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Every continuous action of a compact group on a uniquely arcwise connected continuum has a fixed point. (English) Zbl 1403.54030
The main results. \(1)\) Let \(G\) be a group every element of which is contained in a compact subgroup. Then every continuous action of \(G\) on a uniquely arcwise connected continuum has a fixed point. Moreover, the set of fixed points of the action is an arcwise connected continuum.
\(2)\) Let \(G\) be a compact commutative semigroup. Then every continuous action of \(G\) on a uniquely arcwise connected continuum or on a tree-like continuum has a fixed point.

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
37B45 Continua theory in dynamics
54F15 Continua and generalizations
54H11 Topological groups (topological aspects)
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References:
[1] Acosta, G; Eslami, P; Oversteegen, LG, On open maps between dendrites, Houst. J. Math., 33, 753-770, (2007) · Zbl 1146.54006
[2] Bellamy, DP, A tree-like continuum without the fixed-point property, Houst. J. Math., 6, 1-13, (1980) · Zbl 0447.54039
[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) · Zbl 1119.54328
[4] Borsuk, K, A theorem on fixed points, Bull. Acad. Polon. Sci. Cl. III., 2, 17-20, (1954) · Zbl 0057.39103
[5] Boyce, WM, Commuting functions with no common fixed point, Trans. Am. Math. Soc., 137, 77-92, (1969) · Zbl 0175.34501
[6] Duchesne, B., Monod, N.: Group actions on dendrites and curves (2016) (ArXiv e-prints) · Zbl 1160.54032
[7] Fugate, JB; McLean, TB, Compact groups of homeomorphisms on tree-like continua, Trans. Am. Math. Soc., 267, 609-620, (1981) · Zbl 0486.54028
[8] Fugate, JB; Mohler, L, A note on fixed points in tree-like continua, Topol. Proc., 2, 457-460, (1978) · Zbl 0407.54027
[9] Fugate, JB; Mohler, L, Fixed point theorems for arc-preserving mappings of uniquely arcwise-connected continua, Proc. Am. Math. Soc., 123, 3225-3231, (1995) · Zbl 0848.54029
[10] Gray, WJ, A fixed-point theorem for commuting monotone functions, Can. J. Math., 21, 502-504, (1969) · Zbl 0188.55303
[11] Gray, WJ; Smith, CM, Common fixed points of commuting mappings, Proc. Am. Math. Soc., 53, 223-226, (1975) · Zbl 0311.54050
[12] Horn, WA, Three results for trees, using mathematical induction, J. Res. Nat. Bur. Stand. Sect. B, 76B, 39-43, (1972) · Zbl 0341.05103
[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)
[14] Isbell, JR, Research problems: commuting wrappings of trees, Bull. Am. Math. Soc., 63, 419, (1957)
[15] Kechris, A.S.: Classical descriptive set theory. Graduate Texts in Mathematics, vol. 156. Springer, New York (1995) · Zbl 0819.04002
[16] Mańka, R, Association and fixed points, Fund. Math., 91, 105-121, (1976) · Zbl 0335.54037
[17] McDowell, EL, Coincidence values of commuting functions, Topol. Proc., 34, 365-384, (2009) · Zbl 1176.54033
[18] Mitchell, T, Common fixed-points for equicontinuous semigroups of mappings, Proc. Am. Math. Soc., 33, 146-150, (1972) · Zbl 0233.54029
[19] Mohler, L, The fixed point property for homeomorphisms of \(1\)-arcwise connected continua, Proc. Am. Math. Soc., 52, 451-456, (1975) · Zbl 0307.54035
[20] Mohler, L; Oversteegen, LG, Open and monotone fixed point free maps on uniquely arcwise connected continua, Proc. Am. Math. Soc., 95, 476-482, (1985) · Zbl 0592.54030
[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) · Zbl 0407.54027
[22] Rabin, M, A note on helly’s theorem, Pac. J. Math., 5, 363-366, (1955) · Zbl 0065.15303
[23] Shi, E; Sun, B, Fixed point properties of nilpotent group actions on 1-arcwise connected continua, Proc. Am. Math. Soc., 137, 771-775, (2009) · Zbl 1160.54032
[24] Shi, E; Ye, X, Periodic points for amenable group actions on dendrites, Proc. Am. Math. Soc., 145, 177-184, (2017) · Zbl 1368.37019
[25] Shi, E., Ye, X.: Periodic points for amenable group actions on uniquely arcwise connected continua (2017) (ArXiv e-prints) · Zbl 0057.39103
[26] Sobolewski, M a, A weakly chainable uniquely arcwise connected continuum without the fixed point property, Fund. Math, 228, 81-86, (2015) · Zbl 1314.54021
[27] Walters, P.: An introduction to Ergodic theory. Graduate Texts in Mathematics, vol. 79. Springer, New York, Berlin (1982) · Zbl 0475.28009
[28] Young, GS, Fixed-point theorems for arcwise connected continua, Proc. Am. Math. Soc., 11, 880-884, (1960) · Zbl 0102.37806
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