Papers by Piotr Nowak-Przygodzki
arXiv (Cornell University), Dec 20, 2018
A. We present a version of the equivariant gradient degree defined for equivariant gradient pertu... more A. We present a version of the equivariant gradient degree defined for equivariant gradient perturbations of an equivariant unbounded self-adjoint operator with purely discrete spectrum in Hilbert space. Two possible applications are discussed.
arXiv (Cornell University), Oct 28, 2019
A. We compare the sets of homotopy classes of gradient and proper gradient vector fields in the p... more A. We compare the sets of homotopy classes of gradient and proper gradient vector fields in the plane. Namely, we show that gradient and proper gradient homotopy classifications are essentially different. We provide a complete description of the sets of homotopy classes of gradient maps from R n to R n and proper gradient maps from R 2 to R 2 with the Brouwer degree greater or equal to zero. I The search for new homotopy invariants in the class of gradient maps has a long history. In 1985, to obtain new bifurcation results, E. N. Dancer [4] introduced a new topological invariant for S 1-equivariant gradient maps. In turn, A. Parusiński [8] showed that if two gradient vector fields on the unit disc D n nonvanishing on S n−1 are homotopic, i.e., have the same Brouwer degree, then they are also gradient homotopic. Similarly, the authors of this paper proved in [1, 2] that there is no better invariant than the Brouwer degree for gradient and proper gradient otopies in R n. Recall that otopy was introduced by J. C. Becker and D. H. Gottlieb [3] as a very useful generalization of the concept of homotopy. However, quite surprisingly, M. Starostka [9] showed that for n 2 there exist proper gradient vector fields in R n which are homotopic but not proper gradient homotopic. Roughly speaking, he proved that the identity and the minus identity on the plane are not proper gradient homotopic and then generalized this result to R n. Since his reasoning is nice and elegant, we will present it briefly here. First recall that the linear source and sink in the plane are isolated invariant sets with different homological Conley indices. Namely, since (D r (0), ∂D r (0)) and (D r (0), ∅), where D r (0) denotes the r-disc at the origin, are index pairs for the source and sink respectively, the respective homological
Author eBooks, 2018
Let f be a smooth map of the m-dimensional sphere S m to itself, preserving the longitudinal foli... more Let f be a smooth map of the m-dimensional sphere S m to itself, preserving the longitudinal foliation. We estimate from below the number of fixed points of the iterates of f , reduce the 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.
The New York Journal of Mathematics, 2019
Let V, W be finite dimensional orthogonal representations of a finite group G. The equivariant de... more Let V, W be finite dimensional orthogonal representations of a finite group G. The equivariant degree with values in the Burnside ring of G has been studied extensively by many authors. We present a short proof of the degree product formula for local equivariant maps on V and W .
Homology, Homotopy and Applications, 2021
A. We compare the sets of homotopy classes of gradient and proper gradient vector fields in the p... more A. We compare the sets of homotopy classes of gradient and proper gradient vector fields in the plane. Namely, we show that gradient and proper gradient homotopy classifications are essentially different. We provide a complete description of the sets of homotopy classes of gradient maps from R n to R n and proper gradient maps from R 2 to R 2 with the Brouwer degree greater or equal to zero. I The search for new homotopy invariants in the class of gradient maps has a long history. In 1985, to obtain new bifurcation results, E. N. Dancer [4] introduced a new topological invariant for S 1-equivariant gradient maps. In turn, A. Parusiński [8] showed that if two gradient vector fields on the unit disc D n nonvanishing on S n−1 are homotopic, i.e., have the same Brouwer degree, then they are also gradient homotopic. Similarly, the authors of this paper proved in [1, 2] that there is no better invariant than the Brouwer degree for gradient and proper gradient otopies in R n. Recall that otopy was introduced by J. C. Becker and D. H. Gottlieb [3] as a very useful generalization of the concept of homotopy. However, quite surprisingly, M. Starostka [9] showed that for n 2 there exist proper gradient vector fields in R n which are homotopic but not proper gradient homotopic. Roughly speaking, he proved that the identity and the minus identity on the plane are not proper gradient homotopic and then generalized this result to R n. Since his reasoning is nice and elegant, we will present it briefly here. First recall that the linear source and sink in the plane are isolated invariant sets with different homological Conley indices. Namely, since (D r (0), ∂D r (0)) and (D r (0), ∅), where D r (0) denotes the r-disc at the origin, are index pairs for the source and sink respectively, the respective homological
arXiv: Algebraic Topology, 2018
We prove that the inclusion of the space of gradient local maps into the space of all local maps ... more We prove that the inclusion of the space of gradient local maps into the space of all local maps induces a bijection between the sets of the respective otopy classes of these maps, where by a local map we mean a compact perturbation of identity with a compact preimage of zero.
