Fixed point indices of planar continuous maps

Bull. London Math. Soc.

Ward

2021

Section: Proof Fix N and Let Gmentioning

confidence: 99%

Dold sequences, periodic points, and dynamics

Byszewski

Bull. London Math. Soc.

Ward

2021

“…Even in the simplest case of two-dimensional sphere S 2 the conjecture still remains unsolved, although there has been a lot of progress in some particular cases. During the last few years the exponential growth was obtained under some topological conditions for S 2 and annulus [2,[9][10][11][12]. Let us remark that the Shub's question is also interesting if posed in C 0 topology under some additional conditions on the considered class of maps.…”

Section: Introductionmentioning

confidence: 99%

Periodic Points for Sphere Maps Preserving Monopole Foliations

Qual. Theory Dyn. Syst.

Misiurewicz

Nowak-Przygodzki³

2018

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.

“…The theorem opened a new field of research in which the influence of the form of indices of iterations on the structure of periodic points and the dynamical behaviour of the map was examined. This brought about a series of new results related to (among other) smooth maps [7,15,16], planar homeomorphisms [17,21,25], 3-dimensional homeomorphisms [20,22], simplicial maps [13].…”

Section: Introductionmentioning

confidence: 99%

Generalized Dold Sequences on Partially-Ordered Sets

Gulgowski²,

Lebiedź³

2022

Electron. J. Combin.

Dold 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 generalized arithmetical functions, both defined on a partially ordered set. We also define vector Dold sequences and show their combinatorial interpretation in terms of periodic points.