Search results for: periodic points

Dold sequences, periodic points, and dynamics
PublicationIn this survey we describe how the socalled Dold congruence arises in topology, and how it relates to periodic point counting in dynamical systems.

Periodic Points for Sphere Maps Preserving MonopoleFoliations
PublicationLet S^2 be a twodimensional 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 logdeg(f). This confirms the Shub’s conjecture in...

Periodic points of latitudinal maps of the $m$dimensional sphere
PublicationLet f be a smooth selfmap of the mdimensional sphere Sm. Under the assumption that f preserves latitudinal foliations with the fibres S1, we estimate from below the number of fixed points of the iterates of f. The paper generalizes the results obtained by Pugh and Shub and by Misiurewicz.

Indices of interations and periodic points of simplical maps of smooth type
PublicationW pracy dowodzi się symplicjalnego odpowiednika twierdzenia Chowa, MalletParet i Yorke´a. Otrzymany wynik służy do badania punktów periodycznych odwzorowań symplicjalnych gładkiego typu.

Maps with bounded sequence of indices of interations and finitaly many periodic points
PublicationW pracy badane są związki pomiędzy globalną topologiczną strukturą przestrzeni wyrażoną w terminach charakterystyki EuleraPoincar odwzorowań na niej określonych, a spełniających założenia z tytułu, a lokalnymi własnościami przestrzeni zdeterminowanymi przez zachowanie się tych odwzorowań w punkatach periodycznych.

Minimal number of periodic points for smooth selfmaps of S^3
PublicationW pracy wyznaczona została najmniejsza liczba punktów periodycznych w gładkiej klasie homotopii odwzorowania sfery trójwymiarowej w siebie.

Minimal number of periodic points for smooth selfmaps of RP^3
PublicationNiech f będzie gładkim odwzorowaniem 3wymiarowej rzeczywistej przestrzeni rzutowej w siebie, r będzie ustaloną liczbą naturalną. W artykule wyznaczona została minimalna liczba punktów rperiodycznych w gładkiej klasie homotopii odwzorowania f.

On the growth of the number of periodic points for smooth self maps of a compact manifold
PublicationDla ciągłego przekształcenia jednospójnej rozmaitości wymiaru co najmniej 3 w siebie, wykazujemy, że wzrost liczby punktów rperiodycznych w klasie homotopii może być nie szybszy niż liniowy, dla dowolnego, ustalonego r.

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....

Minimizing the number of periodic points for smooth maps. Nonsimply connected case
PublicationNiech f będzie gładkim odwzorowaniem zamkniętej rozmaitości o wymiarze wiekszym niż 2, a r ustaloną liczbą naturalną. W artykule zdefiniowany został niezmiennik topologiczny równy minimalnej liczbie punktów rperiodycznych w gładkiej klasie homotopii f.

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...

Minimal number of periodic points of smooth boundarypreserving selfmaps of simplyconnected manifolds
PublicationLet M be a smooth compact and simplyconnected manifold with simplyconnected boundary ∂M, r be a fixed odd natural number. We consider f, a C1 selfmap of M, preserving ∂M . Under the assumption that the dimension of M is at least 4, we define an invariant Dr(f;M,∂M) that is equal to the minimal number of rperiodic points for all maps preserving ∂M and C1homotopic to f. As an application, we give necessary and sufficient...

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 for C^1 selfmaps of compact simplyconnected manifolds
PublicationNiech f będzie odwzorowaniem gładkiej zwartej i jednospójnej rozmaitości o wymiarze większym lub równym 3. W pracy zdefiniowany został topologiczny niezmiennik będący najlepszym dolnym oszacowaniem liczby punktów periodycznych w klasie gładkich odwzorowań homotopijnych z f.

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...

Combinatorial scheme of finding minimal number of periodic points for smooth selfmaps of simply connected manifolds
PublicationLet M be a closed smooth connected and simply connected manifold of dimension m at least 3, and let r be a fixed natural number. The topological invariant D^m_r [f], defined by the authors in [Forum Math. 21 (2009), 491509], is equal to the minimal number of rperiodic points in the smooth homotopy class of f, a given selfmap of M. In this paper, we present a general combinatorial scheme of computing D^m_r [f] for arbitrary dimension...

Estimation of the minimal number of periodic points for smooth selfmaps of odd dimensional real projective spaces
PublicationLet f be a smooth selfmap of a closed connected manifold of dimension m⩾3. The authors introduced in [G. Graff, J. Jezierski, Minimizing the number of periodic points for smooth maps. Nonsimply connected case, Topology Appl. 158 (3) (2011) 276290] the topological invariant NJD_r[f], where r is a fixed natural number, which is equal to the minimal number of rperiodic points in the smooth homotopy class of f. In this paper smooth...

