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A generalization of the Lefschetz fixed point theorem and detection of chaos

Author: Roman Srzednicki
Journal: Proc. Amer. Math. Soc. 128 (2000), 1231-1239
MSC (2000): Primary 55M20; Secondary 37B10, 37D45
Published electronically: October 18, 1999
MathSciNet review: 1691005
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Abstract: We consider the problem of existence of fixed points of a continuous map $f:X\to X$ in (possibly) noninvariant subsets. A pair $(C,E)$ of subsets of $X$ induces a map $f^\dag:C/E\to C/E$ given by $f^\dag([x])=[f(x)]$ if $x,f(x)\in C\setminus E$ and $f^\dag([x])=[E]$ elsewhere. The following generalization of the Lefschetz fixed point theorem is proved: If $X$ is metrizable, $C$ and $E$ are compact ANRs, and $f^\dag$ is continuous, then $f$ has a fixed point in $\overline{C\setminus E}$ provided the Lefschetz number of $\widetilde H^\ast (f^\dag)$ is nonzero. Actually, we prove an extension of that theorem to the case of a composition of maps. We apply it to a result on the existence of an invariant set of a homeomorphism such that the dynamics restricted to that set is chaotic.

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Additional Information

Roman Srzednicki
Affiliation: Institute of Mathematics, Jagiellonian University, ul. Reymonta 4, 30-059 Kraków, Poland

Keywords: Fixed point, Lefschetz number, periodic point, chaos, shift
Received by editor(s): October 6, 1997
Received by editor(s) in revised form: June 3, 1998
Published electronically: October 18, 1999
Additional Notes: The author was supported by KBN, Grant 2 P03A 040 10
Communicated by: Linda Keen
Article copyright: © Copyright 2000 American Mathematical Society

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