zbMATH — the first resource for mathematics

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)
Full Text:
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.