On fixed points of zero index in asymptotic fixed point theory

Fenske, Christian; Peitgen, Heinz‐Otto

doi:10.2140/pjm.1976.66.391

Cited by 11 publications

(10 citation statements)

References 15 publications

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“…The following theorem generalizes the result of Fried [4]. The proof is essentially the same but we use Theorem 3 instead of the lemma in [4].…”

Section: F Is Expansive Iff the Diagonal A^xxxisan Isolated Invariantmentioning

confidence: 79%

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Open index pairs, the fixed point index and rationality of zeta functions

Mrożek

1990

Ergod. Th. Dynam. Sys.

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Abstract.We define open index pairs of an isolated invariant set, prove their existence and compute the fixed point index of an isolating neighbourhood in terms of the Lefschetz number of a certain map associated with the open index pair. We use this to establish rationality of zeta functions and Lefschetz zeta functions. IntroductionThe Conley index [2] of an isolated invariant set of a flow has become an important tool in the qualitative study of flows and differential equations. Recently Fried [7] observed that the Conley theory can be successfully applied to extend Manning's theorem [9] on the rationality of the zeta function of a basic set of an Axiom A diffeomorphism to the case of a C 1 diffeomorphism and its expansive, isolated invariant set with nondegenerate periodic points. Fried's proof is based on a lemma concerning the rationality of Lefschetz zeta functions. In order to show this lemma the diffeomorphism is suspended to obtain a C 1 flow and then the Conley-Easton theorem [3] on the existence of smooth isolating blocks for flows is used. This allows the fixed point (Lefschetz) indices to be expressed in terms of traces of certain matrices.In this paper we prove the rationality of the Lefschetz zeta function for an isolated invariant set of maps of compact attraction on an arbitrary metric ANR, which generalizes Fried's lemma. As a consequence we also obtain a generalization of Manning's theorem. The proof goes along similar lines as in Fried's paper [7] with one important difference: we cannot use suspension because otherwise we could obtain a space which is not an ANR. Thus we have to use a substitute of the Conley-Easton theorem for discrete time dynamical systems. In the discrete case it seems natural to consider index pairs instead of isolating blocks. In particular, this approach was chosen in [12] and [13] to carry over the Conley index from the continuous to the discrete case. In order to be able to consider the fixed point index we have to ensure the existence of index pairs being ANRs. Since the existence of such index pairs seems to be settled only in case of smooth flows and diffeomorphisms, we introduce open index pairs and prove their existence. Then we extend the fixed point index formula established for the case of compact ANR index pairs in

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“…The following theorem generalizes the result of Fried [4]. The proof is essentially the same but we use Theorem 3 instead of the lemma in [4].…”

Section: F Is Expansive Iff the Diagonal A^xxxisan Isolated Invariantmentioning

confidence: 79%

“…The proof is essentially the same but we use Theorem 3 instead of the lemma in [4]. We include the proof for sake of completeness.…”

Section: F Is Expansive Iff the Diagonal A^xxxisan Isolated Invariantmentioning

confidence: 99%

Open index pairs, the fixed point index and rationality of zeta functions

Mrożek

1990

Ergod. Th. Dynam. Sys.

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“…Result (0.2) was independently and implicitly established in [16], Theorem 1.1 and [6], Theorem 4.9. Note, that the proof in [6] does not make use of any type of a (mod p Note also, that assumption (0.1.1) is just Rothe's condition for T n (cf. [19]).…”

Section: And U= {Xe Pl Min {R R } < Hxt]mentioning

confidence: 88%

“…Now, by the definition of ~W R we have that To0~(x)= T(x)+x for all x~CF~R\(P\cl BR); thus, ind(P, T°PR, P\cl BR)=ind(P, T°OR, ~R). Observe that ToO~ is compact on a neighborhood of Fix(T°0RI~rR) and therefore the weakcommutativity property of the index yields ind(P, TOQR, qCFR)= ind UC/~R, T°OR, ~#/R)" Finally [6] or Corollary 1.1 in [14] we know that ind(P, T, Br)=0 if 0eP is an ejective fixed point of a completely continuous operator 72.P~P relative to Br. Assuming that Tsatisfies (2.5.1)and (2.5.2) [resp.…”

Section: °Os)((9~) C (9~ Implies That ((Tm~o@s)le)=(idle~) Once Wmentioning

confidence: 97%