Education

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  2018-2021

  licencjat (undergraduate): Mathematics

  Thesis: Minimality, equicontinuity and uniform rigidity of topologically transitive dynamical systems (in Polish)

  advisor: Piotr Oprocha

  AGH University of Krakow

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  2021-2023

  magister (Masters): Mathematics

  Thesis: Topological properties of sin(1/x)-type continua (in Polish)

  advisor: Piotr Oprocha

  AGH University of Krakow

  specialization: Mathematics in Technical and Natural Sciences

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  2023-ongoing (expected graduation 2027)

  doktor (PhD): Mathematics

  Research topic: complex structures in topology and dynamics

  advisor: Piotr Oprocha

  AGH University of Krakow

Reaserch visits and internships

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  January 2024

  University of Maribor

  2 weeks internship

  Maribor, Slovenia

  Funded by: AGH IDUB grant

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  July-August 2024

  ICTP

  5 weeks internship

  Trieste, Italy

  Funded by: AGH IDUB grant

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  April 2025

  Charles University

  1 week stay

  Prague, Czech Republic

  Funded by: AGH IDUB grant

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  June-September 2025

  Fudan University

  3 months internship

  Shanghai, China

  Funded by: AGH IDUB grant

Last publications

• Tranched graphs: consequences for topology and dynamics

  Authors: Michał Kowalewski, Piotr Oprocha

  Fundamenta Mathematicae 273, no. 1, 1–46 • 2026 • doi:10.4064/fm240613-5-12

  We compare quasi-graphs and generalized sin(1/x)-type continua, which are two classes of continua that generalize topological graphs and contain the Warsaw circle as a nontrivial common element. We show that neither class is a subset of the other, provide some characterizations, and present illustrative examples. We unify both approaches by considering the class of tranched graphs, compare it to concepts known from the literature, and describe how the topological structure of its elements restricts possible dynamics.

• Every nondegenerate Peano continuum admits a pure mixing selfmap

  Authors: Klára Karásová, Michał Kowalewski, Piotr Oprocha

  Proc. Amer. Math. Soc. 154, no. 10, 4389–4403 • 2026 • doi:10.1090/proc/17765

  We prove that every Peano continuum (a space that is a continuous image of [0,1]) admits a topologically mixing but not exact map. The constructed map has a dense set of periodic points.

• Observable Dynamics and the Generic Coincidence of Milnor, Statistical, and Physical Attractors

  Authors: Magdalena Foryś-Krawiec, Jana Hantáková, Michał Kowalewski, Piotr Oprocha

  arXiv:2511.09718 (submitted) • 2025

  We study observable dynamics of generic continuous interval maps. We prove that, for a residual subset of C([0,1]), the global Milnor, statistical, and physical attractors coincide with the non-wandering set. Thus, for a typical interval map, the asymptotic dynamics detected by Lebesgue-almost every initial condition recovers all recurrent dynamics. The common attractor is a robust Cantor set of zero Hausdorff dimension. Moreover, every invariant probability measure has a basin of zero Lebesgue measure. We also present examples showing that, outside the generic setting, positive topological entropy may occur outside the global Milnor attractor, and the three attractors need not coincide.

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