Homotopy Methods in Topological Fixed and Periodic Points Theory

Jezierski, Jerzy; Marzantowicz, Wacław

doi:10.1007/1-4020-3931-x

Cited by 92 publications

(80 citation statements)

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“…Proof. The proof follows closely the proof of the classical Wecken theorem, as described, for example, in [13]. We shall only outline the main steps.…”

Section: Given By the Inclusion Of Null(s) In H-null(s) The Homotopymentioning

confidence: 96%

The homotopy coincidence index

Crabb

2010

J. Fixed Point Theory Appl.

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“…Proof. The proof follows closely the proof of the classical Wecken theorem, as described, for example, in [13]. We shall only outline the main steps.…”

Section: Given By the Inclusion Of Null(s) In H-null(s) The Homotopymentioning

confidence: 96%

The homotopy coincidence index

Crabb

2010

J. Fixed Point Theory Appl.

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“…This will be proved in the more precise form, stated below as Theorem 5.6, due to Chow, Mallet-Paret and Yorke [2]. (See also the accounts in [15] and [16].) Our exposition will follow closely that in [5] (but sharpened to include the full result of Chow et al).…”

Section: Lemma 52mentioning

confidence: 99%

“…We shall call them the Dold indices and write them as D k (φ, X). (In [16] the term 'multiplicity' is used, rather than 'index'.) Dold's proof involved consideration of the map π k (φ) : X k → X k , (x 1 , x 2 , .…”

Section: Introductionmentioning

confidence: 99%

Equivariant fixed-point indices of iterated maps

Crabb

2007

J. fixed point theory appl.

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We describe an equivariant version (for actions of a finite group G) of Dold's index theory, [10], for iterated maps. Equivariant Dold indices are defined, in general, for a G-map U → X defined on an open G-subset of a G-ANR X (and satisfying a suitable compactness condition). A local index for isolated fixed-points is introduced, and the theorem of Shub and Sullivan on the vanishing of all but finitely many Dold indices for a continuously differentiable map is extended to the equivariant case. Homotopy Dold indices, arising from the equivariant Reidemeister trace, are also considered. Mathematics Subject Classification (2000). Primary 37C25, 55M20, 55N91; Secondary 55P91.

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“…Section 3 deals with the linearization of a self-map f on the 3-nilmanifold to understand the Nielsen numbers of the iterates of f by using Anosov's theorem. Linearizations for self-maps of arbitrary nilmanifolds and solvmanifolds of type N R are explored and surveyed in for example [5,6]. With these preliminary sections, we determine the Nielsen type numbers in Section 4 and the sets of homotopy minimal periods in Section 5 for maps on the 3-nilmanifolds.…”

Section: Introductionmentioning

confidence: 99%

Nielsen type numbers and homotopy minimal periods for maps on the 3-nilmanifolds

Lee

Zhao

2008

Sci. China Ser. A-Math.

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For all continuous maps on the 3-nilmanifolds, we give explicit formulae for a complete computation of the Nielsen type numbers NPn(f ) and N Φn(f ). The most general cases were explored by Heath and Keppelmann and the complementary part is studied in this paper. While studying the homotopy minimal periods of all maps on the 3-nilmanifolds, we also give a complete description of the homotopy minimal periods for all maps on the 3-nilmanifolds, including a correction to Jezierski and Marzantowicz's result.

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Homotopy Methods in Topological Fixed and Periodic Points Theory

Cited by 92 publications

References 0 publications

The homotopy coincidence index

The homotopy coincidence index

Equivariant fixed-point indices of iterated maps

Nielsen type numbers and homotopy minimal periods for maps on the 3-nilmanifolds

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