@inproceedings{22002,
  abstract     = {Topological simplification is the process of reducing complexity of a function while maintaining its essential features. Its goal is to find a new filter function, which reorders cells of the input complex in a way which eliminates some persistent homological features, without affecting the rest. We present a new approach to simplification based on the concept of forbidden regions and combinatorial dynamics. It allows us to reorder and cancel critical values, whose cancellation is not possible using existing methods because they are not consecutive in the total order. Each such cancellation takes O(c⋅n) time in the worst case, where c is the number of birth-death pairs and n is the size of the input complex.},
  author       = {Leśkiewicz, Jakub and Furmanek, Bartosz and Lipiński, Michał and Morozov, Dmitriy},
  booktitle    = {42nd International Symposium on Computational Geometry},
  isbn         = {9783959774185},
  issn         = {1868-8969},
  keywords     = {persistent homology, topological simplification, depth posets},
  location     = {New Brunswick, NJ, United States},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Topological simplification guided by forbidden regions}},
  doi          = {10.4230/LIPIcs.SoCG.2026.72},
  volume       = {367},
  year         = {2026},
}

@article{20980,
  abstract     = {Morse decompositions partition the flows in a vector field into equivalent structures. Given such a decomposition, one can define a further summary of its flow structure by what is called a connection matrix. These matrices, a generalization of Morse boundary operators from classical Morse theory, capture the connections made by the flows among the critical structures—such as attractors, repellers, and orbits—in a vector field. Recently, in the context of combinatorial dynamics, an efficient persistence-like algorithm to compute connection matrices has been proposed in Dey, Lipiński, Mrozek, and Slechta [SIAM J. Appl. Dyn. Syst., 23 (2024), pp. 81–97]. We show that, actually, the classical persistence algorithm with exhaustive reduction retrieves connection matrices, both simplifying the algorithm of Dey et al. and bringing the theory of persistence closer to combinatorial dynamical systems. We supplement this main result with an observation: the concept of persistence as defined for scalar fields naturally adapts to Morse decompositions whose Morse sets are filtered with a Lyapunov function. We conclude by presenting preliminary experimental results.},
  author       = {Dey, Tamal K. and Haas, Andrew and Lipiński, Michał},
  issn         = {1536-0040},
  journal      = {SIAM Journal on Applied Dynamical Systems},
  number       = {1},
  pages        = {108--130},
  publisher    = {Society for Industrial & Applied Mathematics},
  title        = {{Computing a connection matrix and persistence efficiently from a morse decomposition}},
  doi          = {10.1137/25m1739406},
  volume       = {25},
  year         = {2026},
}

@article{18580,
  abstract     = {Motivated by the study of recurrent orbits and dynamics within a Morse set of a Morse decomposition we introduce the concept of Morse predecomposition of an isolated invariant set within the setting of both combinatorial and classical dynamical systems. While Morse decomposition summarizes solely the gradient part of a dynamical system, the developed generalization extends to the recurrent component as well. In particular, a chain recurrent set, which is indecomposable in terms of Morse decomposition, can be represented more finely in the Morse predecomposition framework. This generalization is achieved by forgoing the poset structure inherent to Morse decomposition and relaxing the notion of connection between Morse sets (elements of Morse decomposition) in favor of what we term ’links’. We prove that a Morse decomposition is a special case of Morse predecomposition indexed by a poset. Additionally, we show how a Morse predecomposition may be condensed back to retrieve a Morse decomposition.},
  author       = {Lipiński, Michał and Mischaikow, Konstantin and Mrozek, Marian},
  issn         = {1662-3592},
  journal      = {Qualitative Theory of Dynamical Systems},
  number       = {1},
  publisher    = {Springer Nature},
  title        = {{Morse predecomposition of an invariant set}},
  doi          = {10.1007/s12346-024-01144-3},
  volume       = {24},
  year         = {2025},
}