## A note on fixed points of abelian actions in dimension one

HTML articles powered by AMS MathViewer

- by J. P. Boroński PDF
- Proc. Amer. Math. Soc.
**147**(2019), 1653-1655 Request permission

## Abstract:

The result of Boyce and Huneke gives rise to a 1-dimensional continuum, which is the intersection of a descending family of disks that admits two commuting homeomorphisms without a common fixed point.## References

- Marcy Barge and Joe Martin,
*The construction of global attractors*, Proc. Amer. Math. Soc.**110**(1990), no. 2, 523–525. MR**1023342**, DOI 10.1090/S0002-9939-1990-1023342-1 - R. H. Bing,
*Snake-like continua*, Duke Math. J.**18**(1951), 653–663. MR**43450** - William M. Boyce,
*Commuting functions with no common fixed point*, Trans. Amer. Math. Soc.**137**(1969), 77–92. MR**236331**, DOI 10.1090/S0002-9947-1969-0236331-5 - O. H. Hamilton,
*A fixed point theorem for pseudo-arcs and certain other metric continua*, Proc. Amer. Math. Soc.**2**(1951), 173–174. MR**39993**, DOI 10.1090/S0002-9939-1951-0039993-2 - John Philip Huneke,
*On common fixed points of commuting continuous functions on an interval*, Trans. Amer. Math. Soc.**139**(1969), 371–381. MR**237724**, DOI 10.1090/S0002-9947-1969-0237724-2 - Piotr Minc and W. R. R. Transue,
*A transitive map on $[0,1]$ whose inverse limit is the pseudoarc*, Proc. Amer. Math. Soc.**111**(1991), no. 4, 1165–1170. MR**1042271**, DOI 10.1090/S0002-9939-1991-1042271-1 - Christopher Mouron,
*Dynamics of commuting homeomorphisms of chainable continua*, Colloq. Math.**121**(2010), no. 1, 63–77. MR**2725702**, DOI 10.4064/cm121-1-6 - Javier Ribón,
*Fixed points of nilpotent actions on $\Bbb {S}^2$*, Ergodic Theory Dynam. Systems**36**(2016), no. 1, 173–197. MR**3436759**, DOI 10.1017/etds.2014.58 - Enhui Shi and Binyong Sun,
*Fixed point properties of nilpotent group actions on 1-arcwise connected continua*, Proc. Amer. Math. Soc.**137**(2009), no. 2, 771–775. MR**2448600**, DOI 10.1090/S0002-9939-08-09522-1 - Benjamin Vejnar,
*Every continuous action of a compact group on a uniquely arcwise connected continuum has a fixed point*, J. Fixed Point Theory Appl.**20**(2018), no. 2, Paper No. 69, 9. MR**3787890**, DOI 10.1007/s11784-018-0552-3

## Additional Information

**J. P. Boroński**- Affiliation: National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 701 03 Ostrava, Czech Republic – and – Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland
- ORCID: 0000-0002-1802-4006
- Email: jan.boronski@osu.cz
- Received by editor(s): March 25, 2018
- Received by editor(s) in revised form: August 15, 2018
- Published electronically: December 12, 2018
- Additional Notes: The author was supported by University of Ostrava grant lRP201824 “Complex topological structures” and the NPU II project LQ1602 IT4Innovations excellence in science.
- Communicated by: Nimish Shah
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 1653-1655 - MSC (2010): Primary 37B05, 37B45
- DOI: https://doi.org/10.1090/proc/14365
- MathSciNet review: 3910429