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

DOI:
https://doi.org/10.1090/S0002-9939-99-05467-2

Published electronically:
October 18, 1999

MathSciNet review:
1691005

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of existence of fixed points of a continuous map in (possibly) noninvariant subsets. A pair of subsets of induces a map given by if and elsewhere. The following generalization of the Lefschetz fixed point theorem is proved: *If is metrizable, and are compact ANRs, and is continuous, then has a fixed point in provided the Lefschetz number of 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

Email:
srzednic@im.uj.edu.pl

DOI:
https://doi.org/10.1090/S0002-9939-99-05467-2

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