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Search results for: BIMODAL SMOOTH MAPS
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Shub’s conjecture for smooth longitudinal maps of S^m
PublicationLet f be a smooth map of the m-dimensional 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.
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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, Mallet-Paret and Yorke conjecture posed in[S.N. Chow, J. Mallet-Parret, 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. 109-131].
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Indices of interations and periodic points of simplical maps of smooth type
PublicationW pracy dowodzi się symplicjalnego odpowiednika twierdzenia Chowa, Mallet-Paret i Yorke´a. Otrzymany wynik służy do badania punktów periodycznych odwzorowań symplicjalnych gładkiego typu.
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Minimal number of periodic points for smooth self-maps of S^3
PublicationW pracy wyznaczona została najmniejsza liczba punktów periodycznych w gładkiej klasie homotopii odwzorowania sfery trójwymiarowej w siebie.
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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 r-periodycznych w klasie homotopii może być nie szybszy niż liniowy, dla dowolnego, ustalonego r.
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Minimal number of periodic points for smooth self-maps of RP^3
PublicationNiech f będzie gładkim odwzorowaniem 3-wymiarowej rzeczywistej przestrzeni rzutowej w siebie, r będzie ustaloną liczbą naturalną. W artykule wyznaczona została minimalna liczba punktów r-periodycznych w gładkiej klasie homotopii odwzorowania f.
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Minimizing the number of periodic points for smooth maps. Non-simply 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 r-periodycznych w gładkiej klasie homotopii f.
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Minimal number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds
PublicationLet M be a smooth compact and simply-connected manifold with simply-connected boundary ∂M, r be a fixed odd natural number. We consider f, a C1 self-map 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 r-periodic points for all maps preserving ∂M and C1-homotopic to f. As an application, we give necessary and sufficient...
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Minimization of the number of periodic points for smooth self-maps of closed simply-connected 4-manifolds
PublicationLet M be a smooth closed simply-connected 4-dimensional manifold, f be a smooth self-map 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 r-periodic points in the smooth homotopy class of f.
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Estimation of the minimal number of periodic points for smooth self-maps of odd dimensional real projective spaces
PublicationLet f be a smooth self-map 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. Non-simply connected case, Topology Appl. 158 (3) (2011) 276-290] the topological invariant NJD_r[f], where r is a fixed natural number, which is equal to the minimal number of r-periodic points in the smooth homotopy class of f. In this paper smooth...
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Computations of the least number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds
PublicationLet $r$ be an odd natural number, $M$ a compact simply-connected smooth manifold, $\dim M\geq 4$, such that its boundary $\partial M$ is also simply-connected. We consider $f$, a $C^1$ self-maps of $M$, preserving $\partial M$. In [G. Graff and J. Jezierski, Geom. Dedicata 187 (2017), 241-258] 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...
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An algorithmic approach to estimating the minimal number of periodic points for smooth self-maps of simply-connected manifolds
PublicationFor a given self-map f of M, a closed smooth connected and simply-connected 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 r-periodic 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),...
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Combinatorial scheme of finding minimal number of periodic points for smooth self-maps 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), 491-509], is equal to the minimal number of r-periodic points in the smooth homotopy class of f, a given self-map of M. In this paper, we present a general combinatorial scheme of computing D^m_r [f] for arbitrary dimension...
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An algorithmic approach to estimating the minimal number of periodic points for smooth self-maps of simply-connected manifolds
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Minimal number of periodic points for smooth self-maps of two-holed 3-dimensional 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 r-periodycznych w klasie wszystkich gładkich odwzorowań homotopijnych z f.
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Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers
PublicationLet f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected 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 self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math....
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Reducing the number of periodic points in the smooth homotopy class of a self-map of a simply-connected manifold with periodic sequence of Lefschetz numbers
PublicationLet f be a smooth self-map of an m-dimensional (m >3) closed connected and simply-connected 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...
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Smooth orthogonal projections on sphere.
PublicationWe construct a decomposition of the identity operator on the sphere S^d as a sum of smooth orthogonal projections subordinate to an open cover of S^d. We give applications of our main result in the study of function spaces and Parseval frames on the sphere.
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Smooth Orthogonal Projections on Riemannian Manifold
PublicationWe construct a decomposition of the identity operator on a Riemannian manifold M as a sum of smooth orthogonal projections subordinate to an open cover of M. This extends a decomposition on the real line by smooth orthogonal projection due to Coifman and Meyer (C. R. Acad. Sci. Paris, S´er. I Math., 312(3), 259–261 1991) and Auscher, Weiss, Wickerhauser (1992), and a similar decomposition when M is the sphere by Bownik and Dziedziul (Const....
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Marcinkiewicz Averages of Smooth Orthogonal Projections on Sphere
PublicationWe construct a single smooth orthogonal projection with desired localization whose average under a group action yields the decomposition of the identity operator. For any full rank lattice \Gamma ⊂ R^d , a smooth projection is localized in a neighborhood of an arbitrary precompact fundamental domain R^d / \Gamma. We also show the existence of a highly localized smooth orthogonal projection, whose Marcinkiewicz average under the...