Year 2025

• Dold Coefficients of Quasi-unipotent Homeomorphisms of Orientable Surfaces

  Publication

  • G. Graff
  • W. Marzantowicz
  • Ł. P. Michalak

  - Qualitative Theory of Dynamical Systems - Year 2025

  The sequence of Dold coefficients $(a_n(f))$ of a self-map $f\colon X \to X$ forms a dual sequence to the sequence of Lefschetz numbers $(L(f^n))$ of iterations of $f$ under the M{\"o}bius inversion formula. The set $\cAP(f) = \{ n \colon a_n(f) \neq 0 \}$ is called the set of algebraic periods of~$f$. Both the set of algebraic periods and sequence of Dold coefficients play an important role in dynamical system and periodic point...

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Year 2024

• Periodic Solutions of Generalized Lagrangian Systems with Small Perturbations

  Publication

  • J. Janczewska

  - Qualitative Theory of Dynamical Systems - Year 2024

  In this paper we study the generalized Lagrangian system with a small perturbation. We assume the main term in the system to have a maximum, but do not suppose any condition for perturbation term. Then we prove the existence of a periodic solution via Ekeland’s principle. Moreover, we prove a convergence theorem for periodic solutions of perturbed systems.

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Year 2023

• Invariant Measures for Uncountable Random Interval Homeomorphisms

  Publication

  • T. Szarek
  • J. Morawiec

  - Qualitative Theory of Dynamical Systems - Year 2023

  A necessary and sufficient condition for the iterated function system { f (·, ω) | ω ∈ } with probability P to have exactly one invariant measure μ∗ with μ∗((0, 1)) = 1 is given. The main novelty lies in the fact that we only require the transformations f (·, ω) to be increasing homeomorphims, without any smoothness condition, nei- ther we impose conditions on the cardinality of . In particular, positive Lyapunov exponents conditions...

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Year 2019

• Periodic Points for Sphere Maps Preserving MonopoleFoliations

  Publication

  • G. Graff
  • M. Misiurewicz
  • P. Nowak-Przygodzki

  - Qualitative Theory of Dynamical Systems - Year 2019

  Let S^2 be a two-dimensional sphere. We consider two types of its foliations with one singularity and maps f:S^2→S^2 preserving these foliations, more and less regular. We prove that in both cases f has at least |deg(f)| fixed points, where deg(f) is a topological degree of f. In particular, the lower growth rate of the number of fixed points of the iterations of f is at least log|deg(f)|. This confirms the Shub’s conjecture in...

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