Every continuous action of a compact group on a uniquely arcwise connected continuum has a fixed point
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We are dealing with the question whether every group or semigroup action (with some additional property) on a continuum (with some additional property) has a fixed point. One of such results was given in 2009 by Shi and Sun. They proved that every nilpotent group action on a uniquely arcwise connected continuum has a fixed point. We are seeking for this type of results with, e.g., commutative, compact, or torsion groups and semigroups acting on dendrites, dendroids, \(\lambda \)-dendroids and uniquely arcwise connected continua. We prove that every continuous action of a compact or torsion group on a uniquely arcwise connected continuum has a fixed point. We also prove that every continuous action of a compact and commutative semigroup on a uniquely arcwise connected continuum or on a tree-like continuum has a fixed point.
KeywordsFixed point group action compact group continuum dendrite dendroid lambda dendroid uniquely arcwise connected tree-like
Mathematics Subject ClassificationPrimary 54H25 Secondary 37B45
I am grateful to J. Boroński and R. Mańka for their comments to the first version of this paper.
- 3.Bogatyĭ, S.A., Frolkina, O.D.: A common fixed point of commuting mappings of a tree. Vestnik Moskov. Univ. Ser. I Mat. Mekh., vol. 69, pp. 3–10 (2002)Google Scholar
- 6.Duchesne, B., Monod, N.: Group actions on dendrites and curves (2016) (ArXiv e-prints)Google Scholar
- 13.Huneke, J.P.: Two commuting continuous functions from the closed unit interval onto the closed unit interval without a common fixed point. In: Topological Dynamics (Symposium, Colorado State Univ., Ft. Collins, Colo., 1967). Benjamin, New York, pp. 291–298 (1968)Google Scholar
- 15.Kechris, A.S.: Classical descriptive set theory. Graduate Texts in Mathematics, vol. 156. Springer, New York (1995)Google Scholar
- 21.Nadler Jr., S.B.: Continuum theory, vol. 158 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York (1992) (An introduction)Google Scholar
- 25.Shi, E., Ye, X.: Periodic points for amenable group actions on uniquely arcwise connected continua (2017) (ArXiv e-prints)Google Scholar
- 27.Walters, P.: An introduction to Ergodic theory. Graduate Texts in Mathematics, vol. 79. Springer, New York, Berlin (1982)Google Scholar