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.

**[Bo]**C. Bowszyc, Fixed point theorems for the pairs of spaces, Bull. Acad. Polon. Sci., Sér. Math. Astronom. Phys. 16 (1968) 845-850. MR**39:7594****[CKM]**M. C. Carbinatto, J. Kwapisz, K. Mischaikow, Horsheshoes and the Conley index spectrum, preprint.**[De]**R. Devaney, Chaotic Dynamical Systems, Addison-Wesley, New York 1989.**[Gi]**M. Gidea, The discrete Conley index for non-invariant sets and detection of chaos, Ph.D. thesis, SUNY at Bufffalo 1997.**[MM1]**K. Mischaikow, M. Mrozek, Chaos in the Lorenz equations: A computer assisted proof, Bull. Amer. Math. Soc. 32 (1995), 66-72. MR**95e:58121****[MM2]**K. Mischaikow, M. Mrozek, Isolating neighborhoods and chaos, Japan J. Indust. Appl. Math 12 (1995), 205-236. MR**96e:58104****[RS]**J. Robbin, D. Salamon, Dynamical system, shape theory and the Conley index, Ergodic Theory Dynamical Systems (1988), 375-393. MR**89h:58094****[Sp]**E. H. Spanier, Algebraic Topology, McGraw-Hill, New York 1966. MR**96a:55001****[Sr]**R. Srzednicki, Generalized Lefschetz Theorem and a fixed point index formula, Topology Appl. 81 (1997), 207-224. MR**98i:55003****[Sz1]**A. Szymczak, The Conley index for decompositions of isolated invariant sets, Fund. Math. 148 (1995), 71-90. MR**96m:58154****[Sz2]**A. Szymczak, The Conley index and symbolic dynamics, Topology 35 (1996), 287-299. MR**97b:58054****[Z1]**P. Zgliczy\'{n}ski, Fixed point index for iterations of maps, topological horseshoe and chaos, Topological Methods Nonlinear Anal. 8 (1996), 169-177.MR**98m:58106****[Z2]**P. Zgliczy\'{n}ski, Computer assisted proofs of chaos in the Rössler equations and in the Hénon map, Nonlinearity 10 (1997), 243-252. MR**98g:58120**

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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