Osaka Journal of Mathematics, 2006
The classical theorem of Shub and Sullivan states that a sequence of local fixed point indices of... more The classical theorem of Shub and Sullivan states that a sequence of local fixed point indices of iterations of a C 1 self-map of R m is periodic. The paper generalizes this result to a wider class of maps.
arXiv: Algebraic Topology, 2018
Let $V, W$ be finite-dimensional orthogonal representations of a finite group $G$. The equivarian... more Let $V, W$ be finite-dimensional orthogonal representations of a finite group $G$. The equivariant degree with values in the Burnside ring of $G$ has been studied extensively by many authors. We present a short proof of the degree product formula for local equivariant maps on $V$ and $W$.
Topological Methods in Nonlinear Analysis, 2003
Let f be a local planar homeomorphism with an isolated fixed point at 0. We study the form of the... more Let f be a local planar homeomorphism with an isolated fixed point at 0. We study the form of the sequence {ind(f n , 0)} n =0 , where ind(f, 0) is a fixed point index at 0.
Central European Journal of Mathematics, 2014
We prove that for n > 1 the space of proper maps P 0(n, k) and the space of local maps F 0(n, ... more We prove that for n > 1 the space of proper maps P 0(n, k) and the space of local maps F 0(n, k) are not homotopy equivalent.
Rocky Mountain Journal of Mathematics, 2007
Journal of Dynamics and Differential Equations, 2011
Let f : U → R 2 be a continuous map, where U is an open subset of R 2. We consider a fixed point ... more Let f : U → R 2 be a continuous map, where U is an open subset of R 2. We consider a fixed point p of f which is neither a sink nor a source and such that { p} is an isolated invariant set. Under these assumption we prove, using Conley index methods and Nielsen theory, that the sequence of fixed point indices of iterations {ind(f n , p)} ∞ n=1 is periodic, bounded from above by 1, and has infinitely many non-positive terms, which is a generalization of Le Calvez and Yoccoz theorem (Annals of Math., 146, 241-293 (1997)) onto the class of non-injective maps. We apply our result to study the dynamics of continuous maps on 2-dimensional sphere.
Discrete & Continuous Dynamical Systems - A, 2006
Abstract: In the case of a $C^1$ self-map of $R^3$ we prove the Chow, Mallet-Paret and Yorke conj... more Abstract: In the case of a $C^1$ self-map of $R^3$ we prove the Chow, Mallet-Paret and Yorke conjecture on the form of sequences of local fixed point indices of iterations and give a complete description of possible sequences of indices. ... Keywords: Fixed point index, iterations, ...
Dynamical Systems, 2008
Let f be a smooth self-map of a compact manifold and be a family of compact subsets of periodic p... more Let f be a smooth self-map of a compact manifold and be a family of compact subsets of periodic points of f. Under some natural condition on the family we find the form of the sequence of indices of iterations, which generalizes the classical theorem of Chow, Mallet ...
Univ. Iagel. Acta Math, 2003
Abstract. Let ind(f, 0) be the local fixed point index at 0. We show that every sequence of integ... more Abstract. Let ind(f, 0) be the local fixed point index at 0. We show that every sequence of integers which satisfies Dold relations can be realized as {ind(fn, 0)}∞ n=1, where f is a continuous self-map of a 2dimensional disk D2. ... 1. Introduction. Let f be a continuous self-map of ...
The New York Journal of Mathematics, 2015
We prove the Hopf theorem for gradient local vector fields on manifolds, i.e., we show that there... more We prove the Hopf theorem for gradient local vector fields on manifolds, i.e., we show that there is a natural bijection between the set of gradient otopy classes of gradient local vector fields and the integers if the manifold is connected Riemannian without boundary.
Topology and its Applications
A. We present a version of the equivariant gradient degree defined for equivariant gradient pertu... more A. We present a version of the equivariant gradient degree defined for equivariant gradient perturbations of an equivariant unbounded self-adjoint operator with purely discrete spectrum in Hilbert space. Two possible applications are discussed.
Monatshefte für Mathematik
Du, Huang and Li showed in 2003 that the class of Dold-Fermat sequences coincides with the class ... more Du, 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.
Qualitative Theory of Dynamical Systems
Let S 2 be a two-dimensional sphere. We consider two types of its foliations with one singularity... more 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 these classes of maps.
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Papers by Piotr Nowak-Przygodzki