An algorithmic approach to estimating the minimal number of periodic points for smooth selfmaps of simplyconnected manifolds
PublicationFor a given selfmap f of M, a closed smooth connected and simplyconnected manifold of dimension m 4, we provide an algorithm for estimating the values of the topological invariant D^m_r [f], which equals the minimal number of rperiodic points in the smooth homotopy class of f. Our results are based on the combinatorial scheme for computing D^m_r [f] introduced by G. Graff and J. Jezierski [J. Fixed Point Theory Appl. 13 (2013),...

Minimal number of periodic points for smooth selfmaps of twoholed 3dimensional closed ball
PublicationDla ciągłego odwzorowania f przestrzeni określonej w tytule w siebie, które posiada rzeczywiste wartości własne na drugiej grupie homologii, wyznaczona została minimalna liczba punktów rperiodycznych w klasie wszystkich gładkich odwzorowań homotopijnych z f.

An algorithmic approach to estimating the minimal number of periodic points for smooth selfmaps of simplyconnected manifolds
Publication 
On delay differential equations with almost periodic boundary conditions started from different points
PublicationDyskutowany jest problem istnienia ekstremalnych rozwiązań dla równań różniczkowych typu opóźnionego przy odpowiednich warunkach brzegowych. Sformułowano odpowiednie twierdzenia porównawcze. W pracy zawarte są również wyniki dotyczące takich równań przy większej ilości argumentów opóźnionych.

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...

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...

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 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 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...

A Strategy to Locate Fixed Points and Global Perturbations of ODE’s: Mixing Topology with Metric Conditions
PublicationIn this paper we discuss a topological treatment for the planar system z' = f (t, z) + g(t, z) where f and g are T periodic in time and g(t, z) is bounded. Namely, we study the effect of g(t, z) in two different frameworks: isochronous centers and time periodic systems having subharmonics. The main tool employed in the proofs consists of a topological strategy to locate fixed points in the class of orientation preserving embedding...

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...

Generalized Dold sequences on partiallyordered sets
PublicationDold sequences constitute an important class of integer sequences that play an important role in combinatorics, number theory, topology and dynamical systems. We generalize the notion of Dold sequence for the case of partially ordered sets and describe their properties. In particular we give two alternative descriptions of generalized Dold sequences: by some class of elementary sequences as well as by different...

DETERMINATION OF VERTICAL DISPLACEMENTS IN RELATIVE MONITORING NETWORKS
PublicationThe problem of determining displacements of objects is an important and current issue, in particular in terms of operational safety. This is a requirement that covers geodetic, periodic control measurements in order to determine horizontal and vertical displacements. The paper is focused on the analysis of vertical displacements. Geodetic measurements and their interpretation allow to reduce the risk of possible structural catastrophes....

Onedimensional chaos in a system with dry friction: analytical approach
PublicationWe introduce a new analytical method, which allows to find chaotic regimes in nonsmooth dynamical systems. A simple mechanical system consisting of a mass and a dry friction element is considered. The corresponding mathematical model is being studied. We show that the considered dynamical system is a skew product over a piecewise smooth mapping of a segment (the socalled base map). For this base map we demonstrate existence of...

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...

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...

NonDestructive Testing of the Longest Span SoilSteel Bridge in Europe—Field Measurements and FEM Calculations
PublicationThe article describes interdisciplinary and comprehensive nondestructive diagnostic tests of final bridge inspection and acceptance proposed for a soilsteel bridge made of corrugated sheets, being the European span length record holder (25.74 m). As an effect of an original concept a detailed and precise information about the structure shortterm response was collected. Periodic diagnostics of bridge deformations was done one...

ELECTRICAL CONDUCTIVITY AND pH IN SURFACE WATER AS TOOL FOR IDENTIFICATION OF CHEMICAL DIVERSITY
PublicationIn the present study, the creeks and lakes located at the western shore of Admiralty Bay were analysed. The impact of various sources of water supply was considered, based on the parameters of temperature, pH and specific electrolytic conductivity (SEC25). All measurements were conducted during a field campaign in JanuaryFebruary 2017. A multivariate dataset was also created and a biplot of SEC25 and pH of the investigated waters...

Dynamics of Sunimodal maps used in population modeling.
Open Research DataSunimodal maps are maps of the interval with negative Schwarzian derivative and having only one turning point (such that the map is increasing to the left of the turning point and decreasing to the right of it). Theory of Sunimodal maps is now a welldeveloped branch of discrete dynamical systems, including famous Singer theorem which implies existence...

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...

Shub’s conjecture for smooth longitudinal maps of S^m
PublicationLet f be a smooth map of the mdimensional sphere Sm to itself, preserving the longitudinal foliation. We estimate from below the number of fixed points of the iterates of f , reduce Shub’s conjecture for longitudinal maps to a lower dimensional classical version, and prove the conjecture in case m = 2 and in a weak form for m = 3.

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,

Patryk Topór
People