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 | 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. Sci. Math. Astronom. Phys.**16**(1968), 845–850 (English, with Loose Russian summary). MR**0246290****[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]**Konstantin Mischaikow and Marian Mrozek,*Chaos in the Lorenz equations: a computer-assisted proof*, Bull. Amer. Math. Soc. (N.S.)**32**(1995), no. 1, 66–72. MR**1276767**, 10.1090/S0273-0979-1995-00558-6**[MM2]**Konstantin Mischaikow and Marian Mrozek,*Isolating neighborhoods and chaos*, Japan J. Indust. Appl. Math.**12**(1995), no. 2, 205–236. MR**1337206**, 10.1007/BF03167289**[RS]**Joel W. Robbin and Dietmar Salamon,*Dynamical systems, shape theory and the Conley index*, Ergodic Theory Dynam. Systems**8***(1988), no. Charles Conley Memorial Issue, 375–393. MR**967645**, 10.1017/S0143385700009494**[Sp]**Edwin H. Spanier,*Algebraic topology*, Springer-Verlag, New York, 1966. Corrected reprint of the 1966 original. MR**1325242****[Sr]**Roman Srzednicki,*Generalized Lefschetz theorem and a fixed point index formula*, Topology Appl.**81**(1997), no. 3, 207–224. MR**1485768**, 10.1016/S0166-8641(97)00037-0**[Sz1]**Andrzej Szymczak,*The Conley index for decompositions of isolated invariant sets*, Fund. Math.**148**(1995), no. 1, 71–90. MR**1354939****[Sz2]**Andrzej Szymczak,*The Conley index and symbolic dynamics*, Topology**35**(1996), no. 2, 287–299. MR**1380498**, 10.1016/0040-9383(95)00029-1**[Z1]**Piotr Zgliczyński,*Fixed point index for iterations of maps, topological horseshoe and chaos*, Topol. Methods Nonlinear Anal.**8**(1996), no. 1, 169–177. MR**1485762****[Z2]**Piotr Zgliczyński,*Computer assisted proof of chaos in the Rössler equations and in the Hénon map*, Nonlinearity**10**(1997), no. 1, 243–252. MR**1430751**, 10.1088/0951-7715/10/1/016

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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:
http://dx.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