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Minimization of the number of periodic points for smooth selfmaps of simplyconnected manifolds with periodic sequence of Lefschetz numbers
PublicationLet f be a smooth selfmap of mdimensional, m ≥ 4, smooth closed connected and simplyconnected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of selfmaps of simplyconnected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math....

Reducing the number of periodic points in the smooth homotopy class of a selfmap of a simplyconnected manifold with periodic sequence of Lefschetz numbers
PublicationLet f be a smooth selfmap of an mdimensional (m >3) closed connected and simplyconnected manifold such that the sequence of the Lefschetz num bers of its iterations is periodic. For a fixed natural r we wish to minimize, in the smooth homotopy class, the number of periodic points with periods less than or equal to r. The resulting number is given by a topological invariant J[f] which is defned in combinatorial terms and is...

Generating sequences of Lefschetz numbers of iterates
PublicationDu, Huang and Li showed in 2003 that the class of Dold–Fermat sequences coincides with the class of Newton sequences, which are defined in terms of socalled generating sequences. The sequences of Lefschetz numbers of iterates form an important subclass of Dold–Fermat (thus also Newton) sequences. In this paper we characterize generating sequences of Lefschetz numbers of iterates.

Minimization of the number of periodic points for smooth selfmaps of closed simplyconnected 4manifolds
PublicationLet M be a smooth closed simplyconnected 4dimensional manifold, f be a smooth selfmap of M with fast grow of Lefschetz numbers and r be a product of different primes. The authors calculate the invariant equal to the minimal number of rperiodic points in the smooth homotopy class of f.

Minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of simplyconnected manifolds of dimension 4 and homology groups with the sum of ranks less or equal to10
Open Research DataAn important problem in periodic point theory is minimization of the number of periodic points with periods <= r in a given class of selfmaps of a space. A closed smooth and simplyconnected manifolds of dimension 4 and its selfmaps f with periodic sequence of Lefschetz numbers are considered. The topological invariant Jr[f] is equal to the minimal...

Minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of simplyconnected manifolds of dimension 6 and homology groups with the sum of ranks less or equal to10
Open Research DataAn important problem in periodic point theory is minimization of the number of periodic points with periods <= r in a given class of selfmaps of a space. A closed smooth and simplyconnected manifolds of dimension 6 and its selfmaps f with periodic sequence of Lefschetz numbers are considered. The topological invariant Jr[f] is equal to the minimal...

Minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of simplyconnected manifolds of dimension 5 and homology groups with the sum of ranks less or equal to10
Open Research DataAn important problem in periodic point theory is minimization of the number of periodic points with periods <= r in a given class of selfmaps of a space. A closed smooth and simplyconnected manifolds of dimension 5 and its selfmaps f with periodic sequence of Lefschetz numbers are considered. The topological invariant Jr[f] is equal to the minimal...

Minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of simplyconnected manifolds of dimension 8 and homology groups with the sum of ranks less or equal to 10
Open Research DataAn important problem in periodic point theory is minimization of the number of periodic points with periods <= r in a given class of selfmaps of a space. A closed smooth and simplyconnected manifolds of dimension 8 and its selfmaps f with periodic sequence of Lefschetz numbers are considered. The topological invariant Jr[f] is equal to the minimal...

Minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of simplyconnected manifolds of dimension 7 and homology groups with the sum of ranks less or equal to10
Open Research DataAn important problem in periodic point theory is minimization of the number of periodic points with periods <= r in a given class of selfmaps of a space. A closed smooth and simplyconnected manifolds of dimension 7 and its selfmaps f with periodic sequence of Lefschetz numbers are considered. The topological invariant Jr[f] is equal to the minimal...

Minimal number of periodic points for smooth selfmaps of simplyconnected manifolds
Open Research DataThe problem of finding the minimal number of periodic points in a given class of selfmaps of a space is one of the central questions in periodic point theory. We consider a closed smooth connected and simplyconnected manifold of dimension at least 4 and its selfmap f. The topological invariant D_r[f] is equal to the minimal number of rperiodic points...

