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Fixed point indices of iterated smooth maps in arbitrary dimension
PublicationWe give a complete description of possible sequences ofindices of iterations of f at an isolated fixed point, answering inaffirmative the Chow, MalletParet and Yorke conjecture posed in[S.N. Chow, J. MalletParret, J.A. Yorke, A periodic point index whichis a bifurcation invariant, in: Geometric Dynamics, Rio de Janeiro,1981, in: Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983,pp. 109131].

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.

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.

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.

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.

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.

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.

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.

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...

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...

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...

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 
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),...

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....

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...

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 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...

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.

Generalized Gradient Equivariant Multivalued Maps, Approximation and Degree
PublicationConsider the Euclidean space Rn with the orthogonal action of a compact Lie group G. We prove that a locally Lipschitz Ginvariant mapping f from Rn to R can be uniformly approximated by Ginvariant smooth mappings g in such a way that the gradient of g is a graph approximation of Clarke’s generalized gradient of f . This result enables a proper development of equivariant gradient degree theory for a class of setvalued gradient...

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...