Concerning periodic points in mappings of continua

Author:
W. T. Ingram

Journal:
Proc. Amer. Math. Soc. **104** (1988), 643-649

MSC:
Primary 54F20; Secondary 54H20, 58F20

MathSciNet review:
962842

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Abstract: In this paper we present some conditions which are sufficient for a mapping to have periodic points.

Theorem. *If* *is a mapping of the space* *into* *and there exist subcontinua* and *of* *such that* (1) *every subcontinuum of* has the fixed point property, (2) *and every subcontinuum of* *are in class* , (3) contains , (4) *contains* , *and* (5) *if* *is a positive integer such that* *intersects* , *then* , *then* *contains periodic points of* *of every period greater than* 1.

Also included is a fixed point lemma:

Lemma. *Suppose* *is a mapping of the space* *into* *and* *is a subcontinuum of* *such that* *contains* . *If* (1) *every subcontinuum of* *has the fixed point property, and* (2) *every subcontinuum of* *is in class* , then there is a point *of* *such that* .

Further we show that: If is a mapping of into and has a periodic point which is not a power of 2, then contains an indecomposable continuum. Moreover, for each positive integer , there is a mapping of into with a periodic point of period and having a hereditarily decomposable inverse limit.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0962842-8

Keywords:
Periodic point,
fixed point property,
class ,
indecomposable continuum,
inverse limit

Article copyright:
© Copyright 1988
American Mathematical Society