Periodic expansion in determining minimal sets of Lefschetz periods for Morse–Smale diffeomorphisms
PublicationWe apply the representation of Lefschetz numbers of iterates in the form of periodic expansion to determine the minimal sets of Lefschetz periods of Morse–Smale diffeomorphisms. Applying this approach we present an algorithmic method of finding the family of minimal sets of Lefschetz periods for Ng, a nonorientable compact surfaces without boundary of genus g. We also partially confirm the conjecture of Llibre and Sirvent (J Diff...

Lefschetz periodic point free selfmaps of compact manifolds
PublicationLet f be a selfmap of a compact connected manifold M. We characterize Lefschetz periodic point free continuous selfmaps of M for several classes of manifolds and generalize the results of Guirao and Llibre [J.L.G. Guirao, J. Llibre, On the Lefschetz periodic point free continuous selfmaps on connected compact manifolds,

Lefschetz periodic point free selfmaps of compact manifolds
PublicationLet f be a selfmap of a compact connected manifold M. We characterize Lefschetz periodic point free continuous selfmaps of M for several classes of manifolds and generalize the results of Guirao and Llibre [J.L.G. Guirao, J. Llibre, On the Lefschetz periodic point free continuous selfmaps on connected compact manifolds, Topology Appl. 158 (16) (2011) 21652169].

Minimal Sets of Lefschetz Periods for MorseSmale Diffeomorphisms of a Connected Sum of g Real Projective Planes
PublicationThe dataset titled Database of the minimal sets of Lefschetz periods for MorseSmale diffeomorphisms of a connected sum of g real projective planes contains all of the values of the topological invariant called the minimal set of Lefschetz periods, computed for MorseSmale diffeomorphisms of a nonorientable compact surface without boundary of genus g (i.e. a connected sum of g real projective planes), where g varies from 1 to...

Estimates for minimal number of periodic points for smooth selfmaps of simplyconnected manifolds
Open Research DataWe consider a closed smooth connected and simplyconnected manifold of dimension at least 4 and its selfmap f. The topological invariant Dr[f] is equal to the minimal number of rperiodic points in the smooth homotopy class of f. We assume that r is odd and all coefficients b(k) of socalled periodic expansion of Lefschetz numbers of iterations are...

Database of the minimal sets of Lefschetz periods for MorseSmale diffeomorphisms of a connected sum of g tori
Open Research DataMorse–Smale diffeomorphisms, structurally stable and having relatively simple dynamics, constitute an important subclass of diffeomorphisms that have been carefully studied during past decades. For a given Morse–Smale diffeomorphism one can consider “Minimal set of Lefschetz periods”, which provides the information about the set of periodic points of...

Topological invariants for equivariant flows: Conley index and degree
PublicationAbout forty years have passed since Charles Conley defined the homotopy index. Thereby, he generalized the ideas that go back to the calculus of variations work of Marston Morse. Within this long time the Conley index has proved to be a valuable tool in nonlinear analysis and dynamical systems. A significant development of applied methods has been observed. Later, the index theory has evolved to cover such areas as discrete dynamical...

Database of the minimal sets of Lefschetz periods for MorseSmale diffeomorphisms of a connected sum of g real projective planes.
Open Research DataMorse–Smale diffeomorphisms, structurally stable and having relatively simple dynamics, constitute an important subclass of diffeomorphisms that were carefully studied during past decades. For a given Morse–Smale diffeomorphism one can consider “Minimal set of Lefschetz periods”, which provides the information about the set of periodic points of considered...

The database of odd algebraic periods for quasiunipotent selfmaps of a space having the same homology group as the connected sum of g tori
Open Research DataThe dataset consists of 20 files indexed by numbers g=1,...,20. Each file provides sets of odd algebraic periods for all quasiunipotent selfmaps of a space having the same homology groups as the connected sum of g tori. Let us remark that each data set covers all algebraical restrictions that come from zeta functions for the sets of minimal Lefschetz...

Computations of the least number of periodic points of smooth boundarypreserving selfmaps of simplyconnected manifolds
PublicationLet $r$ be an odd natural number, $M$ a compact simplyconnected smooth manifold, $\dim M\geq 4$, such that its boundary $\partial M$ is also simplyconnected. We consider $f$, a $C^1$ selfmaps of $M$, preserving $\partial M$. In [G. Graff and J. Jezierski, Geom. Dedicata 187 (2017), 241258] the smooth Nielsen type periodic number $D_r(f;M,\partial M)$ was defined and proved to be equal to the minimal number of $r$periodic points...