniformly sheared along the direction of the flow, of the base can depend on $k$-tuple of times. This note serves as a complementary material to an article on mixing of higher order in flows of the author (details in the note), where we use the new definition of good special flows to show that flows of Fayad on $n$-torus, $n\geq 3$, are mixing of any order.

math.DS↗
arXiv2026

Maximal attractors for perturbations of unimodal maps near a homoclinic tangency

We show that small planar perturbations of families of unimodal maps have maximal (equal to the intersection of the iterates of their trapping region) attractors at parameters near homoclinic tangency of one of the fixed points. The results hold for general unimodal maps exhibiting certain hyperbolic properties. Hence, admissible families considered include Hénon-like families, Lozi-like families, or perturbations of border collision normal forms. Therefore, our results generalize results of Viana in the case of Hénon-like families and Glendinning and Simpson in the case of border collision normal forms.

math.DS↗
arXiv2024

Translation algorithms for graph covers

Graph covers are a way to describe continuous maps (and homeomorphisms) of a Cantor set, more generally than e.g.\ Bratteli-Vershik systems. Every continuous map on a zero-dimensional compact set can be expressed by a graph cover (e.g.\ non-minimality or aperiodicty are no restrictions). We give a survey on the construction, properties and some special cases of graph covers.

math.DS↗
arXiv2023

Strange attractors for the family of orientation preserving Lozi maps

We extend the result of Michal Misiurewicz assuring the existence of strange attractors for the parametrized family $\{f_{(a,b)}\}$ of orientation reversing Lozi maps to the orientation preserving case. That is, we rigorously determine an open subset of the parameter space for which an attractor $\mathcal{A}_{(a,b)}$ of $f_{(a,b)}$ always exists and exhibits chaotic properties. Moreover, we prove that the attractor is maximal in some open parameter region, and arises as the closure of the unstable manifold of a fixed point, on which $f_{(a,b)}|_{\mathcal{A}_{(a,b)}}$ is mixing. We also show that $\mathcal{A}_{(a,b)}$ vary continuously with parameter $(a,b)$ in the Hausdorff metric.

math.DS↗
arXiv2023

Strange attractors for the generalized Lozi-like family

We generalize the Lozi-like family introduced in Misiurewicz and Štimac work from 2017. The generalized Lozi-like family encompasses in particular certain Lozi-like maps, orientation preserving or reversing Lozi maps or large parameter regions of 2-dimensional border collision normal forms. We prove that it possesses a strange attractor, arising as a homoclinic class.

math.DS↗
arXiv2022

On Cartwright-Littlewood Fixed Point Theorem

We prove the following generalization of the Cartwright-Littlewood fixed point theorem. Suppose $ h\colon~{\mathbb R}^{2}\to{\mathbb R}^{2} $ is an orientation preserving planar homeomorphism, and $ X $ is an acyclic continuum. Let $ C $ be a component of $ X \cap h(X) $. If there is a $ c \in C $ such that $ {\mathcal O}_{+} (c) \subseteq C $ or $ {\mathcal O}_{-} (c) \subseteq C $ then $ C $ also contains a fixed point of $ h$. Our result also generalizes earlier results of Ostrovski and Boroński, and answers the Question from Boroński's work in 2017. The proof is inspired by a short proof of the result of Cartwright and Littlewood due to Hamilton.

math.DS↗
arXiv2020

Entropy of Toeplitz systems over residually finite groups

The purpose of this work is to bound sofic topological entropy of Toeplitz systems over residually finite groups and to prove the Krieger Theorem about attaining arbitrary entropy by the Toeplitz systems. To achieve these results, we discuss certain properties of the sofic topological entropy in the context of finitely indexed normal subgroups of the group. It will help us to formulate results almost independently of the natural sofic approximation sequences of residually finite groups.

math.DS↗
arXiv2016

Design and implementation of a user interface for a multi-device spatially-aware mobile system

The aim of this thesis was the design and development of an interactive system enhancing collaborative sensemaking. The system design was based on related work research and preliminary user study. The system is based on multiple spatially-aware mobile devices. The spatial awareness is established with a motion tracking system. The information about position of tablets is received by a server. The server communicates with the devices and manages the content of each device. The implemented system supports managing the elements of information across the devices basing on their relative spatial arrangement. The evaluation of the system was done by a user study.

cs.HC↗